Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 80:461690d60d07
remove ∅-base-def
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 03 Jun 2019 12:29:33 +0900 |
| parents | c07c548b2ac1 |
| children | 96c932d0145d |
| rev | line source |
|---|---|
| 16 | 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 | 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 | 6 |
| 23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 3 | 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 | 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 | 12 open import Data.Empty |
| 13 open import Relation.Nullary | |
| 14 | |
| 15 open import Relation.Binary | |
| 16 open import Relation.Binary.Core | |
| 17 | |
| 27 | 18 -- Ordinal Definable Set |
| 11 | 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 | 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 | 61 od∅ : {n : Level} → OD {n} |
| 62 od∅ {n} = record { def = λ _ → Lift n ⊥ } | |
| 63 | |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
64 postulate |
| 36 | 65 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y |
| 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 |
| 46 | 71 |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
72 ∅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
|
73 ∅1 {n} x (lift ()) |
| 28 | 74 |
| 37 | 75 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 76 ∅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
|
77 c0 : Nat → Ordinal {n} → Set n |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
78 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
|
79 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
|
80 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
|
81 ... | t with t (case1 ≤-refl ) |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
82 c1 lx not | t | () |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
83 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
|
84 c2 Zero not = refl |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
85 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
|
86 ... | t with t (case1 ≤-refl ) |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
87 c2 (Suc lx) not | t | () |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
88 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
|
89 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
|
90 ... | t with t (case2 Φ< ) |
|
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 | t | () |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
92 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 93 ... | 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
|
94 c3 lx (OSuc .lx x₁) d not | t | () |
| 34 | 95 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } ) |
| 41 | 96 ... | t with t (case2 (s< ℵΦ< )) |
| 34 | 97 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
|
98 |
| 36 | 99 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 |
| 100 def-subst df refl refl = df | |
| 101 | |
| 69 | 102 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
| 103 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) | |
| 36 | 104 ... | t = lemma0 (lemma t) where |
| 105 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) | |
| 106 lemma xo<z = o<→c< xo<z | |
| 107 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x) | |
| 69 | 108 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso) |
| 36 | 109 |
| 41 | 110 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where |
| 111 field | |
| 112 mino : Ordinal {n} | |
| 113 min<x : mino o< x | |
| 114 | |
| 57 | 115 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 116 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 117 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 118 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 119 |
| 46 | 120 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 } |
| 121 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
|
122 |
| 51 | 123 -- avoiding lv != Zero error |
| 124 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
| 125 orefl refl = refl | |
| 126 | |
| 127 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
| 128 ==-iso {n} {x} {y} eq = record { | |
| 129 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 130 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 131 where | |
| 132 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
| 133 lemma {x} {z} d = def-subst d oiso refl | |
| 134 | |
| 57 | 135 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 136 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
| 137 | |
| 51 | 138 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
| 139 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
| 140 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 141 lemma ox ox refl = eq-refl | |
| 142 | |
| 143 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 144 o≡→== {n} {x} {.x} refl = eq-refl | |
| 145 | |
| 146 =→¬< : {x : Nat } → ¬ ( x < x ) | |
| 147 =→¬< {Zero} () | |
| 148 =→¬< {Suc x} (s≤s lt) = =→¬< lt | |
| 149 | |
| 150 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 151 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 152 | |
| 153 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
| 154 c≤-refl x = case1 refl | |
| 155 | |
| 156 o<> : {n : Level } ( ox oy : Ordinal {n}) → ox o< oy → oy o< ox → ⊥ | |
| 157 o<> ox oy (case1 x<y) (case1 y<x) = >→¬< x<y y<x | |
| 158 o<> ox oy (case1 x<y) (case2 y<x) with d<→lv y<x | |
| 159 ... | refl = =→¬< x<y | |
| 160 o<> ox oy (case2 x<y) (case1 y<x) with d<→lv x<y | |
| 161 ... | refl = =→¬< y<x | |
| 162 o<> ox oy (case2 x<y) (case2 y<x) with d<→lv x<y | |
| 163 ... | refl = trio<> x<y y<x | |
| 164 | |
| 54 | 165 o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ |
| 51 | 166 o<¬≡ ox ox refl (case1 lt) = =→¬< lt |
| 167 o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt | |
| 168 | |
| 54 | 169 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
