Mercurial > hg > Members > kono > Proof > ZF-in-agda

annotate ordinal-definable.agda @ 39:457e6626e0b1

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 23 May 2019 19:48:51 +0900
parents 20cddbb2fc90
children 9439ff020cbd
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rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
2 module ordinal-definable where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
4 open import zf
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
5 open import ordinal
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
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⊔_ )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
9 open import Relation.Binary.PropositionalEquality
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
11 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
14
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
15 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
16 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
17
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
18 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
19
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
20 record OD {n : Level} : Set (suc n) where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
21 field
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22 def : (x : Ordinal {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
23
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
24 open OD
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
25 open import Data.Unit
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
26
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
27 postulate
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
28 od→ord : {n : Level} → OD {n} → Ordinal {n}
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
29 ord→od : {n : Level} → Ordinal {n} → OD {n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
30
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
31 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
32 _∋_ {n} a x = def a ( od→ord x )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
33
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
34 _c<_ : { n : Level } → ( a x : OD {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
35 x c< a = a ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
36
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
37 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n)
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
38 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
39
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
40 postulate
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
41 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
42 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
43 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
44 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
45 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
46 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
47
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
48 HOD = OD
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
49
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
50 od∅ : {n : Level} → HOD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
51 od∅ {n} = record { def = λ _ → Lift n ⊥ }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
52
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
53 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
54 ∅1 {n} x (lift ())
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
55
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
56 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
57 ∅3 {n} {x} = TransFinite {n} c1 c2 c3 x where
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
58 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
59 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n}
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
60 c1 : ∀ (lx : Nat ) → c0 lx (record { lv = Suc lx ; ord = ℵ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
61 c1 lx not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
62 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
63 c1 lx not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
64 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
65 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
66 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
67 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
68 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
69 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ })
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
70 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
71 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
72 c3 lx (Φ .lx) d not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
73 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } )
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
74 ... | t with t (case2 (s< s<refl ) )
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
75 c3 lx (OSuc .lx x₁) d not | t | ()
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
76 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } )
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
77 ... | t with t (case2 (s< (ℵΦ< {_} {_} {Φ (Suc lx)})))
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
78 c3 .(Suc lx) (ℵ lx) d not | t | ()
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
79
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
80 -- find : {n : Level} → ( x : Ordinal {n} ) → o∅ o< x → Ordinal {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
81 -- exists : {n : Level} → ( x : Ordinal {n} ) → (0<x : o∅ o< x ) → find x 0<x o< x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
82
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
83 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
84 def-subst df refl refl = df
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
85
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
86 transitive : {n : Level } { x y z : OD {n} } → y ∋ x → z ∋ y → z ∋ x
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
87 transitive {n} {x} {y} {z} x∋y z∋y with ordtrans ( c<→o< {n} {x} {y} x∋y ) ( c<→o< {n} {y} {z} z∋y )
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
88 ... | t = lemma0 (lemma t) where
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
89 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x )))
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
90 lemma xo<z = o<→c< xo<z
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
91 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x)
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
92 lemma0 dz = def-subst {n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso)
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
93
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
94 ominimal : {n : Level} → (x : Ordinal {n} ) → o∅ o< x → Ordinal {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
95 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case1 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
