annotate OD.agda @ 365:7f919d6b045b

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 18 Jul 2020 12:29:38 +0900
parents 67580311cc8e
children f74681db63c7
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
364
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 363
diff changeset
1 {-# OPTIONS --allow-unsolved-metas #-}
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
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
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
6 open import zf
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
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
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
10 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
11 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
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
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
20 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
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
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
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
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
40 ==-refl : { x : OD } → x == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
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
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
45 ==-trans : { x y z : OD } → x == y → y == z → x == z
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
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
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
48 ==-sym : { x y : OD } → x == y → y == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
49 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
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
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
57 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
58 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
59 -- correspondence to the OD then the OD looks like a ZF Set.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
60 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
61 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
62 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
63 --
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
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
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
68 --
309
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
70 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
71 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
72 -- bound on each HOD.
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
73 --
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
75 -- we need explict assumption on sup.
309
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
76 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
79 data One : Set n where
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
80 OneObj : One
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
81
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
82 -- Ordinals in OD , the maximum
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
83 Ords : OD
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
84 Ords = record { def = λ x → One }
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
85
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
88 od : OD
304
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
89 odmax : Ordinal
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
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
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
96 -- HOD is isomorphic to Ordinal (by means of Goedel number)
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
97 od→ord : HOD → Ordinal
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
98 ord→od : Ordinal → HOD
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
99 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
100 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
101 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
102 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
103 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
104 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
306
b07fc3ef5aab fix sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 305
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
110 -- odmax minimality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
111 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
112 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
113 -- We can calculate the minimum using sup but it is tedius.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
114 -- Only Select has non minimum odmax.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
115 -- We have the same problem on 'def' itself, but we leave it.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
116
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
117 odmaxmin : Set (suc n)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
118 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 361
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
121 hod-ord< : {x : HOD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
122 hod-ord< {x} = od→ord x o< next (odmax x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
123
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
124 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
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) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
128 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
129 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
130 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
131 next-ord : Ordinal → Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
132 next-ord x = osuc x
301
b012a915bbb5 contradiction found ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
133
179
aa89d1b8ce96 fix comments
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
134 -- Ordinal in OD ( and ZFSet ) Transitive Set
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
135 Ord : ( a : Ordinal ) → HOD
304
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
136 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
137 lemma : {x : Ordinal} → x o< a → x o< a
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
142
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
143 odef : HOD → Ordinal → Set n
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
144 odef A x = def ( od A ) x
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
145
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
146 -- If we have reverse of c<→o<, everything becomes Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
148 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
149 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
151 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
152 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
153
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
154 _∋_ : ( a x : HOD ) → Set n
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
155 _∋_ a x = odef a ( od→ord x )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
156
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
159
361
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
160 cseq : HOD → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
161 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
162 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
163 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
164
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
165 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
166 odef-subst df refl refl = df
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
167
361
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
168 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
187
ac872f6b8692 add Todo
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
169 otrans x<a y<x = ordtrans y<x x<a
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
170
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
171 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
174 -- avoiding lv != Zero error
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
175 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
176 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
177
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
180 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ;
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
181 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) }
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
182 where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
183 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
184 lemma {x} {z} d = odef-subst d oiso refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
185
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
186 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y)
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
187 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
188
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
191 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy)
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
192 lemma ox ox refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
193
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
194 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y)
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
195 o≡→== {x} {.x} refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
199 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
203 lemma : od (ord→od o∅) == od od∅
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
204 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
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
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
208
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
213 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
216
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
219
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
220 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
221 odef-iso refl t = t
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
228
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
229 -- the pair
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
230 _,_ : HOD → HOD → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
232 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
233 lemma {t} (case1 refl) = omax-x _ _
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
234 lemma {t} (case2 refl) = omax-y _ _
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
235
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
236 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
237 pair-xx<xy {x} {y} = ⊆→o≤ lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
238 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
239 lemma {z} (case1 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
240 lemma {z} (case2 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
241
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
242 -- another form of infinite
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
243 -- pair-ord< : {x : Ordinal } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
244 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
245 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
246 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
247 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
248 lemmab1 : od→ord (x , x) o< next ( odmax (x , x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
257 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
258 odmax<od→ord : { x y : HOD } → x ∋ y → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
259 odmax<od→ord {x} {y} x∋y = odmax x o< od→ord x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 360
diff changeset
260
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
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
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
264 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
265 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
266
360
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 358
diff changeset
267 _∩_ : ( A B : HOD ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 358
diff changeset
268 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 358
diff changeset
269 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
270
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
271 record _⊆_ ( A B : HOD ) : Set (suc n) where
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
272 field
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
273 incl : { x : HOD } → A ∋ x → B ∋ x
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
274
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
275 open _⊆_
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
276 infixr 220 _⊆_
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
277
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
278 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
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 )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
280
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
281 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o<
