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

annotate ordinal-definable.agda @ 58:323b561210b5

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 28 May 2019 23:02:50 +0900
parents 419688a279e0
children d13d1351a1fa
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
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
131 ∅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
132 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
133 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
134 ∅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
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
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
155 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
156 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
157
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
158 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
159 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
160 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
161 lemma ox ox 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 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
164 o≡→== {n} {x} {.x} refl = eq-refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
165
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
166 ∅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
167 ∅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
168 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
169 e2 {y} (lift ())
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 45
diff changeset
170 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
171 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
172
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
173 =→¬< : {x : Nat } → ¬ ( x < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
174 =→¬< {Zero} ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
175 =→¬< {Suc x} (s≤s lt) = =→¬< lt
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 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
178 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
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 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
181 c≤-refl x = case1 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 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
184 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
185 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
186 ... | refl = =→¬< x<y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
187 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
188 ... | refl = =→¬< y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
189 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
190 ... | refl = trio<> x<y y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
191
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
192 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
193 o<¬≡ ox ox refl (case1 lt) = =→¬< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
194 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
195
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
196 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥
52
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
197 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
198 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
199 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
200 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
201 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
202 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
203 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
204 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
205
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
206 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y )
52
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 51
diff changeset
207 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
51
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 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
210 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
211
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
212 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
213 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
214
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
215 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
216 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
217 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
218 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
219 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
220
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
221 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
222 c<> {n} {x} {y} x<y y<x with tri-c< x y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
223 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
224 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y )
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
225 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
226
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
227 ∅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
228 ∅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
229 ... | 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
230 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
231
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
232 ∅0 : {n : Level} → { x : Ordinal {n} } → o∅ {n} o< x → ¬ ( ord→od x == od∅ {n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
233 ∅0 {n} {x} lt record { eq→ = eq→ ; eq← = eq← } with ominimal x lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
234 ... | min with eq→ (o<→c< (Minimumo.min<x min))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
235 ... | ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
236
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
237
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
238 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
239 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
240 is-od∅ {n} x | tri≈ ¬a b ¬c = yes ( ∅7 (orefl b) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
241 is-od∅ {n} x | tri< (case1 ()) ¬b ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
242 is-od∅ {n} x | tri< (case2 ()) ¬b ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
243 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
244
53
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
245 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
246 is-∋ {n} x y with tri-c< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
247 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
248 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
249 is-∋ {n} x y | tri> ¬a ¬b c = yes c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 52
diff changeset
250
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
251 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
252 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
253 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
254 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
255
45
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
256 open _∧_
33860eb44e47 od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 44
diff changeset
257
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
258 ∅9 : {n : Level} → {x : OD {n} } → ¬ x == od∅ → o∅ o< od→ord x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
259 ∅9 {_} {x} not = ∅5 lemma where
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 37
diff changeset
260 lemma : ¬ od→ord x ≡ o∅
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
261 lemma eq = not ( ∅7 eq )
37
f10ceee99d00 ¬ ( y c< x ) → x ≡ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
262
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
263 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
40
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
264 OD→ZF {n} = record {
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
265 ZFSet = OD {suc n}
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
266 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
267 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
268 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
269 ; _,_ = _,_
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
270 ; Union = Union
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
271 ; Power = Power
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
272 ; Select = Select
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
273 ; Replace = Replace
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
274 ; 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
275 ; isZF = isZF
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
276 } where
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
277 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
278 Replace X ψ = sup-od ψ
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
279 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
280 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
281 _,_ : OD {suc n} → OD {suc n} → OD {suc n}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
282 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) }
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
283 Union : OD {suc n} → OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
