annotate OD.agda @ 358:811152bf2f47

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 14 Jul 2020 12:39:21 +0900
parents 08d94fec239c
children 2a8a51375e49
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
2 open import Ordinals
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
3 module OD {n : Level } (O : Ordinals {n} ) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
5 open import zf
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
6 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
7 open import Relation.Binary.PropositionalEquality
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
8 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
9 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
10 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
11 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
14 open import logic
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
15 open import nat
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
16
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
17 open inOrdinal O
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
18
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
19 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
20
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
21 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
22 field
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
23 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
24
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
25 open OD
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
26
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
27 open _∧_
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
28 open _∨_
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
29 open Bool
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
30
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
31 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
32 field
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
33 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
34 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
35
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
36 id : {A : Set n} → A → A
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
37 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
39 ==-refl : { x : OD } → x == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
40 ==-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
41
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
42 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
43
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
44 ==-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
45 ==-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
46
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
47 ==-sym : { x y : OD } → x == y → y == x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
48 ==-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
49
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
50
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
51 ⇔→== : { 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
52 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
53 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
54
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
55 -- next assumptions are our axiom
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
56 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
57 -- 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
58 -- correspondence to the OD then the OD looks like a ZF Set.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
59 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
60 -- 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
61 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
62 --
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
63 -- 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
64 -- HOD = { x | TC x ⊆ OD }
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
65 -- 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
66 -- 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
67 --
309
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
68 -- 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
69 -- 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
70 -- 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
71 -- bound on each HOD.
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
72 --
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
73 -- 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
74 -- we need explict assumption on sup.
309
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
75 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
76 -- ==→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
77
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
78 data One : Set n where
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
79 OneObj : One
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
80
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
81 -- Ordinals in OD , the maximum
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
82 Ords : OD
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
83 Ords = record { def = λ x → One }
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
84
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
85 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
86 field
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
87 od : OD
304
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
88 odmax : Ordinal
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
89 <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
90
304c271b3d47 HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 301
diff changeset
91 open HOD
304c271b3d47 HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 301
diff changeset
92
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
93 record ODAxiom : Set (suc n) where
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
94 field
304
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
95 -- 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
96 od→ord : HOD → Ordinal
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
97 ord→od : Ordinal → HOD
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
98 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
99 ⊆→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
100 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
322
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
103 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
104 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
105
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
106 postulate odAxiom : ODAxiom
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
107 open ODAxiom odAxiom
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
108
344
e0916a632971 possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 343
diff changeset
109 -- possible order restriction
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
110 hod-ord< : {x : HOD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
111 hod-ord< {x} = od→ord x o< next (odmax x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
112
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
113 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
114 ¬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
115 ¬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
116
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
117 -- 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
118 ¬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
119 ¬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
120 next-ord : Ordinal → Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
121 next-ord x = osuc x
301
b012a915bbb5 contradiction found ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 300
diff changeset
122
179
aa89d1b8ce96 fix comments
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 176
diff changeset
123 -- Ordinal in OD ( and ZFSet ) Transitive Set
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
124 Ord : ( a : Ordinal ) → HOD
304
2c111bfcc89a HOD using <maxod
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 303
diff changeset
125 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
126 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
127 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
128
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
129 od∅ : HOD
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
130 od∅ = Ord o∅
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
131
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
132 odef : HOD → Ordinal → Set n
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
133 odef A x = def ( od A ) x
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
134
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
135 -- If we have reverse of c<→o<, everything becomes Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
136 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
137 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
138 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
139 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
140 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
141 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
142
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
143 _∋_ : ( a x : HOD ) → Set n
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
144 _∋_ a x = odef a ( od→ord x )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
145
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
146 _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
147 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
148
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
149 cseq : {n : Level} → HOD → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
150 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
151 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
152 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
153
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
154 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
155 odef-subst df refl refl = df
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
156
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
157 otrans : {n : Level} {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
158 otrans x<a y<x = ordtrans y<x x<a
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
159
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
160 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
161 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
162
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
163 -- avoiding lv != Zero error
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
164 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
165 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
166
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
167 ==-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
168 ==-iso {x} {y} eq = record {
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
169 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
170 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
171 where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
172 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
173 lemma {x} {z} d = odef-subst d oiso refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
174
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
175 =-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
176 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
177
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
178 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
179 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
180 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
181 lemma ox ox refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
182
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
183 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
184 o≡→== {x} {.x} refl = ==-refl
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
185
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
186 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
187 o∅≡od∅ = ==→o≡ lemma where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
188 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
189 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
190 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
191 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
192 lemma : od (ord→od o∅) == od od∅
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
193 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
194
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
195 ord-od∅ : od→ord (od∅ ) ≡ o∅
