annotate HOD.agda @ 116:47541e86c6ac

axiom of selection
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 26 Jun 2019 08:05:58 +0900
parents 277c2f3b8acb
children a4c97390d312
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
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
2 module HOD 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
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
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
14 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
15
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
16 record HOD {n : Level} : Set (suc n) where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
17 field
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
18 def : (x : Ordinal {n} ) → Set n
114
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
19 otrans : {x : Ordinal {n} } → def x → { y : Ordinal {n} } → y o< x → def y
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
20
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
21 open HOD
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22 open import Data.Unit
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
23
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
24 open Ordinal
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
25
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
26 record _==_ {n : Level} ( a b : HOD {n} ) : Set n where
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
27 field
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
28 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
29 eq← : ∀ { x : Ordinal {n} } → def b x → def a x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
30
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
31 id : {n : Level} {A : Set n} → A → A
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
32 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
33
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
34 eq-refl : {n : Level} { x : HOD {n} } → x == x
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
35 eq-refl {n} {x} = record { eq→ = id ; eq← = id }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
36
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
37 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
39 eq-sym : {n : Level} { x y : HOD {n} } → x == y → y == x
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
40 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
41
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
42 eq-trans : {n : Level} { x y z : HOD {n} } → x == y → y == z → x == z
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
43 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
44
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
45 -- Ordinal in HOD ( and ZFSet )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
46 Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
47 Ord {n} a = record { def = λ y → y o< a ; otrans = lemma } where
114
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
48 lemma : {x : Ordinal} → x o< a → {y : Ordinal} → y o< x → y o< a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
49 lemma {x} x<a {y} y<x = ordtrans {n} {y} {x} {a} y<x x<a
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
50
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
51 od∅ : {n : Level} → HOD {n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
52 od∅ {n} = Ord o∅
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
53
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
54 postulate
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
55 -- HOD can be iso to a subset of Ordinal ( by means of Godel Set )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
56 od→ord : {n : Level} → HOD {n} → Ordinal {n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
57 ord→od : {n : Level} → Ordinal {n} → HOD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
58 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
59 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
60 c<→o< : {n : Level} {x y : HOD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
61 ord-Ord :{n : Level} {x : Ordinal {n}} → x ≡ od→ord (Ord x)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
62 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
63 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
64 -- supermum as Replacement Axiom
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
65 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
66 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
67 -- contra-position of mimimulity of supermum required in Power Set Axiom
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
68 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
69 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
70 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
71
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
72 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
73 _∋_ {n} a x = def a ( od→ord x )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
74
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
75 _c<_ : { n : Level } → ( x a : HOD {n} ) → Set n
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
76 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
77
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
78 _c≤_ : {n : Level} → HOD {n} → HOD {n} → Set (suc n)
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
79 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
80
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
81 cseq : {n : Level} → HOD {n} → HOD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
82 cseq x = record { def = λ y → osuc y o< (od→ord x) ; otrans = lemma } where
114
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
83 lemma : {ox : Ordinal} → osuc ox o< od→ord x → { oy : Ordinal}→ oy o< ox → osuc oy o< od→ord x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
84 lemma {ox} oox<Ox {oy} oy<ox = ordtrans (osucc oy<ox ) oox<Ox
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
85
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
86 def-subst : {n : Level } {Z : HOD {n}} {X : Ordinal {n} }{z : HOD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
87 def-subst df refl refl = df
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
88
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
89 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
90 o<→c< {n} {x} {y} lt = subst ( λ k → k o< y ) ord-Ord lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
91
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
92 sup-od : {n : Level } → ( HOD {n} → HOD {n}) → HOD {n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
93 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
94
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
95 sup-c< : {n : Level } → ( ψ : HOD {n} → HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
96 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
97 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
98 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
99 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) )
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
100
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
101 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 80
diff changeset
102 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
103 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
104 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
105 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
106 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
107 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
108 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
109 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
110 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
111 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
112 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
113 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
114 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
115 ... | 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
116 c3 lx (OSuc .lx x₁) d not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
117
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
118 transitive : {n : Level } { z y x : HOD {suc n} } → z ∋ y → y ∋ x → z ∋ x
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
119 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y )
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
120 ... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y )
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
121
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
122 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
123 field
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
124 mino : Ordinal {n}
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
125 min<x : mino o< x
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
126
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
127 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
128 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
129 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
130 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n)
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
132 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y }
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
133 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
134
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
135 -- avoiding lv != Zero error
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
136 orefl : {n : Level} → { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
137 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
138
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
139 ==-iso : {n : Level} → { x y : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
140 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
141 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
142 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
143 where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
144 lemma : {x : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
145 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
146
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
147 =-iso : {n : Level } {x y : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
148 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
149
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
150 ord→== : {n : Level} → { x y : HOD {n} } → od→ord x ≡ od→ord y → x == y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