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
282 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
283 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
284 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
285 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
286 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
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 )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
288 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
289 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
290 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
291 lemma (case1 refl) = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
292 lemma (case2 refl) = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
293 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
294 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
295 lemma1 : osuc (od→ord y) o< od→ord (x , x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
296 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c )
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
297
360
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 358
diff changeset
298 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
299 subset-lemma {A} {x} = record {
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
300 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
301 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
302 }
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
303
312
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
304 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
305 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
306 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
307 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 323
diff changeset
311 ε-induction : { ψ : HOD → Set n}
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
312 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
320 -- level trick (what'a shame)
330
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
321 ε-induction1 : { ψ : HOD → Set (suc n)}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
322 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
323 → (x : HOD ) → ψ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
324 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
325 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
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 )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
327 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
328 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
329
365
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
330 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
331 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) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
332 Replace : HOD → (HOD → HOD) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
333 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 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
334 ; odmax = rmax ; <odmax = rmax<} where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
335 rmax : Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
336 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
337 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
338 rmax< lt = proj1 lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
339 Union : HOD → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
340 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
341 ; odmax = osuc (od→ord U) ; <odmax = umax< } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
342 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
343 umax< {y} not = lemma (FExists _ lemma1 not ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
344 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
345 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))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
346 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
347 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))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
348 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
349 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
350 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
351 lemma not with trio< y (od→ord U)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
352 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
353 lemma not | tri≈ ¬a refl ¬c = <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
354 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
355 _∈_ : ( A B : HOD ) → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
356 A ∈ B = B ∋ A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
358 OPwr : (A : HOD ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
359 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
360
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
361 Power : HOD → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
362 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
363 -- {_} : ZFSet → ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
364 -- { x } = ( x , x ) -- better to use (x , x) directly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
365
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
366 data infinite-d : ( x : Ordinal ) → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
367 iφ : infinite-d o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
368 isuc : {x : Ordinal } → infinite-d x →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
369 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
370
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
371 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
372 -- We simply assumes infinite-d y has a maximum.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
373 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
374 -- This means that many of OD may not be HODs because of the od→ord mapping divergence.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
375 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x).
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
376 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
377 postulate
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
378 ωmax : Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
379 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
380
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
381 infinite : HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
382 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
383
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
384 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
385 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
386 u : (y : Ordinal ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
387 u y = Union (ord→od y , (ord→od y , ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
388 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
389 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
390 lemma8 = ho<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
391 --- (x,y) < next (omax x y) < next (osuc y) = next y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
392 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
393 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< )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
394 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
395 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
396 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
397 lemma9 = lemmaa (c<→o< (case1 refl))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
398 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
399 lemma71 = next< lemma81 lemma9
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
400 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
401 lemma1 = ho<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
402 --- main recursion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
403 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
404 lemma {o∅} iφ = x<nx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
405 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))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
406
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
407 ω<next-o∅ : ({x : HOD} → od→ord x o< next (odmax x)) → {y : Ordinal} → infinite-d y → y o< next o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
408 ω<next-o∅ ho< {y} lt = <odmax (infinite' ho<) lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
409
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
410 nat→ω : Nat → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
411 nat→ω Zero = od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
412 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
413
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
414 ω→nat : (n : HOD) → infinite ∋ n → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
415 ω→nat n = lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
416 lemma : {y : Ordinal} → infinite-d y → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
417 lemma iφ = Zero
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
418 lemma (isuc lt) = Suc (lemma lt)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
419
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
420 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
421 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) {!!} iφ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
422 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) {!!} (isuc ( ω∋nat→ω {n}))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
423
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
424 _=h=_ : (x y : HOD) → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
425 x =h= y = od x == od y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
426
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
427 infixr 200 _∈_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
428 -- infixr 230 _∩_ _∪_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
429
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
430 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
431 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
432 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
433
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
434 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
435 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
436 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
437
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
438 empty : (x : HOD ) → ¬ (od∅ ∋ x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
439 empty x = ¬x<0
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
440
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
441 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
442 o<→c< lt = record { incl = λ z → ordtrans z lt }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
443
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
444 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
445 ⊆→o< {x} {y} lt with trio< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
446 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
447 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
448 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
449 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
450
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
451 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
452 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
453 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
454 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
455 union← X z UX∋z = FExists _ lemma UX∋z where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
456 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
457 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
458
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
459 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
460 ψiso {ψ} t refl = t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
461 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
462 selection {ψ} {X} {y} = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
463 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
464 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
465 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
466 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))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
467 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
468 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
469 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
470 lemma : def (in-codomain X ψ) (od→ord (ψ x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
471 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
472 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
473 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
474 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
475 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
476 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
477 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
478 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
479 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
480 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
