annotate HOD.agda @ 122:4d2434513228

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 30 Jun 2019 11:03:34 +0900
parents 453ef0d4ee8a
children 0c2cbf37e002
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
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
25 open _∧_
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
26
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
27 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
28 field
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
29 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
30 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
31
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
32 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
33 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
34
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
35 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
36 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
37
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
39
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
40 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
41 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
42
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
43 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
44 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
45
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
46 ⇔→== : {n : Level} { x y : HOD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
47 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
48 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
49
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
50 -- Ordinal in HOD ( and ZFSet )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
51 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
52 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
53 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
54 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
55
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
56 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
57 od∅ {n} = Ord o∅
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
58
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
59 postulate
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
60 -- 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
61 od→ord : {n : Level} → HOD {n} → Ordinal {n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
62 ord→od : {n : Level} → Ordinal {n} → HOD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
63 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
64 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
65 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
66 ord-Ord :{n : Level} {x : Ordinal {n}} → x ≡ od→ord (Ord x)
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
67 ==→o≡ : {n : Level} → { x y : HOD {suc n} } → (x == y) → x ≡ y
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
68 -- 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
69 -- 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
70 -- supermum as Replacement Axiom
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
71 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
72 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
73 -- contra-position of mimimulity of supermum required in Power Set Axiom
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
74 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
75 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
76 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ )
117
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
77 minimul : {n : Level } → (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n}
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
78 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x )
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
79 x∋minimul : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
80 minimul-1 : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) )
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
81
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
82 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
83 _∋_ {n} a x = def a ( od→ord x )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
84
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
85 _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
86 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
87
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
88 _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
89 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
90
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
91 cseq : {n : Level} → HOD {n} → HOD {n}
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
92 cseq x = record { def = λ y → def x (osuc y) ; otrans = lemma } where
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
93 lemma : {ox : Ordinal} → def x (osuc ox) → { oy : Ordinal}→ oy o< ox → def x (osuc oy)
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
94 lemma {ox} oox<Ox {oy} oy<ox = otrans x oox<Ox (osucc oy<ox )
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
95
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
96 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
97 def-subst df refl refl = df
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
98
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
99 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
100 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
101
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
102 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
103 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
104
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
105 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
106 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
107 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
108 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
109 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
110
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
111 ∅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
112 ∅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
113 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
114 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
115 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
116 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
117 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
118 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
119 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
120 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
121 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
122 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
123 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
124 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
125 ... | 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
126 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
127
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
128 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
129 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
130 ... | 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
131
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
132 ∅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
133 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
134 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
135 ∅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
136
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
137 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
138 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
139
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
140 -- avoiding lv != Zero error
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
141 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
142 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
143
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
144 ==-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
145 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
146 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
147 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
148 where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
149 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
150 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
151
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
152 =-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
153 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
154
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
155 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
156 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
157 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
158 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
159
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
160 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
161 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
162
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
163 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
164 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
165
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
166 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
167 c≤-refl x = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
168
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
169 ∋→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
170 ∋→o< {n} {a} {x} lt = t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
171 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
172 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
173
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
174 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
175 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
176 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
177 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
178 lemma lt with o<→c< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
179 lemma lt | t = o<¬≡ refl t
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
180 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
181 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
182
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