151 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
152 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
153 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
154
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
155 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
156 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
157
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
158 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
159 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
160
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
161 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
162 c≤-refl x = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
163
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
164 ∋→o< : {n : Level} → { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
165 ∋→o< {n} {a} {x} lt = t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
166 t : (od→ord x) o< (od→ord a)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
167 t = c<→o< {suc n} {x} {a} lt
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
168
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
169 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
170 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
171 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
172 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
173 lemma lt with o<→c< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
174 lemma lt | t = o<¬≡ refl t
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
175 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
176 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
177
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
178 o<→¬c> : {n : Level} → { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x )
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
179 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
180
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
181 o≡→¬c< : {n : Level} → { x y : HOD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
182 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt)
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
183
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
184 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ ; otrans = λ () } == od∅ {n}
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
185 eq→ ∅0 {w} (lift ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
186 eq← ∅0 {w} (case1 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
187 eq← ∅0 {w} (case2 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
188
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
189 ∅< : {n : Level} → { x y : HOD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} )
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
190 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
60
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
191 ∅< {n} {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
192
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
193 -- ∅6 : {n : Level} → { x : HOD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
194 -- ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
195
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
196 def-iso : {n : Level} {A B : HOD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
197 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
198
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
199 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
200 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
201 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
202 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
203
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
204 open _∧_
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
205
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
206 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
207 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (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
208
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
209 csuc : {n : Level} → HOD {suc n} → HOD {suc n}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
210 csuc x = ord→od ( osuc ( od→ord x ))
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
211
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
212 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
213
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
214 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
215 ZFSubset A x = record { def = λ y → def A y ∧ def x y ; otrans = {!!} }
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
216
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
217 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
218 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
219
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
220 -- Constructible Set on α
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
221 L : {n : Level} → (α : Ordinal {suc n}) → HOD {suc n}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
222 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
223 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) )
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
224 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
225 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} }
89
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
226
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
227 omega : { n : Level } → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
228 omega = record { lv = Suc Zero ; ord = Φ 1 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
229
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
230 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
231 HOD→ZF {n} = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
232 ZFSet = HOD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
233 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
234 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
235 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
236 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
237 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
238 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
239 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
240 ; Replace = Replace
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
241 ; infinite = Ord omega
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
242 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
243 } where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
244 Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
245 Replace X ψ = sup-od ψ
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
246 Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {suc n}
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
247 Select X ψ = record { def = λ x → ((y : Ordinal {suc n} ) → X ∋ ord→od y → ψ (ord→od y)) ∧ (X ∋ ord→od x ) ; otrans = lemma } where
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
248 lemma : {x : Ordinal} → ((y : Ordinal) → X ∋ ord→od y → ψ (ord→od y)) ∧ (X ∋ ord→od x) →
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
249 {y : Ordinal} → y o< x → ((y₁ : Ordinal) → X ∋ ord→od y₁ → ψ (ord→od y₁)) ∧ (X ∋ ord→od y)
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
250 lemma {x} select {y} y<x = record { proj1 = proj1 select ; proj2 = y<X } where
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
251 y<X : X ∋ ord→od y
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
252 y<X = otrans X (proj2 select) (o<-subst y<x (sym diso) (sym diso) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
253 _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
254 x , y = Ord (omax (od→ord x) (od→ord y))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
255 Union : HOD {suc n} → HOD {suc n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
256 Union U = cseq U
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
257 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
258 Power : HOD {suc n} → HOD {suc n}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
259 Power A = Def A
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
260 ZFSet = HOD {suc n}
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
261 _∈_ : ( A B : ZFSet ) → Set (suc n)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
262 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
263 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
264 _⊆_ A B {x} = A ∋ x → B ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
265 _∩_ : ( A B : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
266 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) )
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
267 -- _∪_ : ( A B : ZFSet ) → ZFSet
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
268 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) )
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
269 {_} : ZFSet → ZFSet
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
270 { x } = ( x , x )
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
271
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
272 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
273 -- infixr 230 _∩_ _∪_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
274 infixr 220 _⊆_
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
275 isZF : IsZF (HOD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
276 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
277 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
278 ; pair = pair
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
279 ; union-u = λ _ z _ → csuc z
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
280 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
281 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
282 ; empty = empty
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
283 ; power→ = power→
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
284 ; power← = power←
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
285 ; extensionality = extensionality
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
286 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
287 ; regularity = regularity
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
288 ; infinity∅ = infinity∅
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
289 ; infinity = λ _ → infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
290 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