| 52 | 170 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
| 171 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso ) | |
| 172 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
| 173 ... | () | |
| 174 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
| 175 yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso ) | |
| 176 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) | |
| 177 ... | () | |
| 178 | |
| 79 | 179 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y |
| 180 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
| 181 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
| 182 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
| 183 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
| 184 | |
| 80 | 185 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) |
| 186 ¬x<0 {n} {x} (case1 ()) | |
| 187 ¬x<0 {n} {x} (case2 ()) | |
| 188 | |
| 189 -- o∅≡od∅0 : {n : Level} → ord→od (o∅ {suc n}) == od∅ {suc n} | |
| 190 -- eq→ (o∅≡od∅0 {n} ) {x} y with c<→o< {suc n} {ord→od x} {ord→od (o∅ {suc n})} (def-subst {suc n} {_} {_} {ord→od o∅} {od→ord (ord→od x)} y refl (sym diso) ) | |
| 191 -- eq→ (o∅≡od∅0 {n}) {x} y | lt = ⊥-elim ( ¬x<0 (o<-subst lt ord-iso diso ) ) | |
| 192 -- eq← (o∅≡od∅0 {n}) {x} (lift ()) | |
| 193 -- | |
| 194 -- o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} | |
| 195 -- o∅≡od∅ {n} = trans (cong (λ k → ord→od k ) ( ==→o≡ {n} (eq-trans o∅≡od∅0 (subst (λ k → od∅ == k ) (sym oiso) eq-refl )) ) ) oiso | |
| 196 | |
| 197 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} | |
| 198 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
| 199 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
| 200 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
| 201 lemma lt with def-subst (o<→c< lt) oiso refl | |
| 202 lemma lt | lift () | |
| 203 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso | |
| 204 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 205 | |
| 51 | 206 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
| 52 | 207 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
| 51 | 208 |
| 209 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) | |
| 210 o<→¬c> {n} {x} {y} olt clt = o<> (od→ord x) (od→ord y) olt (c<→o< clt ) where | |
| 211 | |
| 212 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
| 213 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) | |
| 214 | |
| 215 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) | |
| 216 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
| 217 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a ) | |
| 218 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) | |
| 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) | |
| 220 | |
| 54 | 221 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ |
| 222 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
| 223 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
| 224 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
| 225 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
| 226 | |
| 60 | 227 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
| 228 ∅< {n} {x} {y} d eq with eq→ eq d | |
| 229 ∅< {n} {x} {y} d eq | lift () | |
| 57 | 230 |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
231 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
232 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x |
| 51 | 233 |
| 76 | 234 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
| 235 def-iso refl t = t | |
| 236 | |
| 53 | 237 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
| 238 is-∋ {n} x y with tri-c< x y | |
| 239 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
| 240 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
| 241 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
| 242 | |
| 57 | 243 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 244 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 245 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 246 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 247 | |
|
45
33860eb44e47
od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
44
diff
changeset
|
248 open _∧_ |
|
33860eb44e47
od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
44
diff
changeset
|
249 |
| 66 | 250 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
| 251 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
| 252 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
| 253 lemma ox ne with is-o∅ ox | |
| 254 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
| 255 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
| 80 | 256 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ |
| 66 | 257 lemma o∅ ne | yes refl | () |
| 80 | 258 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (∅5 ¬p)) |
| 69 | 259 |
| 79 | 260 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
|
59
d13d1351a1fa
lemma = cong₂ (λ x not → minimul x not ) oiso { }6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
58
diff
changeset
|
261 |
| 54 | 262 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 40 | 263 OD→ZF {n} = record { |
| 54 | 264 ZFSet = OD {suc n} |
|
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
265 ; _∋_ = _∋_ |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
266 ; _≈_ = _==_ |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
267 ; ∅ = od∅ |
| 28 | 268 ; _,_ = _,_ |
| 269 ; Union = Union | |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
270 ; Power = Power |
| 28 | 271 ; Select = Select |
| 272 ; Replace = Replace | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
273 ; infinite = ord→od ( 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
|
274 ; isZF = isZF |
| 28 | 275 } where |
| 54 | 276 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
|
277 Replace X ψ = sup-od ψ |
| 54 | 278 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
| 279 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
| 280 _,_ : 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