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
196 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
197
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
198 ∅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
199 eq→ ∅0 {w} (lift ())
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
200 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
201
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
202 ∅< : { 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
203 ∅< {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
204 ∅< {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
205
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
206 ∅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
207 ∅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
208
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
209 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
210 odef-iso refl t = t
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
211
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
212 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
213 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
214 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
215 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
216 is-o∅ x | tri> ¬a ¬b c = no ¬b
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
217
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
218 -- the pair
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
219 _,_ : HOD → HOD → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
220 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
221 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
222 lemma {t} (case1 refl) = omax-x _ _
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
223 lemma {t} (case2 refl) = omax-y _ _
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
224
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
225 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
226 pair-xx<xy {x} {y} = ⊆→o≤ lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
227 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
228 lemma {z} (case1 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
229 lemma {z} (case2 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
230
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
231 -- another form of infinite
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
232 -- pair-ord< : {x : Ordinal } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
233 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
234 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
235 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
236 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
237 lemmab1 : od→ord (x , x) o< next ( odmax (x , x))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
238 lemmab1 = ho<
188
1f2c8b094908 axiom of choice → p ∨ ¬ p
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
239
344
e0916a632971 possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 343
diff changeset
240 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
241 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
242 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
243 lemma (case1 refl) = y∋x
e0916a632971 possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 343
diff changeset
244 lemma (case2 refl) = y∋x
e0916a632971 possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 343
diff changeset
245
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
246 -- 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
247 -- 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
248
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
249 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
250 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
251
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
252 ZFSubset : (A x : HOD ) → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
253 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; odmax = omin (odmax A) (odmax x) ; <odmax = lemma } where -- roughly x = A → Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
254 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
255 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
256
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
257 record _⊆_ ( A B : HOD ) : Set (suc n) where
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
258 field
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
259 incl : { x : HOD } → A ∋ x → B ∋ x
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
260
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
261 open _⊆_
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
262 infixr 220 _⊆_
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
263
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
264 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
265 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
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
267 -- 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
268 ⊆→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
269 → ({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
270 → {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
271 ⊆→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
272 ⊆→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
273 ⊆→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
274 ⊆→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
275 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
276 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
277 lemma (case1 refl) = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
278 lemma (case2 refl) = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
279 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
280 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
281 lemma1 : osuc (od→ord y) o< od→ord (x , x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
282 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
283
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
284 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
285 subset-lemma {A} {x} = record {
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
286 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
287 ; 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
288 }
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
289
312
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
290 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
291 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
292 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
293 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
294
261
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
295 open import Data.Unit
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
296
324
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 323
diff changeset
297 ε-induction : { ψ : HOD → Set n}
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
298 → ( {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
299 → (x : HOD ) → ψ x
261
d9d178d1457c ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 260
diff changeset
300 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
301 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
302 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
303 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
304 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
305
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
306 -- level trick (what'a shame)
330
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
307 ε-induction1 : { ψ : HOD → Set (suc n)}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
308 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
309 → (x : HOD ) → ψ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 328
diff changeset
310 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
311 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
312 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
313 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
314 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
315
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
316 HOD→ZF : ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
317 HOD→ZF = record {
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
318 ZFSet = HOD
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
319 ; _∋_ = _∋_
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
320 ; _≈_ = _=h=_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
321 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
322 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
323 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
324 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
325 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
326 ; Replace = Replace
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
327 ; infinite = infinite
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
328 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
329 } where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
330 ZFSet = HOD -- is less than Ords because of maxod
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
331 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 307
diff changeset
332 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
333 Replace : HOD → (HOD → HOD) → HOD
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
334 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 }
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
335 ; odmax = rmax ; <odmax = rmax<} where
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
336 rmax : Ordinal
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
337 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
338 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
339 rmax< lt = proj1 lt
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
340 _∩_ : ( A B : ZFSet ) → ZFSet
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
341 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
342 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
343 Union : HOD → HOD
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
344 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) }
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
345 ; odmax = osuc (od→ord U) ; <odmax = umax< } where
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
346 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U)
319
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
347 umax< {y} not = lemma (FExists _ lemma1 not ) where
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
348 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
349 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))
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
350 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
351 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))
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
352 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y)
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
353 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
354 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U)
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
355 lemma not with trio< y (od→ord U)
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
356 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
357 lemma not | tri≈ ¬a refl ¬c = <-osuc
eef432aa8dfb Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 318
diff changeset
358 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
359 _∈_ : ( 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
360 A ∈ B = B ∋ A
312
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
361
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
362 OPwr : (A : HOD ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
363 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
364
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
365 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
366 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
367 -- {_} : ZFSet → ZFSet
335
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 330
diff changeset
368 -- { 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
369
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
370 data infinite-d : ( x : Ordinal ) → Set n where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
371 iφ : infinite-d o∅
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
372 isuc : {x : Ordinal } → infinite-d x →
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
373 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
374
328
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
375 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
376 -- We simply assumes infinite-d y has a maximum.