96 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
97 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case1 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
98 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case2 Φ<) = record { lv = Zero ; ord = Φ 0 }
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
99 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case1 (s≤s x)) = record { lv = lv ; ord = Φ lv }
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
100 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
101 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case1 (s≤s x)) = record { lv = (Suc lv) ; ord = ord }
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
102 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
103 ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case1 (s≤s z≤n)) = record { lv = lv ; ord = Φ lv }
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
104 ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
105
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
106 ∅4 : {n : Level} → ( x : OD {n} ) → x ≡ od∅ {n} → od→ord x ≡ o∅ {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
107 ∅4 {n} x refl = ∅3 lemma1 where
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
108 lemma0 : (y : Ordinal {n}) → def ( od∅ {n} ) y → ⊥
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
109 lemma0 y (lift ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
110 lemma1 : (y : Ordinal {n}) → y o< od→ord od∅ → ⊥
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
111 lemma1 y y<o = lemma0 y ( def-subst {n} {ord→od (od→ord od∅ )} {od→ord (ord→od y)} (o<→c< y<o) oiso diso )
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
112
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
113 ∅5 : {n : Level} → ( x : Ordinal {n} ) → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
114 ∅5 {n} record { lv = Zero ; ord = (Φ .0) } not = ⊥-elim (not refl)
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
115 ∅5 {n} record { lv = Zero ; ord = (OSuc .0 ord) } not = case2 Φ<
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
116 ∅5 {n} record { lv = (Suc lv) ; ord = ord } not = case1 (s≤s z≤n)
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
117
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
118 postulate extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
119
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
120 -- ∅7 : {n : Level} → { x : OD {n} } → ((z : OD {n}) → ¬ ( z c< x )) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
121 -- ∅7 {n} {x} not = cong ( λ k → record { def = k } ) lemma2 where
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
122
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
123 ∅6 : {n : Level } ( x : Ordinal {suc n}) → o∅ o< x → ¬ x ≡ o∅
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
124 ∅6 {n} x lt eq with trio< {n} (o∅ {suc n}) x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
125 ∅6 {n} x lt refl | tri< a ¬b ¬c = ¬b refl
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
126 ∅6 {n} x lt refl | tri≈ ¬a b ¬c = ¬a lt
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
127 ∅6 {n} x lt refl | tri> ¬a ¬b c = ¬b refl
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
128
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
129 ∅8 : {n : Level} → ( x : Ordinal {n} ) → ¬ x o< o∅ {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
130 ∅8 {n} x (case1 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
131 ∅8 {n} x (case2 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
132
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
133 ∅7'' : {n : Level} → o∅ {suc n} ≡ od→ord (od∅ {suc n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
134 ∅7'' {n} = sym ( ∅3 lemma ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
135 ⊥-leq : lift {_} {n} ⊥ ≡ lift {_} {n} ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
136 ⊥-leq = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
137 lemma : {n : Level} (y : Ordinal {suc n}) → ¬ (y o< od→ord od∅)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
138 lemma {n} y lt = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
139 -- with ⊥-leq
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
140 -- ... | refl = ⊥-elim {!!} --- ∅1 (ord→od y) ( def-subst {suc n} {od∅} {y} {!!} {!!} {!!} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
141
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
142 ∅10 : {n : Level} → (x : OD {n} ) → ¬ ( ( y : OD {n} ) → Lift (suc n) ( x ∋ y)) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
143 ∅10 {n} x not = cong ( λ k → record { def = k }) ( extensionality {n} ( λ y → {!!} ) ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
144 ⊥-leq : lift {_} {n} ⊥ ≡ lift {_} {n} ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
145 ⊥-leq = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
146 lemma : (y : Ordinal {n} ) → def x y ≡ def od∅ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
147 lemma y with def x y | def od∅ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
148 ... | s | t = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
149
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
150 open Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
151
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
152
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
153 ∅77 : {n : Level} → (x : OD {suc n} ) → ¬ ( ord→od (o∅ {suc n}) ∋ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
154 ∅77 {n} x lt with od→ord x | c<→o< {suc n} {x} {ord→od (o∅ {suc n})} lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
155 ... | s | t = lemma0 s t where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
156 lemma : ( ox : Ordinal {suc n} ) → ox o< o∅ {suc n} → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
157 lemma = ∅8
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
158 lemma1 : { y : Ordinal {suc n} } { la lA : Nat } { oa : OrdinalD {suc n} la }{ oA : OrdinalD {suc n} lA }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
159 → ( record {lv = la ; ord = oa } ≡ record {lv = lA ; ord = oA } )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
160 → (la < lv y ) ∨ ( oa d< ord y ) → (lA < lv y ) ∨ ( oA d< ord y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
161 lemma1 refl x = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
162 lemma0 : ( ox : Ordinal {suc n} ) → ox o< od→ord (ord→od (o∅ {suc n})) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
163 lemma0 = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
164
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
165 -- lemma0 ox t = lemma ox ( lemma1 (diso {suc n} {o∅ {suc n}}) t )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
166 --- ... | s | t = lemma s t (diso {suc n} {o∅ {suc n}}) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
167 -- with od→ord x | c<→o< {n} {x} {ord→od (o∅ {n})} lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
168 -- ∅8 {n} (od→ord x) ( def-subst {n} {ord→od (od→ord x) } {{!!}} ( c<→o< lt ) {!!} {!!} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
169
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