291 ; replacement = replacement
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
292 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
293 open _∧_
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
294 open Minimumo
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
295 pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
87
296388c03358 split omax?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
296 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
296388c03358 split omax?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
297 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
298 empty : (x : HOD {suc n} ) → ¬ (od∅ ∋ x)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
299 empty x (case1 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
300 empty x (case2 ())
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
301 ---
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
302 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
303 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
304 --
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
305 -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
306 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
307 -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
308 --
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
309 power→ : (A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
310 power→ A t P∋t {x} t∋x = proj1 lemma-s where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
311 minsup : HOD
99
74330d0cdcbc Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
312 minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))
74330d0cdcbc Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
313 lemma-t : csuc minsup ∋ t
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
314 lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl )
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
315 lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
316 lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso )
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
317 lemma-s | case1 eq = def-subst {!!} oiso refl
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
318 lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) t∋x
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
319 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
320 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
321 -- Power A is a sup of ZFSubset A t, so Power A ∋ t
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
322 --
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
323 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
99
74330d0cdcbc Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
324 power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t}
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
325 {!!} refl lemma1 where
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
326 lemma-eq : ZFSubset A t == t
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
327 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
328 eq← lemma-eq {z} w = record { proj2 = w ;
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
329 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
99
74330d0cdcbc Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
330 lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
331 lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) {!!}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
332 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x)))
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
333 lemma = sup-o<
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
334 union-lemma-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
335 union-lemma-u {X} {z} U>z = lemma <-osuc where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
336 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
337 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} {!!} refl refl
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
338 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
339 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
340 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx))
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
341 union→ X y u xx | tri< a ¬b ¬c | ()
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
342 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where
73
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
343 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
344 lemma refl lt = lt
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
345 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx ))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
346 union← : (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z )
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
347 union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z }
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
348 ψiso : {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y → ψ y
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
349 ψiso {ψ} t refl = t
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
350 selection : {X : HOD } {ψ : (x : HOD ) → Set (suc n)} {y : HOD} → (((y₁ : HOD) → X ∋ y₁ → ψ y₁) ∧ (X ∋ y)) ⇔ (Select X ψ ∋ y)
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
351 selection {X} {ψ} {y} = record { proj1 = λ s → record {
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
352 proj1 = λ y1 y1<X → proj1 s (ord→od y1) y1<X ; proj2 = subst (λ k → def X k ) (sym diso) (proj2 s) }
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
353 ; proj2 = λ s → record { proj1 = λ y1 dy1 → subst (λ k → ψ k ) oiso ((proj1 s) (od→ord y1) (def-subst {suc n} {_} {_} {X} {_} dy1 refl (sym diso )))
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
354 ; proj2 = def-subst {suc n} {_} {_} {X} {od→ord y} (proj2 s ) refl diso } } where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
355 replacement : {ψ : HOD → HOD} (X x : HOD) → Replace X ψ ∋ ψ x
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
356 replacement {ψ} X x = sup-c< ψ {x}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
357 ∅-iso : {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅)
60
6a1f67a4cc6e dead end
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 59
diff changeset
358 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
359 minimul : (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
360 minimul x not = {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
361 regularity : (x : HOD) (not : ¬ (x == od∅)) →
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
362 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
363 proj1 (regularity x not ) = {!!}
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
364 proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
365 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
366 reg {y} t = {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
367 extensionality : {A B : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
368 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
369 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
370 xx-union : {x : HOD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
371 xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
372 xxx-union : {x : HOD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
373 xxx-union {x} = cong ( λ k → Ord k ) lemma where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
374 lemma1 : {x : HOD {suc n}} → od→ord x o< od→ord (x , x)
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
375 lemma1 {x} = c<→o< ( proj1 (pair x x ) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
376 lemma2 : {x : HOD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
377 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
378 lemma : {x : HOD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
379 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
380 uxxx-union : {x : HOD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) }
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
381 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where
90
38d139b5edac def ord conversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 89
diff changeset
382 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
383 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
384 uxxx-2 : {x : HOD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) }
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
385 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
386 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
387 uxxx-ord : {x : HOD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x)
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
388 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
389 infinite : HOD {suc n}
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
390 infinite = Ord omega
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
391 infinity∅ : Ord omega ∋ od∅ {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
392 infinity∅ = {!!}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
393 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
394 infinity x lt = {!!} where
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
395 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
396 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
397 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
398 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ()))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
399 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
400 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
401 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
402 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
403 -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ { t }) ]
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
404 record Choice (z : HOD {suc n}) : Set (suc (suc n)) where
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
405 field
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
406 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
407 t : {x : HOD {suc n}} ( x∈z : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
408 choice : { x : HOD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == { t x∈z x }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
409 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
410 -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
411 -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
412