|
281 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } |
| 54 | 282 Union : OD {suc n} → OD {suc n} |
|
71
d088eb66564e
add osuc ( next larger element of Ordinal )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
70
diff
changeset
|
283 Union U = record { def = λ y → osuc y o< (od→ord U) } |
| 77 | 284 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
| 54 | 285 Power : OD {suc n} → OD {suc n} |
| 77 | 286 Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
| 54 | 287 ZFSet = OD {suc n} |
| 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 | 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 _⊆_ |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
299 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( 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 |
| 72 | 303 ; union-u = union-u |
| 304 ; union→ = union→ | |
| 305 ; union← = union← | |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
306 ; empty = empty |
| 76 | 307 ; power→ = power→ |
| 308 ; power← = power← | |
| 309 ; extensionality = extensionality | |
|
30
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
310 ; minimul = minimul |
| 51 | 311 ; regularity = regularity |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
312 ; infinity∅ = infinity∅ |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
313 ; infinity = {!!} |
|
55
9c0a5e28a572
regurality elimination case
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
|
314 ; 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
|
315 ; replacement = {!!} |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
316 } where |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
317 open _∧_ |
| 41 | 318 open Minimumo |
| 54 | 319 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
|
320 proj1 (pair A B ) = case1 refl |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
321 proj2 (pair A B ) = case2 refl |
| 54 | 322 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
|
323 empty x () |
| 77 | 324 --- Power X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → ( def X x ∧ def (ord→od y) x ) } |
| 76 | 325 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
| 77 | 326 power→ A t P∋t {x} t∋x = proj1 (P∋t (od→ord x) ) |
| 327 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
| 328 power← A t t→A z = record { proj1 = lemma2 ; proj2 = lemma1 } where | |
| 329 lemma1 : def (ord→od (od→ord t)) z | |
| 76 | 330 lemma1 = {!!} |
| 77 | 331 lemma2 : def A z |
| 76 | 332 lemma2 = {!!} |
| 70 | 333 union-u : (X z : OD {suc n}) → Union X ∋ z → OD {suc n} |
| 72 | 334 union-u X z U>z = ord→od ( osuc ( od→ord z ) ) |
| 335 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → union-u X z U>z ∋ z | |
| 336 union-lemma-u {X} {z} U>z = lemma <-osuc where | |
| 337 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
| 338 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl diso | |
| 73 | 339 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
| 72 | 340 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
|
74
819da8c08f05
ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
73
diff
changeset
|
341 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
|
819da8c08f05
ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
73
diff
changeset
|
342 union→ X y u xx | tri< a ¬b ¬c | () |
| 73 | 343 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
| 344 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX | |
| 345 lemma refl lt = lt | |
| 346 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) | |
| 72 | 347 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z ) |
| 73 | 348 union← X z X∋z = record { proj1 = def-subst {suc n} (o<→c< X∋z) oiso refl ; proj2 = union-lemma-u X∋z } |
| 54 | 349 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
| 350 ψiso {ψ} t refl = t | |
| 351 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
| 352 selection {ψ} {X} {y} = record { | |
| 353 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 354 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 355 } | |
| 60 | 356 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
| 357 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
| 54 | 358 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
| 68 | 359 minimul x not = od∅ |
| 57 | 360 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 361 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
| 66 | 362 proj1 (regularity x not ) = ¬∅=→∅∈ not |
| 363 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where | |
| 364 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
| 365 reg {y} t with proj1 t | |
| 366 ... | x∈∅ = x∈∅ | |
| 76 | 367 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 368 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 369 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
370 next : (x : OD) → Union (x , (x , x)) == ord→od (osuc (od→ord x)) |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
371 eq→ (next x ) {y} z = {!!} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
372 eq← (next x ) {y} z = {!!} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
373 infinite : OD {suc n} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
374 infinite = ord→od ( record { lv = Suc Zero ; ord = ℵ Zero } ) |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
375 infinity∅ : ord→od ( record { lv = Suc Zero ; ord = ℵ Zero } ) ∋ od∅ {suc n} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
376 infinity∅ = def-subst {suc n} {_} {od→ord (ord→od o∅)} {infinite} {od→ord od∅} |
| 80 | 377 (o<→c< ( case1 (s≤s z≤n ))) refl (cong (λ k → od→ord k) o∅≡od∅ ) |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
378 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
379 infinity x ∞∋x = {!!} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
380 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
381 replacement {ψ} X x = {!!} |
|
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
382 |