annotate HOD.agda @ 140:312e27aa3cb5

remove otrans again. start over
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 07 Jul 2019 23:02:47 +0900
parents 53077af367e9
children 21b2654985c4
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
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
19
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
20 open HOD
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
21 open import Data.Unit
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
23 open Ordinal
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
24 open _∧_
44
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
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
45 ⇔→== : {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
46 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
47 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
48
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
49 -- Ordinal in HOD ( and ZFSet )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
50 Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n}
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
51 Ord {n} a = record { def = λ y → y o< a }
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
52
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
53 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
54 od∅ {n} = Ord o∅
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
55
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
56 postulate
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
57 -- 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
58 od→ord : {n : Level} → HOD {n} → Ordinal {n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
59 ord→od : {n : Level} → Ordinal {n} → HOD {n}
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
60 c<→o< : {n : Level} {x y : HOD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
61 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
62 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
63 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
64 ==→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
65 -- 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
66 -- 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
67 -- supermum as Replacement Axiom
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
68 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
69 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
70 -- contra-position of mimimulity of supermum required in Power Set Axiom
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
71 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
72 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
73 -- 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
74 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
75 -- 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
76 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
77 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) )
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
78 -- we should prove this in agda, but simply put here
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
79 ===-≡ : {n : Level} { x y : HOD {suc n}} → x == y → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
80
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
81 Ord-ord : {n : Level } {ox : Ordinal {suc n}} → Ord ox ≡ ord→od ox
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
82 Ord-ord {n} {px} = trans (sym oiso) (cong ( λ k → ord→od k ) (sym ord-Ord))
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
83
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
84 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
85 _∋_ {n} a x = def a ( od→ord x )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
86
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
87 _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
88 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
89
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
90 _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
91 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
92
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
93 cseq : {n : Level} → HOD {n} → HOD {n}
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
94 cseq x = record { def = λ y → def x (osuc y) } where
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
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
111 o<→o> : {n : Level} → { x y : Ordinal {n} } → (Ord x == Ord y) → x o< y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
112 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with o<-subst (yx (case1 lt)) ord-Ord refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
113 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
114 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
115 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with o<-subst (yx (case2 lt)) ord-Ord refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
116 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
117 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
118
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
119 Ord==→≡ : {n : Level} { x y : Ordinal {suc n}} → Ord x == Ord y → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
120 Ord==→≡ {n} {x} {y} eq with trio< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
121 Ord==→≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq a )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
122 Ord==→≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
123 Ord==→≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) c )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
124
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
125
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
126 ∅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
127 ∅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
128 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
129 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
130 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
131 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
132 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
133 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
134 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
135 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
136 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
137 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
138 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
139 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
140 ... | 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
141 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
142
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
143 ∅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
144 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
145 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
146 ∅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
147
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
148 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
149 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
150
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
151 -- avoiding lv != Zero error
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
152 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
153 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
154
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
155 ==-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
156 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
157 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
158 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
159 where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
160 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
161 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
162
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
163 =-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
164 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
165
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
166 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
167 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
168 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
169 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
170
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
171 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
172 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
173
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
174 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
175 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
176
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
177 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
178 c≤-refl x = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
179
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
180 ∋→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
181 ∋→o< {n} {a} {x} lt = t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
182 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
183 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
184
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
185 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
186 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
187 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
188 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
189 lemma lt with o<→c< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
190 lemma lt | t = o<¬≡ refl t
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
191 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
192 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
193
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
194 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
195 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
196
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
197 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
198 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
199
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
200 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n}
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
201 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
202 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
203 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
204
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
205 ∅< : {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
206 ∅< {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
207 ∅< {n} {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
208
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
209 ∅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
210 ∅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
211
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
212 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
213 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
214
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
