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

annotate ordinal-definable.agda @ 51:83b13f1f4f42

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 27 May 2019 15:00:45 +0900
parents 7cb32d22528c
children a9007b02eaa1
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
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
2 module ordinal-definable where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
4 open import zf
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
5 open import ordinal
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
9 open import Relation.Binary.PropositionalEquality
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
11 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
14
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
15 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
16 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
17
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
18 -- Ordinal Definable Set
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
19
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
20 record OD {n : Level} : Set (suc n) where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
21 field
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
22 def : (x : Ordinal {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
23
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
24 open OD
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
25 open import Data.Unit
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
26
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
27 open Ordinal
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
28
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
29 postulate
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
30 od→ord : {n : Level} → OD {n} → Ordinal {n}
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
31 ord→od : {n : Level} → Ordinal {n} → OD {n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
32
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
33 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
34 _∋_ {n} a x = def a ( od→ord x )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
35
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
36 _c<_ : { n : Level } → ( a x : OD {n} ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
37 x c< a = a ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
38
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
39 record _==_ {n : Level} ( a b : OD {n} ) : Set n where
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
40 field
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
41 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
42 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
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
44 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
45 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
46
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
47 eq-refl : {n : Level} { x : OD {n} } → x == x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
48 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
49
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
50 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
51
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
52 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
53 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
54
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
55 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
56 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
57
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
58 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n)
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
59 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
60
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
61 od∅ : {n : Level} → OD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
62 od∅ {n} = record { def = λ _ → Lift n ⊥ }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
63
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
64 postulate
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
65 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
66 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
67 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
68 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
69 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
70 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
71 ∅-base-def : {n : Level} → def ( ord→od (o∅ {n}) ) ≡ def (od∅ {n})
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
72
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
73 o∅→od∅ : {n : Level} → ord→od (o∅ {n}) ≡ od∅ {n}
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
74 o∅→od∅ {n} = cong ( λ k → record { def = k }) ( ∅-base-def )
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
75
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
76 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} )
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
77 ∅1 {n} x (lift ())
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
78
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
79 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
80 ∅3 {n} {x} = TransFinite {n} c1 c2 c3 x where
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
81 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
82 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
83 c1 : ∀ (lx : Nat ) → c0 lx (record { lv = Suc lx ; ord = ℵ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
84 c1 lx not with not ( record { lv = lx ; ord = Φ lx } )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
85 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
86 c1 lx not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
87 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
88 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
89 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
90 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
91 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
92 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
93 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
94 ... | t with t (case2 Φ< )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
95 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
96 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
97 ... | 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
98 c3 lx (OSuc .lx x₁) d not | t | ()
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
99 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } )
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
100 ... | t with t (case2 (s< ℵΦ< ))
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
101 c3 .(Suc lx) (ℵ lx) d not | t | ()
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
102
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
103 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
104 def-subst df refl refl = df
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
105
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
106 transitive : {n : Level } { x y z : OD {n} } → y ∋ x → z ∋ y → z ∋ x
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
107 transitive {n} {x} {y} {z} x∋y z∋y with ordtrans ( c<→o< {n} {x} {y} x∋y ) ( c<→o< {n} {y} {z} z∋y )
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
108 ... | t = lemma0 (lemma t) where
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
109 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x )))
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
110 lemma xo<z = o<→c< xo<z
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
111 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x)
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
112 lemma0 dz = def-subst {n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso)
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
113
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
114 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
115 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
116 mino : Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
117 min<x : mino o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
118
