Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 129:2a5519dcc167
ord power set
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 02 Jul 2019 09:28:26 +0900 |
| parents | 0c2cbf37e002 |
| children | 3849614bef18 |
| rev | line source |
|---|---|
| 112 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 16 | 3 open import Level |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
4 module ordinal-definable where |
| 3 | 5 |
|
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
6 open import zf |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
7 open import ordinal |
| 3 | 8 |
| 23 | 9 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
|
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
10 open import Relation.Binary.PropositionalEquality |
|
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 open import Relation.Binary | |
| 15 open import Relation.Binary.Core | |
| 16 | |
| 27 | 17 -- Ordinal Definable Set |
| 11 | 18 |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
19 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
|
20 field |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
21 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
|
22 |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
23 open OD |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
24 open import Data.Unit |
|
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
25 |
|
44
fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
diff
changeset
|
26 open Ordinal |
|
fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
diff
changeset
|
27 |
| 112 | 28 -- Ordinal in OD ( and ZFSet ) |
| 29 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} | |
| 30 Ord {n} a = record { def = λ y → y o< a } | |
| 31 | |
| 32 -- od∅ : {n : Level} → OD {n} | |
| 33 -- od∅ {n} = record { def = λ _ → Lift n ⊥ } | |
| 34 od∅ : {n : Level} → OD {n} | |
| 35 od∅ {n} = Ord o∅ | |
| 36 | |
|
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
37 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
|
38 field |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
39 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
|
40 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
|
41 |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
42 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
|
43 id x = 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 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
|
46 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
|
47 |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
48 open _==_ |
|
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 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
|
51 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
|
52 |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
53 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
|
54 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
|
55 |
| 112 | 56 ord→od : {n : Level} → Ordinal {n} → OD {n} |
| 57 ord→od a = Ord a | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
58 |
| 112 | 59 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
| 60 o<→c< {n} {x} {y} lt = lt | |
| 40 | 61 |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
62 postulate |
| 100 | 63 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 95 | 64 od→ord : {n : Level} → OD {n} → Ordinal {n} |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
65 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
| 95 | 66 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x |
| 67 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
| 100 | 68 -- supermum as Replacement Axiom |
| 95 | 69 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 70 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 112 | 71 -- a property of supermum required in Power Set Axiom |
| 98 | 72 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 73 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
74 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) |
| 95 | 75 |
| 76 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n | |
| 77 _∋_ {n} a x = def a ( od→ord x ) | |
| 78 | |
| 79 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
80 x c< a = a ∋ x |
| 103 | 81 |
| 95 | 82 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
| 83 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) | |
| 84 | |
| 85 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 | |
| 86 def-subst df refl refl = df | |
| 87 | |
| 88 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} | |
| 112 | 89 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 90 |
| 91 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) | |
| 112 | 92 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )} |
| 93 ( o<→c< sup-o< ) refl (cong ( λ k → od→ord (ψ k) ) oiso) | |
| 94 | |
| 95 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) | |
| 96 ∅1 {n} x (case1 ()) | |
| 97 ∅1 {n} x (case2 ()) | |
| 28 | 98 |
| 37 | 99 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 81 | 100 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
|
30
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
101 c0 : Nat → Ordinal {n} → Set n |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
102 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
|
103 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
|
104 c2 Zero not = refl |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
105 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
|
106 ... | t with t (case1 ≤-refl ) |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
107 c2 (Suc lx) not | t | () |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
108 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
|
109 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
|
110 ... | t with t (case2 Φ< ) |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
111 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
|
112 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 113 ... | 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
|
114 c3 lx (OSuc .lx x₁) d not | t | () |
|
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
115 |
| 69 | 116 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
117 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 ) |
| 112 | 118 ... | t = lemma0 (lemma t) where |
| 119 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x) | |
| 120 lemma xo<z = o<→c< xo<z | |
| 121 lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x) | |
| 122 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl | |
| 36 | 123 |
| 57 | 124 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 125 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 126 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 127 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 128 |
| 46 | 129 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 } |
| 130 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
|
131 |
| 51 | 132 -- avoiding lv != Zero error |
| 133 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
| 134 orefl refl = refl | |
| 135 | |
| 136 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