284 Union x = record { def = λ y → {z : Ordinal {suc n}} → def x z → def (ord→od z) y }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
285 Power : OD {suc n} → OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
286 Power x = record { def = λ y → (z : Ordinal {suc n} ) → ( def x y ∧ def (ord→od z) y ) }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
287 ZFSet = OD {suc n}
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
288 _∈_ : ( 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
289 A ∈ B = B ∋ A
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
290 _⊆_ : ( 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
291 _⊆_ 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
292 _∩_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
293 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
294 _∪_ : ( A B : ZFSet ) → ZFSet
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
295 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
296 infixr 200 _∈_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
297 infixr 230 _∩_ _∪_
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
298 infixr 220 _⊆_
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
299 isZF : IsZF (OD {suc 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
300 isZF = record {
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
301 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
302 ; pair = pair
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
303 ; union→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
304 ; union← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
305 ; empty = empty
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
306 ; power→ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
307 ; power← = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
308 ; extentionality = {!!}
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
309 ; minimul = minimul
51
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 50
diff changeset
310 ; regularity = regularity
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
311 ; infinity∅ = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
312 ; infinity = {!!}
55
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
313 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
314 ; replacement = {!!}
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
315 } where
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
316 open _∧_
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
317 open Minimumo
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
318 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
319 proj1 (pair A B ) = case1 refl
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
320 proj2 (pair A B ) = case2 refl
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
321 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
322 empty x ()
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
323 union→ : (X x y : OD {suc n} ) → (X ∋ x) → (x ∋ y) → (Union X ∋ y)
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
324 union→ X x y X∋x x∋y = {!!} where
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
325 lemma : {z : Ordinal {suc n} } → def X z → z ≡ od→ord y
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
326 lemma {z} X∋z = {!!}
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
327 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
328 ψiso {ψ} t refl = t
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
329 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
330 selection {ψ} {X} {y} = record {
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
331 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
332 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
333 }
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
334 minord : (x : OD {suc n} ) → ¬ (x == od∅ )→ Minimumo (od→ord x)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
335 minord x not = ominimal (od→ord x) (∅9 not)
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
336 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
41
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
337 minimul x not = ord→od ( mino (minord x not))
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
338 minimul<x : (x : OD {suc n} ) → (not : ¬ x == od∅ ) → x ∋ minimul x not
42
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
339 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
340 lemma0 : mino (minord x not) o< (od→ord x) → def x (od→ord (ord→od (mino (minord x not))))
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
341 lemma0 m<x = def-subst {suc n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
342 regularity-ord : (x : Ordinal ) (not : ¬ (ord→od x == od∅)) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
343 (ord→od x ∋ minimul (ord→od x) not) ∧ (Select (minimul (ord→od x) not) (λ x₁ → (minimul (ord→od x) not ∋ x₁) ∧ ((ord→od x) ∋ x₁)) == od∅)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
344 proj1 ( regularity-ord x not ) = minimul<x (ord→od x) not
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
345 proj2 ( regularity-ord x not ) = reg1 where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
346 reg2 : {y : Ordinal} → ( def (minimul (ord→od x) not) y ∧ (minimul (ord→od x) not ∋ ord→od y) ∧ ((ord→od x) ∋ ord→od y) ) → ⊥
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
347 reg2 {y} t with proj1 t | proj1 (proj2 t) | proj2 (proj2 t)
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
348 ... | p1 | p2 | p3 with is-∋ (ord→od x) ( ord→od y)
56
aad8cdce8845 almost ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
349 reg2 {y} t | p1 | p2 | p3 | no ¬p = ⊥-elim (¬p p3 ) -- ¬ x ∋ ord→od y empty x case
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
350 reg2 {y} t | p1 | p2 | p3 | yes p with is-∋ (minimul (ord→od x) not) (ord→od y)
56
aad8cdce8845 almost ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
351 reg2 {y} t | p1 | p2 | p3 | yes p | no ¬p = ⊥-elim (¬p p2 ) -- minimum contains nothing q.e.d.
aad8cdce8845 almost ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
352 reg2 {y} t | p1 | p2 | p3 | yes p | yes p₁ = {!!}
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
353 reg0 : {y : Ordinal {suc n}} → def (Select (minimul (ord→od x) not) (λ z → (minimul (ord→od x) not ∋ z) ∧ ((ord→od x) ∋ z))) y → def od∅ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
354 reg0 {y} t with trio< y (mino (minord (ord→od x) not))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
355 reg0 {y} t | tri< a ¬b ¬c with reg2 {y} t
54
33fb8228ace9 fix selection axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 53
diff changeset
356 ... | ()
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
357 reg0 {y} t | tri≈ ¬a refl ¬c = lemma y ( mino (minord (ord→od x) not) ) refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
358 (def-subst {suc n} {ord→od y} {mino (minord (ord→od x) not)} (proj1 t) refl (sym diso))
55
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
359 where
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
360 lemma : ( ox oy : Ordinal {suc n} ) → ox ≡ oy → ord→od ox c< ord→od oy → Lift (suc n) ⊥
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
361 lemma ox oy refl lt = lift ( o≡→¬c< {suc n} {ord→od oy} {ord→od oy} refl lt )
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
362 reg0 {y} t | tri> ¬a ¬b c with o<> y (mino (minord (ord→od x) not)) (lemma {!!}) c where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
363 lemma : def (ord→od (mino (minord (ord→od x) not))) y → y o< mino (minord (ord→od x) not)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
364 lemma d with c<→o< {suc n} {ord→od y} {ord→od (mino (minord (ord→od x) not))}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
365 (def-subst {suc n} {ord→od (mino (minord (ord→od x) not))} {y} {!!} refl (sym diso))
55
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
366 lemma d | clt = o<-subst clt ord-iso ord-iso
9c0a5e28a572 regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
367 ... | ()
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
368 reg1 : Select (minimul (ord→od x) not) (λ x₁ → (minimul (ord→od x) not ∋ x₁) ∧ ((ord→od x) ∋ x₁)) == od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
369 reg1 = record { eq→ = reg0 ; eq← = λ () } where
57
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
370 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
371 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
372 regularity : (x : OD) (not : ¬ (x == od∅)) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
373 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
374 proj1 (regularity x not ) = minimul<x x not
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
375 proj2 (regularity x not ) = record { eq→ = reg4 ; eq← = λ () } where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
376 reg4 : {xd : Ordinal } → def (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁))) xd → def od∅ xd
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
377 reg4 {xd} = {!!} (eq→ (proj1 (regularity-ord {!!} {!!} )) )
42
4d5fc6381546 regurality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
378
58
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
379
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
380
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
381
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
382