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
119 ominimal : {n : Level} → (x : Ordinal {n} ) → o∅ o< x → Minimumo {n} x
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
120 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case1 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
121 ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
122 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case1 ())
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
123 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case2 Φ<) = record { mino = record { lv = Zero ; ord = Φ 0 } ; min<x = case2 Φ< }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
124 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case1 (s≤s x)) = record { mino = record { lv = lv ; ord = Φ lv } ; min<x = case1 (s≤s ≤-refl)}
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
125 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case2 ())
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
126 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case1 (s≤s x)) = record { mino = record { lv = (Suc lv) ; ord = ord } ; min<x = case2 s<refl}
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
127 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case2 ())
44
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
128 ominimal {n} record { lv = (Suc lx) ; ord = (ℵ .lx) } (case1 (s≤s z≤n)) = record { mino = record { lv = Suc lx ; ord = Φ (Suc lx) } ; min<x = case2 ℵΦ< }
fcac01485f32 od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
129 ominimal {n} record { lv = (Suc lx) ; ord = (ℵ .lx) } (case2 ())
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
130
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131 ∅5 : {n : Level} → ( x : Ordinal {n} ) → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
132 ∅5 {n} record { lv = Zero ; ord = (Φ .0) } not = ⊥-elim (not refl)
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
133 ∅5 {n} record { lv = Zero ; ord = (OSuc .0 ord) } not = case2 Φ<
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
134 ∅5 {n} record { lv = (Suc lv) ; ord = ord } not = case1 (s≤s z≤n)
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
135
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
136 ∅8 : {n : Level} → ( x : Ordinal {n} ) → ¬ x o< o∅ {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
137 ∅8 {n} x (case1 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
138 ∅8 {n} x (case2 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
139
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
140 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
141 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
142
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
143 -- avoiding lv != Zero error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
144 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
145 orefl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
146
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
147 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
148 ==-iso {n} {x} {y} eq = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
149 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
150 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
151 where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
152 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
153 lemma {x} {z} d = def-subst d oiso refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
154
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
155 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
156 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
157 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
158 lemma ox ox refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
159
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
160 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
161 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
162
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
163 ∅7 : {n : Level} → { x : OD {n} } → od→ord x ≡ o∅ {n} → x == od∅ {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
164 ∅7 {n} {x} eq = record { eq→ = e1 (orefl eq) ; eq← = e2 } where
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
165 e2 : {y : Ordinal {n}} → def od∅ y → def x y
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
166 e2 {y} (lift ())
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
167 e1 : {ox y : Ordinal {n}} → ox ≡ o∅ {n} → def x y → def od∅ y
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
168 e1 {o∅} {y} refl x>y = lift ( ∅8 y (o<-subst (c<→o< {n} {ord→od y} {x} (def-subst {n} {x} {y} x>y refl (sym diso))) ord-iso eq ))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
169
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
170 =→¬< : {x : Nat } → ¬ ( x < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
171 =→¬< {Zero} ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
172 =→¬< {Suc x} (s≤s lt) = =→¬< lt
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
177 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x
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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
180 o<> : {n : Level } ( ox oy : Ordinal {n}) → ox o< oy → oy o< ox → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
181 o<> ox oy (case1 x<y) (case1 y<x) = >→¬< x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
182 o<> ox oy (case1 x<y) (case2 y<x) with d<→lv y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
183 ... | refl = =→¬< x<y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
184 o<> ox oy (case2 x<y) (case1 y<x) with d<→lv x<y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
185 ... | refl = =→¬< y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
186 o<> ox oy (case2 x<y) (case2 y<x) with d<→lv x<y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
187 ... | refl = trio<> x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
188
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
189 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
190 o<¬≡ ox ox refl (case1 lt) = =→¬< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
191 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
192
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
193 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
194 o<→¬== {n} {x} {y} (case1 lt) eq = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
195 o<→¬== {n} {x} {y} (case2 lt) eq = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
196
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
197 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
198 o<→¬c> {n} {x} {y} olt clt = o<> (od→ord x) (od→ord y) olt (c<→o< clt ) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
200 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
201 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
202
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
203 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
204 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
205 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
206 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
207 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso diso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
208
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
209 ∅2 : {n : Level} → { x : OD {n} } → o∅ {n} o< od→ord x → ¬ ( x == od∅ {n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
210 ∅2 {n} {x} lt record { eq→ = eq→ ; eq← = eq← } with ominimal (od→ord x ) lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
211 ... | min with eq→ ( def-subst (o<→c< (Minimumo.min<x min)) oiso refl )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
212 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
213
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
214
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
215 is-od∅ : {n : Level} → ( x : OD {suc n} ) → Dec ( x == od∅ {suc n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
216 is-od∅ {n} x with trio< {n} (od→ord x) (o∅ {suc n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
217 is-od∅ {n} x | tri≈ ¬a b ¬c = yes ( ∅7 (orefl b) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
218 is-od∅ {n} x | tri< (case1 ()) ¬b ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
