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annotate ordinal-definable.agda @ 43:0d9b9db14361

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equalitu and internal parametorisity
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 24 May 2019 22:22:16 +0900
parents 4d5fc6381546
children fcac01485f32
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
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
27 postulate
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
28 od→ord : {n : Level} → OD {n} → Ordinal {n}
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
29 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
30
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
31 _∋_ : { 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
32 _∋_ {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
33
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
34 _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
35 x c< a = a ∋ x
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
36
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
37 -- _=='_ : {n : Level} → Set (suc n) -- Rel (OD {n}) (suc n)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38 -- _=='_ {n} = ( a b : OD {n} ) → ( ∀ { x : OD {n} } → a ∋ x → b ∋ x ) ∧ ( ∀ { x : OD {n} } → a ∋ x → b ∋ x )
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
39
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
40 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
41 field
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
42 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
43 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
44
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
45 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
46 id x = x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
47
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
48 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
49 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
50
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
51 open _==_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
52
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
53 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
54 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
55
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
56 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
57 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
58
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
59 _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
60 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
61
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
62 od∅ : {n : Level} → OD {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
63 od∅ {n} = record { def = λ _ → Lift n ⊥ }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
64
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
65 postulate
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
66 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
67 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
68 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
69 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
70 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
71 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
72 ∅-base-def : {n : Level} → def ( ord→od (o∅ {n}) ) ≡ def (od∅ {n})
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
73
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
74 ∅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
75 ∅1 {n} x (lift ())
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
76
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
77 ∅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
78 ∅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
79 c0 : Nat → Ordinal {n} → Set n
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
80 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
81 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
82 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
83 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
84 c1 lx not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
85 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
86 c2 Zero not = refl
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
87 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
88 ... | t with t (case1 ≤-refl )
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
89 c2 (Suc lx) not | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
90 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
91 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
92 ... | t with t (case2 Φ< )
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 | t | ()
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
94 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
95 ... | 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
96 c3 lx (OSuc .lx x₁) d not | t | ()
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
97 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
98 ... | t with t (case2 (s< ℵΦ< ))
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
99 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
100
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
101 -- find : {n : Level} → ( x : Ordinal {n} ) → o∅ o< x → Ordinal {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
102 -- exists : {n : Level} → ( x : Ordinal {n} ) → (0<x : o∅ o< x ) → find x 0<x o< x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
103
36
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
104 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
105 def-subst df refl refl = df
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
106
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
107 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
108 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
109 ... | t = lemma0 (lemma t) where
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
110 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
111 lemma xo<z = o<→c< xo<z
4d64509067d0 transitive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
112 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
113 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
114
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
115 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
116 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
117 mino : Ordinal {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
118 min<x : mino o< x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
119
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
120 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
121 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
122 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
123 ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case1 ())
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
124 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
125 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
126 ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case2 ())
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
127 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
128 ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case2 ())
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
129 ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case1 (s≤s z≤n)) = record { mino = record { lv = Suc lv ; ord = Φ (Suc lv) } ; min<x = case2 ℵΦ< }
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
130 ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case2 ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
132 ∅4 : {n : Level} → ( x : OD {n} ) → x ≡ od∅ {n} → od→ord x ≡ o∅ {n}
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
133 ∅4 {n} x refl = ∅3 lemma1 where
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
134 lemma0 : (y : Ordinal {n}) → def ( od∅ {n} ) y → ⊥
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
135 lemma0 y (lift ())
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
136 lemma1 : (y : Ordinal {n}) → y o< od→ord od∅ → ⊥
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
137 lemma1 y y<o = lemma0 y ( def-subst {n} {ord→od (od→ord od∅ )} {od→ord (ord→od y)} (o<→c< y<o) oiso diso )
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
138
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
139 ∅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
140 ∅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
141 ∅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
142 ∅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
143
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
144 postulate extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
145
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
146 ∅6 : {n : Level } ( x : Ordinal {suc n}) → o∅ o< x → ¬ x ≡ o∅
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
147 ∅6 {n} x lt eq with trio< {n} (o∅ {suc n}) x
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
148 ∅6 {n} x lt refl | tri< a ¬b ¬c = ¬b refl
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
149 ∅6 {n} x lt refl | tri≈ ¬a b ¬c = ¬a lt
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
150 ∅6 {n} x lt refl | tri> ¬a ¬b c = ¬b refl
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
151
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
152 ∅8 : {n : Level} → ( x : Ordinal {n} ) → ¬ x o< o∅ {n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
153 ∅8 {n} x (case1 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
154 ∅8 {n} x (case2 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
155
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
156 -- ∅10 : {n : Level} → (x : OD {n} ) → ¬ ( ( y : OD {n} ) → Lift (suc n) ( x ∋ y)) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
157 -- ∅10 {n} x not = ?