183 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
184 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
185
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
186 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
187 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
188
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
189 ∅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
190 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
191 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
192 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
193
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
194 ∅< : {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
195 ∅< {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
196 ∅< {n} {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
197
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
198 ∅6 : {n : Level} → { x : HOD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
199 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x )
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
200
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
201 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
202 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
203
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
204 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
205 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
206 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
207 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
208
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
209
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
210 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
211 -- 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
212
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
213 csuc : {n : Level} → HOD {suc n} → HOD {suc n}
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
214 csuc x = Ord ( osuc ( od→ord x ))
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
215
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
216 -- 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
217
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
218 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
219 ZFSubset A x = record { def = λ y → def A y ∧ def x y ; otrans = lemma } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
220 lemma : {z : Ordinal} → def A z ∧ def x z → {y : Ordinal} → y o< z → def A y ∧ def x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
221 lemma {z} d {y} y<z = record { proj1 = otrans A (proj1 d) y<z ; proj2 = otrans x (proj2 d) y<z }
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
222
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
223 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n}
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
224 Def {n} A = Ord ( 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
225
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
226 -- Constructible Set on α
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
227 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
228 -- L (Φ 0) = Φ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
229 -- L (OSuc lv n) = { Def ( L n ) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
230 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
231 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
232 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
233 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
234 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
235 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx }))))
89
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
236
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
237 LS : {n : Level} → {la : Nat } → {oa : OrdinalD {suc n} la }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
238 → L {n} (record { lv = la; ord = OSuc la oa }) ∋ L {n} (record { lv = la; ord = oa })
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
239 LS {n} {la} {oa} = {!!} where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
240 lemma0 : {n : Level} → (ox : Ordinal {suc n}) → od→ord (ZFSubset (Ord ox) (ord→od ox)) ≡ ox
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
241 lemma0 = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
242 lemma : {n : Level} → (ox : Ordinal {suc n}) → ox o< sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x )))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
243 lemma {n} ox = o<-subst {suc n} {_} {_} {ox} {sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x )))}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
244 (sup-o< {suc n} {λ x → od→ord ( ZFSubset (Ord ox) (ord→od x ))} {ox} ) (lemma0 ox) refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
245
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
246 L0 : {n : Level} → {la : Nat } → {oa : OrdinalD {suc n} (Suc la) }{ob : OrdinalD {suc n} la }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
247 → L (record { lv = Suc la; ord = oa }) ∋ L (record { lv = la; ord = ob })
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
248 L0 = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
249
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
250 L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : HOD {suc n}) → L α ∋ x → L β ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
251 L1 {n} record { lv = .0 ; ord = (Φ .0) } record { lv = .(Suc _) ; ord = ord₁ } (case1 (s≤s z≤n)) x (case1 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
252 L1 {n} record { lv = .0 ; ord = (Φ .0) } record { lv = .(Suc _) ; ord = ord₁ } (case1 (s≤s z≤n)) x (case2 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
253 L1 {n} record { lv = .0 ; ord = (OSuc .0 ord₂) } record { lv = (Suc lx) ; ord = ord₁ } (case1 (s≤s z≤n)) x α∋x = lemma α∋x where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
254 lemma : (od→ord x) o< (sup-o (λ x₁ → od→ord (ZFSubset (L (record { lv = 0 ; ord = ord₂ })) (ord→od x₁))))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
255 → L (record { lv = Suc lx ; ord = ord₁ }) ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
256 lemma lt = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
257 L1 {n} record { lv = (Suc _) ; ord = ord₂ } record { lv = (Suc (Suc _)) ; ord = ord₁ } (case1 (s≤s (s≤s x₁))) x α∋x = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
258 L1 {n} record { lv = lx ; ord = (Φ lx) } record { lv = lx ; ord = (OSuc lx _) } (case2 Φ<) x α∋x = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
259 L1 {n} record { lv = lx ; ord = (OSuc lx _) } record { lv = lx ; ord = (OSuc lx _) } (case2 (s< x₁)) x α∋x = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
260
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
261 omega : { n : Level } → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
262 omega = record { lv = Suc Zero ; ord = Φ 1 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
263
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
264 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
265 HOD→ZF {n} = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
266 ZFSet = HOD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
267 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
268 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
269 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
270 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
271 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
272 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
273 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
274 ; Replace = Replace
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
275 ; infinite = Ord omega
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
276 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
277 } where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
278 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
279 Replace X ψ = sup-od ψ
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
280 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
281 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
282 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
283 {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
284 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
285 y<X : X ∋ ord→od y
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
286 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
287 _,_ : 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
288 x , y = Ord (omax (od→ord x) (od→ord y))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
289 Union : HOD {suc n} → HOD {suc n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
290 Union U = cseq U
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
291 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
292 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
293 Power A = Def A
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
294 ZFSet = HOD {suc n}
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
295 _∈_ : ( 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
296 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
297 _⊆_ : ( 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
298 _⊆_ A B {x} = A ∋ x → B ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
299 _∩_ : ( A B : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
300 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
301 -- _∪_ : ( A B : ZFSet ) → ZFSet
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
302 -- 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
303 {_} : ZFSet → ZFSet
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
304 { 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
305
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
306 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
307 -- infixr 230 _∩_ _∪_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
308 infixr 220 _⊆_
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
309 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
310 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
311 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
312 ; pair = pair
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
313 ; union-u = λ X z UX∋z → union-u {X} {z} UX∋z
72
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
314 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