219 is-od∅ {n} x | tri< (case2 ()) ¬b ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
220 is-od∅ {n} x | tri> ¬a ¬b c = no ( ∅2 c )
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
221
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
222 open _∧_
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
223
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
224 ∅9 : {n : Level} → (x : OD {n} ) → ¬ x == od∅ → o∅ o< od→ord x
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
225 ∅9 x not = ∅5 ( od→ord x) lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
226 lemma : ¬ od→ord x ≡ o∅
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
227 lemma eq = not ( ∅7 eq )
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
228
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
229 OD→ZF : {n : Level} → ZF {suc n} {n}
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
230 OD→ZF {n} = record {
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
231 ZFSet = OD {n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
232 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
233 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
234 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
235 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
236 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
237 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
238 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
239 ; Replace = Replace
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
240 ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
241 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
242 } where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
243 Replace : OD {n} → (OD {n} → OD {n} ) → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
244 Replace X ψ = sup-od ψ
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
245 Select : OD {n} → (OD {n} → Set n ) → OD {n}
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
246 Select X ψ = record { def = λ x → def X x → ψ ( ord→od x ) }
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
247 _,_ : OD {n} → OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
248 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
249 Union : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
250 Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
251 Power : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
252 Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) }
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
253 ZFSet = OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
254 _∈_ : ( A B : ZFSet ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
255 A ∈ B = B ∋ A
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
256 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
257 _⊆_ A B {x} = A ∋ x → B ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
258 _∩_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
259 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) )
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
260 _∪_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
261 A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) )
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
262 infixr 200 _∈_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
263 infixr 230 _∩_ _∪_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
264 infixr 220 _⊆_
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
265 isZF : IsZF (OD {n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } })
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
266 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
267 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
268 ; pair = pair
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
269 ; union→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
270 ; union← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
271 ; empty = empty
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
272 ; power→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
273 ; power← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
274 ; extentionality = {!!}
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
275 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
276 ; regularity = regularity
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
277 ; infinity∅ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
278 ; infinity = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
279 ; selection = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
280 ; replacement = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
281 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
282 open _∧_
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
283 open Minimumo
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
284 pair : (A B : OD {n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
285 proj1 (pair A B ) = case1 refl
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
286 proj2 (pair A B ) = case2 refl
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
287 empty : (x : OD {n} ) → ¬ (od∅ ∋ x)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
288 empty x ()
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
289 union→ : (X x y : OD {n} ) → (X ∋ x) → (x ∋ y) → (Union X ∋ y)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
290 union→ X x y X∋x x∋y = {!!} where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
291 lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
292 lemma {z} X∋z = {!!}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
293 minord : (x : OD {n} ) → ¬ (x == od∅ )→ Minimumo (od→ord x)
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
294 minord x not = ominimal (od→ord x) (∅9 x not)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
295 minimul : (x : OD {n} ) → ¬ (x == od∅ )→ OD {n}
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
296 minimul x not = ord→od ( mino (minord x not))
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
297 minimul<x : (x : OD {n} ) → (not : ¬ x == od∅ ) → x ∋ minimul x not
42
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
298 minimul<x x not = lemma0 (min<x (minord x not)) where
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
299 lemma0 : mino (minord x not) o< (od→ord x) → def x (od→ord (ord→od (mino (minord x not))))
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
300 lemma0 m<x = def-subst {n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl
49
7c23969befc9 fix Select
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 47
diff changeset
301 regularity : (x : OD) (not : ¬ (x == od∅)) →
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
302 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
303 proj1 ( regularity x non ) = minimul<x x non
47
264784731a67 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 46
diff changeset
304 proj2 ( regularity x non ) = reg1 where
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
305 reg4 : {y : Ordinal} → (y ≡ od→ord (ord→od (mino (ominimal (od→ord x) (∅5 (od→ord x) (λ eq → non (∅7 eq))))))) ∨ (y ≡ od→ord x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
306 reg4 {y} = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
307 reg2 : {y : Ordinal} → ( def (minimul x non) y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
308 → ( def (minimul x non) y → (minimul x non ∋ ord→od y) ∧ (x ∋ ord→od y) ) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
309 reg2 {y} or t with t or
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
310 reg2 s t | record { proj1 = proj1 ; proj2 = proj2 } = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
311 reg0 : {y : Ordinal} → def (Select (minimul x non) (λ z → (minimul x non ∋ z) ∧ (x ∋ z))) y → def od∅ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
312 reg0 {y} t with (reg2 {y}) {!!} t
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
313 reg0 {y} t | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
314 reg1 : Select (minimul x non) (λ x₁ → (minimul x non ∋ x₁) ∧ (x ∋ x₁)) == od∅
49
7c23969befc9 fix Select
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 47
diff changeset
315 reg1 = record { eq→ = reg0 ; eq← = λ () }
42
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
316