170 ∅7' : {n : Level} → ord→od (o∅ {n}) ≡ od∅ {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
171 ∅7' {n} = cong ( λ k → record { def = k }) ( extensionality {n} lemma ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
172 lemma : ( x : Ordinal {n} ) → def (ord→od o∅) x ≡ def od∅ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
173 lemma x = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
174
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
175 ∅7 : {n : Level} → ( x : OD {n} ) → od→ord x ≡ o∅ {n} → x ≡ od∅ {n}
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
176 ∅7 {n} x eq = trans ( trans (sym oiso)( cong ( λ k → ord→od k ) eq ) ) ∅7'
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
177
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
178 ∅9 : {n : Level} → (x : OD {n} ) → ¬ x ≡ od∅ → o∅ o< od→ord x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
179 ∅9 x not = ∅5 ( od→ord x) lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
180 lemma : ¬ od→ord x ≡ o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
181 lemma eq = not ( ∅7 x eq )
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
182
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
183 HOD→ZF : {n : Level} → ZF {suc n} {suc n}
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
184 HOD→ZF {n} = record {
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
185 ZFSet = OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
186 ; _∋_ = λ a x → Lift (suc n) ( a ∋ x )
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
187 ; _≈_ = _≡_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
188 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
189 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
190 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
191 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
192 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
193 ; Replace = Replace
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
194 ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
195 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
196 } where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
197 Replace : OD {n} → (OD {n} → OD {n} ) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
198 Replace X ψ = sup-od ψ
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
199 Select : OD {n} → (OD {n} → Set (suc n) ) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
200 Select X ψ = record { def = λ x → select ( ord→od x ) } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
201 select : OD {n} → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
202 select x with ψ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
203 ... | t = Lift n ⊤
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
204 _,_ : OD {n} → OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
205 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
206 Union : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
207 Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
208 Power : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
209 Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
210 ZFSet = OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
211 _∈_ : ( A B : ZFSet ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
212 A ∈ B = B ∋ A
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
213 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
214 _⊆_ A B {x} = A ∋ x → B ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
215 _∩_ : ( A B : ZFSet ) → ZFSet
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
216 A ∩ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∧ (Lift (suc n) ( B ∋ x ) ))
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
217 _∪_ : ( A B : ZFSet ) → ZFSet
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
218 A ∪ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∨ (Lift (suc n) ( B ∋ x ) ))
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
219 infixr 200 _∈_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
220 infixr 230 _∩_ _∪_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
221 infixr 220 _⊆_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
222 isZF : IsZF (OD {n}) (λ a x → Lift (suc n) ( a ∋ x )) _≡_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } })
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
223 isZF = record {
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
224 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
225 ; pair = pair
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
226 ; union→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
227 ; union← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
228 ; empty = empty
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
229 ; power→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
230 ; power← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
231 ; extentionality = {!!}
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
232 ; minimul = minimul
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
233 ; regularity = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
234 ; infinity∅ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
235 ; infinity = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
236 ; selection = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
237 ; replacement = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
238 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
239 open _∧_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
240 pair : (A B : OD {n} ) → Lift (suc n) ((A , B) ∋ A) ∧ Lift (suc n) ((A , B) ∋ B)
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
241 proj1 (pair A B ) = lift ( case1 refl )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
242 proj2 (pair A B ) = lift ( case2 refl )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
243 empty : (x : OD {n} ) → ¬ Lift (suc n) (od∅ ∋ x)
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
244 empty x (lift (lift ()))
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
245 union→ : (X x y : OD {n} ) → Lift (suc n) (X ∋ x) → Lift (suc n) (x ∋ y) → Lift (suc n) (Union X ∋ y)
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
246 union→ X x y (lift X∋x) (lift x∋y) = lift {!!} where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
247 lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
248 lemma {z} X∋z = {!!}
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
249 minimul : (x : OD {n} ) → ¬ x ≡ od∅ → OD {n}
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
250 minimul x not = ord→od ( ominimal (od→ord x) (∅9 x not) )
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
251 regularity : (x : OD) → (not : ¬ x ≡ od∅ ) →
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
252 Lift (suc n) (x ∋ minimul x not ) ∧
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
253 (Select x (λ x₁ → Lift (suc n) ( minimul x not ∋ x₁) ∧ Lift (suc n) (x ∋ x₁)) ≡ od∅)
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
254 proj1 ( regularity x non ) = {!!}
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
255 proj2 ( regularity x non ) = {!!}