annotate HOD.agda @ 135:b60b6e8a57b0

otrans in repl
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 07 Jul 2019 00:51:12 +0900
parents b52683497a27
children 3cc848664a86
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}
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
63 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
64 oiso : {n : Level} {x : HOD {n}} → ord→od ( od→ord x ) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
65 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
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) )
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
81 -- we should prove this in agda, but simply put here
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
82 ===-≡ : {n : Level} { x y : HOD {suc n}} → x == y → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
83
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
84 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
85 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
86
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
87 _∋_ : { n : Level } → ( a x : HOD {n} ) → Set n
95
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
88 _∋_ {n} a x = def a ( od→ord x )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
89
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
90 _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
91 x c< a = a ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
92
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
93 _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
94 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
95
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
96 cseq : {n : Level} → HOD {n} → HOD {n}
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
97 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
98 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
99 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
100
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
101 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
102 def-subst df refl refl = df
f3da2c87cee0 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
103
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
104 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
105 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
106
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
107 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
108 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
109
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
110 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
111 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
112 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
113 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
114 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
115
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
116 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
117 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
118 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
119 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
120 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
121 ... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
122 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
124 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
125 Ord==→≡ {n} {x} {y} eq with trio< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
126 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
127 Ord==→≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
128 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
129
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
130
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131 ∅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
132 ∅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
133 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
134 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
135 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
136 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
137 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
138 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
139 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
140 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
141 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
142 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
143 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
144 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
145 ... | 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
146 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
147
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
148 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
149 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
150 ... | 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
151
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
152 ∅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
153 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
154 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
155 ∅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
156
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
157 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
158 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
159
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
160 -- avoiding lv != Zero error
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
161 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
162 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
163
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
164 ==-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
165 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
166 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
167 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
168 where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
169 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
170 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
171
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
172 =-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
173 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
174
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
175 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
176 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
177 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
178 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
179
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
180 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
181 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
182
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
183 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
184 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
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 c≤-refl : {n : Level} → ( x : HOD {n} ) → x c≤ x
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
187 c≤-refl x = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
188
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
189 ∋→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
190 ∋→o< {n} {a} {x} lt = t where
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
191 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
192 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
193
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
194 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
195 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
196 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
197 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
198 lemma lt with o<→c< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
199 lemma lt | t = o<¬≡ refl t
80
461690d60d07 remove ∅-base-def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 79
diff changeset
200 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
201 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
202
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
203 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
204 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
205
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
206 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
207 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
208
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
209 ∅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
210 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
211 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
212 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
213
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
214 ∅< : {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
215 ∅< {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
216 ∅< {n} {x} {y} d eq | lift ()
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
217
120
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
218 ∅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
219 ∅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
220
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
221 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
222 def-iso refl t = t
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
223
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
224 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
225 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
226 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
227 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
228
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
229
79
c07c548b2ac1 add some lemma
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
230 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 93
diff changeset
231 -- 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
232
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
233 csuc : {n : Level} → HOD {suc n} → HOD {suc n}
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
234 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
235
134
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
236 in-codomain : {n : Level} → (X : HOD {suc n} ) → ( ψ : HOD {suc n} → HOD {suc n} ) → HOD {suc n}
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