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
158
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
159 open Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
160
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
161 -- ∋-subst : {n : Level} {X Y x y : OD {suc n} } → X ≡ x → Y ≡ y → X ∋ Y → x ∋ y
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
162 -- ∋-subst refl refl x = x
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
163
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
164 -- ∅77 : {n : Level} → (x : OD {suc n} ) → ¬ ( ord→od (o∅ {suc n}) ∋ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
165 -- ∅77 {n} x lt = {!!} where
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
166
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
167 ∅7' : {n : Level} → ord→od (o∅ {n}) ≡ od∅ {n}
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
168 ∅7' {n} = cong ( λ k → record { def = k }) ( ∅-base-def ) where
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
169
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
170 ∅7 : {n : Level} → ( x : OD {n} ) → od→ord x ≡ o∅ {n} → x == od∅ {n}
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
171 ∅7 {n} x eq = record { eq→ = e1 ; eq← = e2 } where
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
172 e0 : {y : Ordinal {n}} → y o< o∅ {n} → def od∅ y
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
173 e0 {y} (case1 ())
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
174 e0 {y} (case2 ())
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
175 e1 : {y : Ordinal {n}} → def x y → def od∅ y
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
176 e1 {y} y<x = e0 ( o<-subst ( c<→o< {n} {x} y<x ) refl {!!} )
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
177 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
178 e2 {y} (lift ())
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
179
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
180 ∅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
181 ∅9 x not = ∅5 ( od→ord x) lemma where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
182 lemma : ¬ od→ord x ≡ o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
183 lemma eq = not ( ∅7 x eq )
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
184
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
185 OD→ZF : {n : Level} → ZF {suc n} {n}
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
186 OD→ZF {n} = record {
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
187 ZFSet = OD {n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
188 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
189 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
190 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
191 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
192 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
193 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
194 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
195 ; Replace = Replace
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
196 ; 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
197 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
198 } where
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
199 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
200 Replace X ψ = sup-od ψ
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
201 Select : OD {n} → (OD {n} → Set n ) → OD {n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
202 Select X ψ = record { def = λ x → select ( ord→od x ) } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
203 select : OD {n} → Set n
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
204 select x = ψ x
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
205 _,_ : 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
206 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
207 Union : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
208 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
209 Power : OD {n} → OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
210 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
211 ZFSet = OD {n}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
212 _∈_ : ( A B : ZFSet ) → Set n
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
213 A ∈ B = B ∋ A
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
214 _⊆_ : ( 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
215 _⊆_ 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
216 _∩_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
217 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
218 _∪_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
219 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
220 infixr 200 _∈_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
221 infixr 230 _∩_ _∪_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
222 infixr 220 _⊆_
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
223 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
224 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
225 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
226 ; pair = pair
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
227 ; union→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
228 ; union← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
229 ; empty = empty
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
230 ; power→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
231 ; power← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
232 ; extentionality = {!!}
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
233 ; minimul = minimul
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
234 ; regularity = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
235 ; infinity∅ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
236 ; infinity = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
237 ; selection = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
238 ; replacement = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
239 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
240 open _∧_
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
241 open Minimumo
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
242 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
243 proj1 (pair A B ) = case1 refl
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
244 proj2 (pair A B ) = case2 refl
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
245 empty : (x : OD {n} ) → ¬ (od∅ ∋ x)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
246 empty x ()
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
247 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
248 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
249 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
250 lemma {z} X∋z = {!!}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
251 minord : (x : OD {n} ) → ¬ (x == od∅ )→ Minimumo (od→ord x)
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
252 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
253 minimul : (x : OD {n} ) → ¬ (x == od∅ )→ OD {n}
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
254 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
255 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
256 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
257 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
258 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
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
259 regularity : (x : OD) (not : ¬ (x == od∅)) → (x ∋ minimul x not) ∧
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
260 (Select (minimul x not , x) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
261 -- regularity : (x : OD) → (not : ¬ x == od∅ ) →
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
262 -- ((x ∋ minimul x not ) ∧ {!!} ) -- (Select x (λ x₁ → (( minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)))
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
263 proj1 ( regularity x non ) = minimul<x x non
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
264 proj2 ( regularity x non ) = {!!} where -- cong ( λ k → record { def = k } ) ( extensionality ( λ y → lemma0 y) ) where
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
265 not-min : ( z : Ordinal {n} ) → ¬ ( def (Select x (λ y → (minimul x non ∋ y) ∧ (x ∋ y))) z )
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
266 not-min z = {!!}
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
267 lemma0 : ( z : Ordinal {n} ) → def (Select x (λ y → (minimul x non ∋ y) ∧ (x ∋ y))) z ≡ Lift n ⊥
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
268 lemma0 z = {!!}
42
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
269