| 137 ==-iso {n} {x} {y} eq = record { | |
| 138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 140 where | |
| 141 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
| 142 lemma {x} {z} d = def-subst d oiso refl | |
| 143 | |
| 57 | 144 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 145 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
| 146 | |
| 51 | 147 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
| 148 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
| 149 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 150 lemma ox ox refl = eq-refl | |
| 151 | |
| 152 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 153 o≡→== {n} {x} {.x} refl = eq-refl | |
| 154 | |
| 155 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 156 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 157 | |
| 158 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
| 159 c≤-refl x = case1 refl | |
| 160 | |
| 112 | 161 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
| 162 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with | |
| 163 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl ) | |
| 164 ... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) | |
| 165 ... | () | |
| 166 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
| 167 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl ) | |
| 168 ... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) | |
| 169 ... | () | |
| 170 | |
| 171 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y | |
| 172 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
| 173 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
| 174 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
| 175 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
| 176 | |
| 177 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) | |
| 178 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where | |
| 179 lemma : ord→od x == record { def = λ z → z o< x } | |
| 180 eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where | |
| 181 t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) | |
| 182 t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) | |
| 183 eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl | |
| 184 | |
| 185 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } | |
| 186 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) | |
| 187 | |
| 188 ==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y | |
| 189 ==→o≡1 eq = ==→o≡ (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq ) | |
| 190 | |
| 191 ==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y | |
| 192 ==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡ eq) z>x | |
| 193 | |
| 194 ==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z | |
| 195 ==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x | |
| 196 | |
| 91 | 197 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
| 198 ∋→o< {n} {a} {x} lt = t where | |
| 199 t : (od→ord x) o< (od→ord a) | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
200 t = c<→o< {suc n} {x} {a} lt |
| 91 | 201 |
| 112 | 202 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x |
| 203 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where | |
| 204 t : def (ord→od (od→ord a)) (od→ord x) | |
| 205 t = o<→c< {suc n} {od→ord x} {od→ord a} lt | |
| 206 | |
| 80 | 207 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
| 208 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) | |
| 209 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where | |
| 210 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ | |
| 112 | 211 lemma lt with def-subst (o<→c< lt) oiso refl |
| 212 lemma lt | case1 () | |
| 213 lemma lt | case2 () | |
| 80 | 214 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso |
| 215 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 216 | |
| 112 | 217 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
| 218 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
219 |
| 51 | 220 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
221 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
| 51 | 222 |
| 223 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y | |
| 112 | 224 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) |
| 54 | 225 |
| 112 | 226 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) |
| 227 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) | |
| 228 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a ) | |
| 229 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) | |
| 230 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 refl) | |
| 231 | |
| 232 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ | |
| 233 c<> {n} {x} {y} x<y y<x with tri-c< x y | |
| 234 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
| 235 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) | |
| 236 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
237 |
| 60 | 238 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
| 112 | 239 ∅< {n} {x} {y} d eq with eq→ eq d |
| 240 ∅< {n} {x} {y} d eq | case1 () | |
| 241 ∅< {n} {x} {y} d eq | case2 () | |
| 57 | 242 |
| 112 | 243 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 244 ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x | |
| 51 | 245 |
| 76 | 246 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 |
| 247 def-iso refl t = t | |
| 248 | |
| 112 | 249 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) |
| 250 is-∋ {n} x y with tri-c< x y | |
| 251 is-∋ {n} x y | tri< a ¬b ¬c = no ¬c | |
| 252 is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c | |
| 253 is-∋ {n} x y | tri> ¬a ¬b c = yes c | |
| 254 | |
| 57 | 255 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 256 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 257 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 258 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 259 | |
|
45
33860eb44e47
od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
44
diff
changeset
|
260 open _∧_ |
|
33860eb44e47
od∅' {n} = ord→od (o∅ {n})
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
44
diff
changeset
|
261 |
| 112 | 262 -- |
| 263 -- This menas OD is Ordinal here | |
| 264 -- | |
| 265 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} | |
| 266 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
| 267 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
| 268 lemma ox ne with is-o∅ ox | |
| 269 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
| 270 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
| 271 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ | |
| 272 lemma o∅ ne | yes refl | () | |
| 273 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) ) | |
| 274 | |
| 79 | 275 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 276 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
|
59
d13d1351a1fa
lemma = cong₂ (λ x not → minimul x not ) oiso { }6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
58
diff
changeset
|
277 |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
278 csuc : {n : Level} → OD {suc n} → OD {suc n} |
| 123 | 279 csuc x = Ord ( osuc ( od→ord x )) |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