328
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
377 --
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
378 -- This means that many of OD may not be HODs because of the od→ord mapping divergence.
346
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 344
diff changeset
379 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x).
328
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
380 --
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
381 postulate
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
382 ωmax : Ordinal
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
383 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
384
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
385 infinite : HOD
328
72f3e3b44c27 intoduce ωmax
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 327
diff changeset
386 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
387
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
388 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
389 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: 338
diff changeset
390 u : (y : Ordinal ) → HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
391 u y = Union (ord→od y , (ord→od y , ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
392 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
393 lemma {o∅} iφ = x<nx
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
394 lemma (isuc {y} x) = lemma2 where
358
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 348
diff changeset
395 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
396 lemma0 : y o< next o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
397 lemma0 = lemma x
340
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 339
diff changeset
398 lemma8 : 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: 339
diff changeset
399 lemma8 = ho<
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
400 --- (x,y) < next (omax x y) < next (osuc y) = next y
341
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 340
diff changeset
401 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y)
343
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 342
diff changeset
402 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< )
340
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 339
diff changeset
403 lemma81 : od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 339
diff changeset
404 lemma81 = 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: 339
diff changeset
405 lemma9 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y))
342
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 341
diff changeset
406 lemma9 = lemmaa (c<→o< (case1 refl))
340
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 339
diff changeset
407 lemma71 : 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: 339
diff changeset
408 lemma71 = next< lemma81 lemma9
339
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
409 lemma1 : 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: 338
diff changeset
410 lemma1 = ho<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 338
diff changeset
411 lemma2 : od→ord (u y) o< next o∅
340
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 339
diff changeset
412 lemma2 = next< lemma0 (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1))
338
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
413
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 335
diff changeset
414
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
415 _=h=_ : (x y : HOD) → Set n
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
416 x =h= y = od x == od y
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
417
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
418 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
419 -- infixr 230 _∩_ _∪_
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
420 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
421 isZF = record {
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
422 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
247
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
423 ; pair→ = pair→
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
424 ; pair← = pair←
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
425 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
426 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
427 ; empty = empty
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
428 ; power→ = power→
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
429 ; power← = power←
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
430 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
274
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
431 ; ε-induction = ε-induction
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
432 ; infinity∅ = infinity∅
160
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 159
diff changeset
433 ; infinity = infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
434 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
435 ; replacement← = replacement←
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
436 ; replacement→ = λ {ψ} → replacement→ {ψ}
274
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
437 -- ; choice-func = choice-func
29a85a427ed2 ε-induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
438 -- ; choice = choice
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
439 } where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
440
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
441 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
442 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
443 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
247
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
444
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
445 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
446 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
447 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
247
d09437fcfc7c fix pair
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
448
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
449 empty : (x : HOD ) → ¬ (od∅ ∋ x)
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
450 empty x = ¬x<0
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
451
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
452 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
453 o<→c< lt = record { incl = λ z → ordtrans z lt }
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
454
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
455 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
456 ⊆→o< {x} {y} lt with trio< x y
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
457 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
458 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
459 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
155
53371f91ce63 union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 154
diff changeset
460 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
151
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
461
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
462 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
463 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
464 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
465 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
466 union← X z UX∋z = FExists _ lemma UX∋z where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
467 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
468 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
469
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
470 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
471 ψiso {ψ} t refl = t
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
472 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
473 selection {ψ} {X} {y} = record {
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
474 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
475 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
476 }
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
477 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: 310
diff changeset
478 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt )
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
479 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
480 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
481 lemma : def (in-codomain X ψ) (od→ord (ψ x))
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
482 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
483 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
150
ebcbfd9d9c8e fix some
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 148
diff changeset
484 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
485 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
486 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
487 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
488 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
489 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
490 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
491 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
492
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
493 ---
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
494 --- Power Set
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
495 ---
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
496 --- First consider ordinals in HOD
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
497 ---
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
498 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
499 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
500 --
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
501 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
502 ∩-≡ {a} {b} inc = record {
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
503 eq→ = λ {x} x<a → record { proj2 = x<a ;
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
504 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
505 eq← = λ {x} x<a∩b → proj2 x<a∩b }
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
506 --
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
507 -- Transitive Set case
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
508 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t
300
e70980bd80c7 -- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
509 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t
e70980bd80c7 -- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
510 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
511 --
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
512 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
513 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
514 lemma refl (lemma1 lemma-eq )where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
515 lemma-eq : ZFSubset (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
516 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
517 eq← lemma-eq {z} w = record { proj2 = w ;
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
518 proj1 = odef-subst {_} {_} {(Ord a)} {z}