215 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
216 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
217 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
218 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
219
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
220
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
221 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
222 -- 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
223
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
224 csuc : {n : Level} → HOD {suc n} → HOD {suc n}
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
225 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
226
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
227 -- 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
228
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
229 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
230 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
231
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
232 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n}
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
233 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
234
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
235 OrdSubset : {n : Level} → (A x : Ordinal {suc n} ) → ZFSubset (Ord A) (Ord x) ≡ Ord ( minα A x )
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
236 OrdSubset {n} A x = ===-≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
237 lemma1 : {y : Ordinal} → def (ZFSubset (Ord A) (Ord x)) y → def (Ord (minα A x)) y
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
238 lemma1 {y} s with trio< A x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
239 lemma1 {y} s | tri< a ¬b ¬c = proj1 s
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
240 lemma1 {y} s | tri≈ ¬a refl ¬c = proj1 s
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
241 lemma1 {y} s | tri> ¬a ¬b c = proj2 s
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
242 lemma2 : {y : Ordinal} → def (Ord (minα A x)) y → def (ZFSubset (Ord A) (Ord x)) y
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
243 lemma2 {y} lt with trio< A x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
244 lemma2 {y} lt | tri< a ¬b ¬c = record { proj1 = lt ; proj2 = ordtrans lt a }
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
245 lemma2 {y} lt | tri≈ ¬a refl ¬c = record { proj1 = lt ; proj2 = lt }
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
246 lemma2 {y} lt | tri> ¬a ¬b c = record { proj1 = ordtrans lt c ; proj2 = lt }
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
247
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
248 -- Constructible Set on α
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
249 -- 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
250 -- L (Φ 0) = Φ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
251 -- L (OSuc lv n) = { Def ( L n ) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
252 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
253 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
254 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
255 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
256 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
257 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
258
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
259 L00 : {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: 122
diff changeset
260 L00 {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: 122
diff changeset
261 (sup-o< {suc n} {λ x → od→ord ( ZFSubset (Ord ox) (ord→od x ))} {ox} ) (lemma0 ox) refl where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
262 lemma1 : {n : Level } {ox z : Ordinal {suc n}} → ( def (Ord ox) z ∧ def (ord→od ox) z ) ⇔ def ( Ord ox ) z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
263 lemma1 {n} {ox} {z} = record { proj1 = proj1 ; proj2 = λ t → record { proj1 = t ; proj2 = subst (λ k → def k z ) Ord-ord t }}
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
264 lemma0 : {n : Level} → (ox : Ordinal {suc n}) → od→ord (ZFSubset (Ord ox) (ord→od ox)) ≡ ox
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
265 lemma0 {n} ox = trans (cong (λ k → od→ord k) (===-≡ (⇔→== lemma1) )) (sym ord-Ord)
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
266
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
267 -- L0 : {n : Level} → (α : Ordinal {suc n}) → α o< β → L (osuc α) ∋ L α
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
268 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : HOD {suc n}) → L α ∋ x → L β ∋ x
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
269
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
270 omega : { n : Level } → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
271 omega = record { lv = Suc Zero ; ord = Φ 1 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
272
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
273 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
274 HOD→ZF {n} = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
275 ZFSet = HOD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
276 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
277 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
278 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
279 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
280 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
281 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
282 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
283 ; Replace = Replace
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
284 ; infinite = Ord omega
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
285 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
286 } where
136
3cc848664a86 ... should use Select in Replace
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 135
diff changeset
287 Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {suc n}
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
288 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
289 Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n}
139
53077af367e9 dead end?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 138
diff changeset
290 Replace X ψ = Select ( Ord (sup-o ( λ x → od→ord (ψ (ord→od x ))))) ( λ x → ¬ (∀ (y : Ordinal ) → ¬ ( def X y ∧ ( x == ψ (Ord y) ))))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
291 _,_ : 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
292 x , y = Ord (omax (od→ord x) (od→ord y))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
293 Union : HOD {suc n} → HOD {suc n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
294 Union U = cseq U
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
295 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
296 ZFSet = HOD {suc n}
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
297 _∈_ : ( 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
298 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
299 _⊆_ : ( 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
300 _⊆_ A B {x} = A ∋ x → B ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
301 _∩_ : ( A B : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
302 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) )
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
303 Power : HOD {suc n} → HOD {suc n}
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
304 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
305 -- _∪_ : ( A B : ZFSet ) → ZFSet
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
306 -- 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
307 {_} : ZFSet → ZFSet
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
308 { 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
309
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
310 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
311 -- infixr 230 _∩_ _∪_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
312 infixr 220 _⊆_
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
313 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
314 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
315 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
316 ; pair = pair
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
317 ; 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
318 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
319 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
320 ; empty = empty
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
321 ; power→ = power→
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
322 ; power← = power←
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
323 ; extensionality = extensionality
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
324 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
325 ; regularity = regularity
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
326 ; infinity∅ = infinity∅
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
327 ; infinity = λ _ → infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
328 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
329 ; replacement← = replacement←
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
330 ; replacement→ = replacement→
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
331 } where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
332
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
333 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
334 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
335 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
336
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
337 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
338 empty x (case1 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
339 empty x (case2 ())
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
340
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
341 ---
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
342 --- 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
343 --- 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
344 --
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
345 -- 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
346 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
128
69a845b82854 ... dead end?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
347 -- In case of later, ZFSubset A ∋ t and t ∋ x implies A ∋ x by transitivity
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
348 --
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
349 POrd : {a : Ordinal } {t : HOD} → Def (Ord a) ∋ t → Def (Ord a) ∋ Ord (od→ord t)
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
350 POrd {a} {t} P∋t = o<→c< P∋t
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
351 ord-power→ : (a : Ordinal ) ( t : HOD) → Def (Ord a) ∋ t → {x : HOD} → t ∋ x → Ord a ∋ x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
352 ord-power→ a t P∋t {x} t∋x with osuc-≡< (sup-lb (POrd P∋t))
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