315 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
316 ; empty = empty
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
317 ; power→ = power→
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
318 ; power← = power←
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
319 ; extensionality = extensionality
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
320 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
321 ; regularity = regularity
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
322 ; infinity∅ = infinity∅
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
323 ; infinity = λ _ → infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
324 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
325 ; replacement = replacement
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
326 } where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
327 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
328 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
329 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
330 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
331 empty x (case1 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
332 empty x (case2 ())
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
333 ---
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
334 --- 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
335 --- 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
336 --
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
337 -- 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
338 -- 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
339 -- 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
340 --
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
341 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
342 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
343 minsup : HOD
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
344 minsup = ZFSubset A ( Ord ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))
99
74330d0cdcbc Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 98
diff changeset
345 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
346 lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl )
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
347 lemma-s : ZFSubset A ( Ord ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
348 lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl (sym ord-Ord) )
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
349 lemma-s | case1 eq = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
350 lemma-s | case2 lt = transitive {n} {minsup} {t} {x} {!!} t∋x
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
351 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
352 -- 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
353 -- 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
354 --
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
355 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
356 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
357 {!!} refl lemma1 where
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
358 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
359 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
360 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
361 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
362 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
363 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
364 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
365 lemma = sup-o<
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
366 union-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → HOD {suc n}
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
367 union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
368 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
369 union→ X z u xx with trio< ( od→ord u ) ( osuc ( od→ord z ))
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
370 union→ X z u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx))
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
371 union→ X z u xx | tri< a ¬b ¬c | ()
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
372 union→ X z u xx | tri≈ ¬a b ¬c = def-subst {suc n} {_} {_} {X} {osuc (od→ord z)} (proj1 xx) refl b
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
373 union→ X z u xx | tri> ¬a ¬b c = otrans X (proj1 xx) c
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
374 union← : (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ union-u {X} {z} X∋z ) ∧ (union-u {X} {z} X∋z ∋ z )
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
375 union← X z X∋z = record { proj1 = lemma ; proj2 = <-osuc } where
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
376 lemma : X ∋ union-u {X} {z} X∋z
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
377 lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} X∋z refl ord-Ord
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
378 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
379 ψiso {ψ} t refl = t
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
380 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
381 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
382 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
383 ; 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
384 ; 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
385 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
386 replacement {ψ} X x = sup-c< ψ {x}
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
387 ∅-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
388 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
389 regularity : (x : HOD) (not : ¬ (x == od∅)) →
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
390 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
117
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
391 proj1 (regularity x not ) = x∋minimul x not
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
392 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
393 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
394 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
395 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
396 lemma3 = proj1 s x₁ (proj2 s)
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
397 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
398 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
399 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
400 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
401 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
402 open import Relation.Binary.PropositionalEquality
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
403 uxxx-ord : {x : HOD {suc n}} → {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ⇔ ( y o< osuc (od→ord x) )
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
404 uxxx-ord {x} {y} = subst (λ k → k ⇔ ( y o< osuc (od→ord x) )) (sym lemma) ( osuc2 y (od→ord x)) where
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
405 lemma : {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ≡ osuc y o< osuc (osuc (od→ord x))
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
406 lemma {y} = let open ≡-Reasoning in begin
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
407 def (Union (x , (x , x))) y
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
408 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
409 def ( Ord ( omax (od→ord x) (od→ord (Ord (omax (od→ord x) (od→ord x) )) ))) ( osuc y )
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
410 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
411 osuc y o< omax (od→ord x) (od→ord (Ord (omax (od→ord x) (od→ord x) )) )
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
412 ≡⟨ cong (λ k → osuc y o< omax (od→ord x) k ) (sym ord-Ord) ⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
413 osuc y o< omax (od→ord x) (omax (od→ord x) (od→ord x) )
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
414 ≡⟨ cong (λ k → osuc y o< k ) (omxxx (od→ord x) ) ⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
415 osuc y o< osuc (osuc (od→ord x))
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
416
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
417 infinite : HOD {suc n}
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
418 infinite = Ord omega
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
419 infinity∅ : Ord omega ∋ od∅ {suc n}
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
420 infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
421 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
422 infinity x lt = o<-subst ( lemma (od→ord x) lt ) eq refl where
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
423 eq : osuc (od→ord x) ≡ od→ord (Union (x , (x , x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
424 eq = let open ≡-Reasoning in begin
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
425 osuc (od→ord x)
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
426 ≡⟨ ord-Ord ⟩
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
427 od→ord (Ord (osuc (od→ord x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
428 ≡⟨ cong ( λ k → od→ord k ) ( sym (==→o≡ ( ⇔→== uxxx-ord ))) ⟩
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
429 od→ord (Union (x , (x , x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
430
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
431 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
432 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
433 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
434 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
435 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
436 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
437 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
438 -- ∀ 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
439 -- ∀ 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
440 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
441 field
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
442 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
443 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
444 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
445 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
446 -- 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
447 -- 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
448