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
519 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
520 lemma1 : {a : Ordinal } { t : HOD }
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
521 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
522 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
312
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
523 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a)))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 311
diff changeset
524 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: 311
diff changeset
525 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord (ZFSubset (Ord a) (ord→od x)))
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
526 lemma = sup-o< _ lemma2
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
527
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
528 --
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
529 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
530 -- then replace of all elements of the Power set by A ∩ y
144
3ba37037faf4 Union again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 143
diff changeset
531 --
300
e70980bd80c7 -- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
532 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
533
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
534 -- we have oly double negation form because of the replacement axiom
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
535 --
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
536 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 257
diff changeset
537 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
538 a = od→ord A
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
539 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
540 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
541 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
542 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
543 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
544 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
545 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
546 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
547
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
548 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
549 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
550 a = od→ord A
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
551 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
552 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
553 lemma3 : OPwr (Ord a) ∋ t
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
554 lemma3 = ord-power← a t lemma0
152
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
555 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
556 lemma4 = let open ≡-Reasoning in begin
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
557 A ∩ ord→od (od→ord t)
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
558 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
559 A ∩ t
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
560 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
152
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
561 t
996a67042f50 power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 151
diff changeset
562
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
563 sup1 : Ordinal
314
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 313
diff changeset
564 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord (ZFSubset (Ord (od→ord A)) (ord→od x)))
313
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 312
diff changeset
565 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: 312
diff changeset
566 lemma9 = <-osuc
315
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
567 lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
568 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
569 lemmad : Ord (osuc (od→ord A)) ∋ t
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
570 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)))
315
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
571 lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A)
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
572 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
573 lemmaf : {x : Ordinal} → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
574 lemmaf {x} lt = proj1 lt
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
575 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
576 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt }
315
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
577 lemmae : od→ord (ZFSubset (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: 314
diff changeset
578 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac)
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
579 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t)
315
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
580 lemma7 with osuc-≡< lemmad
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 314
diff changeset
581 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
582 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
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
583 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
584 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
585 diso
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
586 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt )))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
587 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
588 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
589 lemmai = let open ≡-Reasoning in begin
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
590 od→ord (Ord (od→ord A))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
591 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
592 od→ord (Ord (od→ord t))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
593 ≡⟨ sym diso ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
594 od→ord (ord→od (od→ord (Ord (od→ord t))))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
595 ≡⟨ sym eq1 ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
596 od→ord (ord→od (od→ord t))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
597 ≡⟨ diso ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
598 od→ord t
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
599
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
600 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
601 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
602 lemmak = let open ≡-Reasoning in begin
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
603 od→ord (ord→od (od→ord (Ord (od→ord t))))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
604 ≡⟨ diso ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
605 od→ord (Ord (od→ord t))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
606 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
607 od→ord (Ord (od→ord A))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
608
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
609 lemmaj : od→ord t o< od→ord (Ord (od→ord A))
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
610 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt
310
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
611 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: 309
diff changeset
612 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
613 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 )
318
9e0c97ab3a4a Replace max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 317
diff changeset
614 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
151
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
615 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
b5a337fb7a6d recovering...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 150
diff changeset
616 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
317
57df07b63cae Power done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 316
diff changeset
617 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A )))
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 141
diff changeset
618
311
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 310
diff changeset
619
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
620 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
621 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
622 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
623 lemma lt y<x with osuc-≡< lt
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
624 lemma lt y<x | case1 refl = c<→o< y<x
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
625 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
262
53744836020b CH trying ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 261
diff changeset
626
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
627 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
628 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
629
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
630 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
631 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
632 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
633
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
634 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
635 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
636 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
637
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
638 infinity∅ : infinite ∋ od∅
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
639 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
640 lemma : o∅ ≡ od→ord od∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
641 lemma = let open ≡-Reasoning in begin
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
642 o∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
643 ≡⟨ sym diso ⟩
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
644 od→ord ( ord→od o∅ )
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
645 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
646 od→ord od∅
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
647
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
648 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
649 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
161
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
650 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
651 ≡ od→ord (Union (x , (x , x)))
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
652 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
4c704b7a62e4 ininite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
653
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
654
303
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
655 Union = ZF.Union HOD→ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
656 Power = ZF.Power HOD→ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
657 Select = ZF.Select HOD→ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
658 Replace = ZF.Replace HOD→ZF
7963b76df6e1 ¬odmax based HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
659 isZF = ZF.isZF HOD→ZF