353 ... | case1 eq = proj1 (def-subst (Ltx t∋x) (sym (subst₂ (λ j k → j ≡ k ) oiso oiso ( cong (λ k → ord→od k) (sym eq) ))) refl ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
354 Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
355 Ltx {n} {x} {t} lt = c<→o< lt
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
356 ... | case2 lt = {!!} where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
357 sp = sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x)))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
358 minsup : HOD
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
359 minsup = ZFSubset (Ord a) ( ord→od ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x)))))
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
360 Ltx : {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
361 Ltx {n} {x} {t} lt = c<→o< lt
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
362 -- lemma1 hold because minsup is Ord (minα a sp)
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
363 lemma1 : od→ord (Ord (od→ord t)) o< od→ord minsup → minsup ∋ Ord (od→ord t)
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
364 lemma1 lt with OrdSubset a ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
365 ... | eq with subst ( λ k → ZFSubset (Ord a) k ≡ Ord (minα a sp)) Ord-ord eq
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
366 ... | eq1 = def-subst {suc n} {_} {_} {minsup} {od→ord (Ord (od→ord t))} (o<→c< lt) lemma2 (sym ord-Ord) where
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
367 lemma2 : Ord (od→ord (ZFSubset (Ord a) (ord→od sp))) ≡ minsup
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
368 lemma2 = let open ≡-Reasoning in begin
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
369 Ord (od→ord (ZFSubset (Ord a) (ord→od sp)))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
370 ≡⟨ cong (λ k → Ord (od→ord k)) eq1 ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
371 Ord (od→ord (Ord (minα a sp)))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
372 ≡⟨ cong (λ k → Ord (od→ord k)) Ord-ord ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
373 Ord (od→ord (ord→od (minα a sp)))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
374 ≡⟨ cong (λ k → Ord k) diso ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
375 Ord (minα a sp)
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
376 ≡⟨ sym eq1 ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
377 minsup
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
378
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
379 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
380 -- 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
381 -- 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
382 --
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
383 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
384 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t}
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
385 lemma refl (lemma1 lemma-eq )where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
386 lemma-eq : ZFSubset (Ord a) t == t
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
387 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
388 eq← lemma-eq {z} w = record { proj2 = w ;
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
389 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
126
1114081eb51f power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
390 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
391 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : HOD {suc n}}
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
392 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
393 lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (===-≡ eq ))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
394 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
98
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
395 lemma = sup-o<
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
396
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
397 -- Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
398 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
399 power→ = {!!}
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
400 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
401 power← A t t→A = {!!} where
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
402 a = od→ord A
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
403 ψ : HOD → HOD
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
404 ψ y = Def (Ord a) ∩ y
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
405
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
406 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
407 union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
408 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
409 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
410 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
411 union→ X z u xx | tri< a ¬b ¬c | ()
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
412 union→ X z u xx | tri≈ ¬a b ¬c = def-subst {suc n} {_} {_} {X} {osuc (od→ord z)} (proj1 xx) refl b
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
413 union→ X z u xx | tri> ¬a ¬b c = {!!}
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
414 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
415 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
416 lemma : X ∋ union-u {X} {z} X∋z
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
417 lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} X∋z refl ord-Ord
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
418
138
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 137
diff changeset
419 -- ψiso : {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y → ψ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 137
diff changeset
420 -- ψiso {ψ} t refl = t
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
421 selection : {ψ : HOD → Set (suc n)} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
422 selection {X} {ψ} {y} = {!!}
139
53077af367e9 dead end?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 138
diff changeset
423 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
424 replacement← {ψ} X x lt = {!!}
139
53077af367e9 dead end?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 138
diff changeset
425 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x == ψ y))
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
426 replacement→ {ψ} X x lt = contra-position lemma {!!} where
138
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 137
diff changeset
427 lemma : ( (y : HOD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 137
diff changeset
428 lemma not y not2 = not (ord→od y) (subst₂ ( λ k j → k == j ) oiso (cong (λ k → ψ k ) Ord-ord ) (proj2 not2 ))
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
429
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
430 ∅-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
431 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
432 regularity : (x : HOD) (not : ¬ (x == od∅)) →
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
433 (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
434 proj1 (regularity x not ) = x∋minimul x not
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
435 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
436 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
437 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
438 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 139
diff changeset
439 lemma3 = {!!}
117
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
440 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
441 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
442
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
443 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
444 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
445 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
446
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
447 open import Relation.Binary.PropositionalEquality
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
448 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
449 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
450 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
451 lemma {y} = let open ≡-Reasoning in begin
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
452 def (Union (x , (x , x))) y
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
453 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
454 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
455 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
456 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
457 ≡⟨ 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
458 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
459 ≡⟨ 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
460 osuc y o< osuc (osuc (od→ord x))
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
461
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
462 infinite : HOD {suc n}
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
463 infinite = Ord omega
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
464 infinity∅ : Ord omega ∋ od∅ {suc n}
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
465 infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
466 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
467 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
468 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
469 eq = let open ≡-Reasoning in begin
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
470 osuc (od→ord x)
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
471 ≡⟨ ord-Ord ⟩
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
472 od→ord (Ord (osuc (od→ord x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
473 ≡⟨ 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
474 od→ord (Union (x , (x , x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
475
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
476 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
477 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
478 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
479 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
480 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
481 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
482 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
483 -- ∀ 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
484 -- ∀ 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
485 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
486 field
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
487 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
488 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
489 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
490 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
491 -- 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
492 -- 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
493