237 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (Ord y )))))
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
238 ; otrans = lemma } where
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
239 lemma : {x : Ordinal} → ¬ ((y : Ordinal) → ¬ def X y ∧ (x ≡ od→ord (ψ (Ord y)))) →
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
240 {y : Ordinal} → y o< x → ¬ ((y₁ : Ordinal) → ¬ def X y₁ ∧ (y ≡ od→ord (ψ (Ord y₁))))
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
241 lemma {x} notx {y} y<x noty = {!!}
134
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
242
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
243 -- 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
244
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
245 ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
246 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
247 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
248 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
249
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
250 Def : {n : Level} → (A : HOD {suc n}) → HOD {suc n}
121
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 120
diff changeset
251 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
252
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
253 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
254 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
255 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
256 lemma1 {y} s with trio< A x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
257 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
258 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
259 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
260 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
261 lemma2 {y} lt with trio< A x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
262 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
263 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
264 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
265
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
266 -- Constructible Set on α
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
267 -- 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
268 -- L (Φ 0) = Φ
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
269 -- L (OSuc lv n) = { Def ( L n ) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
270 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
271 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
272 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
273 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
274 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
275 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
276
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
277 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
278 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
279 (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
280 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
281 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
282 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
283 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
284
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 122
diff changeset
285 -- 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
286 -- 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
287
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
288 omega : { n : Level } → Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
289 omega = record { lv = Suc Zero ; ord = Φ 1 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
290
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
291 HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
292 HOD→ZF {n} = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
293 ZFSet = HOD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
294 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
295 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
296 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
297 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
298 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
299 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
300 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
301 ; Replace = Replace
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
302 ; infinite = Ord omega
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
303 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
304 } where
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
305 Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n}
135
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
306 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x ;
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
307 otrans = lemma } where
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
308 lemma : {x : Ordinal} → (x o< sup-o (λ x₁ → od→ord (ψ (ord→od x₁)))) ∧ def (in-codomain X ψ) x →
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
309 {y : Ordinal} → y o< x → (y o< sup-o (λ x₁ → od→ord (ψ (ord→od x₁)))) ∧ def (in-codomain X ψ) y
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
310 lemma {x} repl {y} y<x = record { proj1 = ordtrans y<x (proj1 repl)
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
311 ; proj2 = otrans (in-codomain X ψ) (proj2 repl) y<x }
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
312 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
313 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
314 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
315 {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
316 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
317 y<X : X ∋ ord→od y
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
318 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
319 _,_ : 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
320 x , y = Ord (omax (od→ord x) (od→ord y))
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
321 Union : HOD {suc n} → HOD {suc n}
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 112
diff changeset
322 Union U = cseq U
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
323 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
324 ZFSet = HOD {suc n}
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
325 _∈_ : ( 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
326 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
327 _⊆_ : ( 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
328 _⊆_ A B {x} = A ∋ x → B ∋ x
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
329 _∩_ : ( A B : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
330 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
331 Power : HOD {suc n} → HOD {suc n}
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
332 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
333 -- _∪_ : ( A B : ZFSet ) → ZFSet
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
334 -- 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
335 {_} : ZFSet → ZFSet
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 100
diff changeset
336 { 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
337
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
338 infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
339 -- infixr 230 _∩_ _∪_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
340 infixr 220 _⊆_
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
341 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
342 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
343 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
344 ; pair = pair
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
345 ; 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
346 ; union→ = union→
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
347 ; union← = union←
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
348 ; empty = empty
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
349 ; power→ = power→
76
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
350 ; power← = power←
8e8f54e7a030 extensionality done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
351 ; extensionality = extensionality
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
352 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
353 ; regularity = regularity
78
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
diff changeset
354 ; infinity∅ = infinity∅
93
d382a7902f5e replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
355 ; infinity = λ _ → infinity
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
356 ; 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
357 ; replacement← = replacement←
b60b6e8a57b0 otrans in repl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 134
diff changeset
358 ; replacement→ = replacement→
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
359 } where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
360
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
361 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
362 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
363 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
364
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
365 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
366 empty x (case1 ())
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
367 empty x (case2 ())
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
368
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
369 ---
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
370 --- 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
371 --- 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
372 --
109
dab56d357fa3 remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 103
diff changeset
373 -- 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
374 -- 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
375 -- 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
376 --
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
377 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
378 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
379 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
380 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
381 ... | 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