481
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
482 ---
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
483 --- Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
484 ---
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
485 --- First consider ordinals in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
486 ---
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
487 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
488 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
489 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
490 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
491 ∩-≡ {a} {b} inc = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
492 eq→ = λ {x} x<a → record { proj2 = x<a ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
493 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
494 eq← = λ {x} x<a∩b → proj2 x<a∩b }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
495 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
496 -- Transitive Set case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
497 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
498 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
499 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
500 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
501 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
502 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
503 lemma refl (lemma1 lemma-eq )where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
504 lemma-eq : ((Ord a) ∩ t) =h= t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
505 eq→ lemma-eq {z} w = proj2 w
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
506 eq← lemma-eq {z} w = record { proj2 = w ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
507 proj1 = odef-subst {_} {_} {(Ord a)} {z}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
508 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
509 lemma1 : {a : Ordinal } { t : HOD }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
510 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
511 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 ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
512 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
513 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)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
514 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)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
515 lemma = sup-o< _ lemma2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
516
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
517 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
518 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
519 -- then replace of all elements of the Power set by A ∩ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
520 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
521 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
522
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
523 -- we have oly double negation form because of the replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
524 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
525 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
526 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
527 a = od→ord A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
528 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
529 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
530 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
531 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
532 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
533 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
534 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
535 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
536
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
537 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
538 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
539 a = od→ord A
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
540 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
541 lemma0 {x} t∋x = c<→o< (t→A t∋x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
542 lemma3 : OPwr (Ord a) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
543 lemma3 = ord-power← a t lemma0
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
544 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
545 lemma4 = let open ≡-Reasoning in begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
546 A ∩ ord→od (od→ord t)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
547 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
548 A ∩ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
549 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
550 t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
551
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
552 sup1 : Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
553 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
554 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
555 lemma9 = <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
556 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
557 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
558 lemmad : Ord (osuc (od→ord A)) ∋ t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
559 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)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
560 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
561 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
562 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
563 lemmaf {x} lt = proj1 lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
564 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
565 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
566 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
567 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
568 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
569 lemma7 with osuc-≡< lemmad
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
570 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
571 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
572 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
573 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
574 diso
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
575 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
576 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
577 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
578 lemmai = let open ≡-Reasoning in begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
579 od→ord (Ord (od→ord A))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
580 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
581 od→ord (Ord (od→ord t))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
582 ≡⟨ sym diso ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
583 od→ord (ord→od (od→ord (Ord (od→ord t))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
584 ≡⟨ sym eq1 ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
585 od→ord (ord→od (od→ord t))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
586 ≡⟨ diso ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
587 od→ord t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
588
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
589 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
590 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
591 lemmak = let open ≡-Reasoning in begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
592 od→ord (ord→od (od→ord (Ord (od→ord t))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
593 ≡⟨ diso ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
594 od→ord (Ord (od→ord t))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
595 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
596 od→ord (Ord (od→ord A))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
597
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
598 lemmaj : od→ord t o< od→ord (Ord (od→ord A))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
599 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
600 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
601 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
602 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
603 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
604 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
605 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
606 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
607
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
609 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
610 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
611 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
612 lemma lt y<x with osuc-≡< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
613 lemma lt y<x | case1 refl = c<→o< y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
614 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
615
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
616 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
617 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
618
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
619 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
620 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
621 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
622
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
623 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
624 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
625 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
626
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
627 infinity∅ : infinite ∋ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
628 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
629 lemma : o∅ ≡ od→ord od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
630 lemma = let open ≡-Reasoning in begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
631 o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
632 ≡⟨ sym diso ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
633 od→ord ( ord→od o∅ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
634 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
635 od→ord od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
636
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
637 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
638 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
639 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
640 ≡ od→ord (Union (x , (x , x)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
641 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
642
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
643 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
644 isZF = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
645 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
646 ; pair→ = pair→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
647 ; pair← = pair←
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
648 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
649 ; union← = union←
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
650 ; empty = empty
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
651 ; power→ = power→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
652 ; power← = power←
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
653 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
654 ; ε-induction = ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
655 ; infinity∅ = infinity∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
656 ; infinity = infinity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
657 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
658 ; replacement← = replacement←
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
659 ; replacement→ = λ {ψ} → replacement→ {ψ}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
660 -- ; choice-func = choice-func
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
661 -- ; choice = choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
662 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
663
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
664 HOD→ZF : ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
665 HOD→ZF = record {
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
666 ZFSet = HOD
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
667 ; _∋_ = _∋_
365
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
668 ; _≈_ = _=h=_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
669 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
670 ; _,_ = _,_
365
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
671 ; Union = Union
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
672 ; Power = Power
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
673 ; Select = Select
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
674 ; Replace = Replace
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
675 ; infinite = infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
676 ; isZF = isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
677 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 364
diff changeset
678
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
679