382 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
383 Ltx {n} {x} {t} lt = c<→o< lt
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
384 ... | case2 lt = otrans (Ord a) (proj1 (lemma1 lt) )
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
385 (c<→o< {suc n} {x} {Ord (od→ord t)} (Ltx t∋x) ) where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
386 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
387 minsup : HOD
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
388 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
389 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
390 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
391 -- lemma1 hold because minsup is Ord (minα a sp)
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 126
diff changeset
392 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
393 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
394 ... | 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
395 ... | 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
396 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
397 lemma2 = let open ≡-Reasoning in begin
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
398 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
399 ≡⟨ cong (λ k → Ord (od→ord k)) eq1 ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
400 Ord (od→ord (Ord (minα a sp)))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
401 ≡⟨ 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
402 Ord (od→ord (ord→od (minα a sp)))
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
403 ≡⟨ cong (λ k → Ord k) diso ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
404 Ord (minα a sp)
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
405 ≡⟨ sym eq1 ⟩
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
406 minsup
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
407
100
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
408 --
a402881cc341 add comment
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 99
diff changeset
409 -- 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
410 -- 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
411 --
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
412 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
413 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
414 lemma refl (lemma1 lemma-eq )where
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
415 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
416 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
417 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
418 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
126
1114081eb51f power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 125
diff changeset
419 ( 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
420 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
421 → (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
422 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
423 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
424 lemma = sup-o<
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
425
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
426 -- 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
427 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
428 power→ = {!!}
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
429 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
430 power← A t t→A = {!!} where
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
431 a = od→ord A
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
432 ψ : HOD → HOD
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
433 ψ y = Def (Ord a) ∩ y
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
434
118
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
435 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
436 union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) )
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
437 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
438 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
439 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
440 union→ X z u xx | tri< a ¬b ¬c | ()
122
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 121
diff changeset
441 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
442 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
443 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
444 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
445 lemma : X ∋ union-u {X} {z} X∋z
78fe704c3543 Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 117
diff changeset
446 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
447
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
448 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
449 ψiso {ψ} t refl = t
116
47541e86c6ac axiom of selection
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 115
diff changeset
450 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
451 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
452 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
453 ; 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
454 ; proj2 = def-subst {suc n} {_} {_} {X} {od→ord y} (proj2 s ) refl diso } } where
134
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
455 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
456 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
457 lemma : def (in-codomain X ψ) (od→ord (ψ x))
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
458 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym (trans Ord-ord oiso ))} ))
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
459 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (ψ x == y))
b52683497a27 replacement in HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 130
diff changeset
460 replacement→ {ψ} X x lt not = ⊥-elim ( not (ψ x) (ord→== refl ) )
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 128
diff changeset
461
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
462 ∅-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
463 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
464 regularity : (x : HOD) (not : ¬ (x == od∅)) →
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 114
diff changeset
465 (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
466 proj1 (regularity x not ) = x∋minimul x not
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
467 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
468 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
469 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
470 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
471 lemma3 = proj1 s x₁ (proj2 s)
a4c97390d312 minimum assuption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
diff changeset
472 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
473 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
474
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
475 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
476 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
477 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
478
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
479 open import Relation.Binary.PropositionalEquality
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
480 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
481 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
482 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
483 lemma {y} = let open ≡-Reasoning in begin
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
484 def (Union (x , (x , x))) y
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
485 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
486 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
487 ≡⟨⟩
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
488 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
489 ≡⟨ 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
490 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
491 ≡⟨ 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
492 osuc y o< osuc (osuc (od→ord x))
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
493
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
494 infinite : HOD {suc n}
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
495 infinite = Ord omega
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 109
diff changeset
496 infinity∅ : Ord omega ∋ od∅ {suc n}
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
497 infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
498 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
499 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
500 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
501 eq = let open ≡-Reasoning in begin
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
502 osuc (od→ord x)
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
503 ≡⟨ ord-Ord ⟩
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
504 od→ord (Ord (osuc (od→ord x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
505 ≡⟨ 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
506 od→ord (Union (x , (x , x)))
ac214eab1c3c inifinite done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 119
diff changeset
507
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 90
diff changeset
508 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
509 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
510 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
511 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
512 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
513 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
514 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
515 -- ∀ 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
516 -- ∀ 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
517 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
518 field
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
519 u : {x : HOD {suc n}} ( x∈z : x ∈ z ) → HOD {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
520 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
521 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
522 -- choice : {x : HOD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
523 -- 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
524 -- 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
525