280 |
| 96 | 281 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
282 |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
283 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
| 112 | 284 ZFSubset A x = record { def = λ y → def A y ∧ def x y } |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
285 |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
286 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
| 123 | 287 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 96 | 288 |
| 289 -- Constructible Set on α | |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
290 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
291 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
292 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
293 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
| 112 | 294 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } |
| 89 | 295 |
| 112 | 296 Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 297 Ord→ZF {n} = record { | |
| 54 | 298 ZFSet = OD {suc n} |
|
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
299 ; _∋_ = _∋_ |
|
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
300 ; _≈_ = _==_ |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
301 ; ∅ = od∅ |
| 28 | 302 ; _,_ = _,_ |
| 303 ; Union = Union | |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
304 ; Power = Power |
| 115 | 305 ; Select = Select |
| 28 | 306 ; Replace = Replace |
| 112 | 307 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
308 ; isZF = isZF |
| 28 | 309 } where |
| 54 | 310 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
|
311 Replace X ψ = sup-od ψ |
| 112 | 312 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
| 116 | 313 Select X ψ = record { def = λ x → ((y : Ordinal {suc n}) → def X y → ψ ( ord→od y )) ∧ def X x } |
| 54 | 314 _,_ : OD {suc n} → OD {suc n} → OD {suc n} |
| 112 | 315 x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } |
| 54 | 316 Union : OD {suc n} → OD {suc n} |
| 112 | 317 Union U = record { def = λ y → osuc y o< (od→ord U) } |
| 77 | 318 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
| 54 | 319 Power : OD {suc n} → OD {suc n} |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
320 Power A = Def A |
| 54 | 321 ZFSet = OD {suc n} |
| 322 _∈_ : ( 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
|
323 A ∈ B = B ∋ A |
| 54 | 324 _⊆_ : ( 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
|
325 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 112 | 326 -- _∩_ : ( A B : ZFSet ) → ZFSet |
| 327 -- A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) | |
| 96 | 328 -- _∪_ : ( A B : ZFSet ) → ZFSet |
| 329 -- 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
|
330 infixr 200 _∈_ |
| 96 | 331 -- infixr 230 _∩_ _∪_ |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
332 infixr 220 _⊆_ |
| 115 | 333 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
334 isZF = record { |
|
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
335 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
|
336 ; pair = pair |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
337 ; union-u = λ _ z _ → csuc z |
| 72 | 338 ; union→ = union→ |
| 339 ; union← = union← | |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
340 ; empty = empty |
| 76 | 341 ; power→ = power→ |
| 342 ; power← = power← | |
| 343 ; extensionality = extensionality | |
|
30
3b0fdb95618e
problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
29
diff
changeset
|
344 ; minimul = minimul |
| 51 | 345 ; regularity = regularity |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
346 ; infinity∅ = infinity∅ |
| 93 | 347 ; infinity = λ _ → infinity |
| 115 | 348 ; selection = λ {X} {ψ} {y} → selection {ψ} {X} {y} |
| 93 | 349 ; replacement = replacement |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
350 } where |
| 54 | 351 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 352 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 353 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 54 | 354 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
355 empty x (case1 ()) |
|
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
356 empty x (case2 ()) |
| 100 | 357 --- |
| 358 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 359 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A | |
| 360 -- | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
361 -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t |
| 100 | 362 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x |
| 363 -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity | |
| 364 -- | |
| 76 | 365 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
366 power→ A t P∋t {x} t∋x = proj1 lemma-s where |
| 98 | 367 minsup : OD |
|
99
74330d0cdcbc
Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
368 minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) |
|
74330d0cdcbc
Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
369 lemma-t : csuc minsup ∋ t |
| 112 | 370 lemma-t = o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) |
| 98 | 371 lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
372 lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso ) |
| 112 | 373 lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl |
| 374 lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) t∋x | |
| 100 | 375 -- |
| 376 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 377 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 378 -- | |
| 77 | 379 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
|
99
74330d0cdcbc
Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
380 power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} |
| 112 | 381 ( o<→c< {suc n} {od→ord (ZFSubset A (ord→od (od→ord t)) )} {sup-o (λ x → od→ord (ZFSubset A (ord→od x)))} |
| 382 lemma ) refl lemma1 where | |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
383 lemma-eq : ZFSubset A t == t |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
384 eq→ lemma-eq {z} w = proj2 w |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
385 eq← lemma-eq {z} w = record { proj2 = w ; |
|
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
386 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
|
99
74330d0cdcbc
Power Set done with min-sup assumption
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
387 lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t |
| 112 | 388 lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq)) |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
389 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
| 98 | 390 lemma = sup-o< |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
391 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z |
| 72 | 392 union-lemma-u {X} {z} U>z = lemma <-osuc where |
| 393 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
| 112 | 394 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl refl |
| 73 | 395 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
| 72 | 396 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
397 union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) |
|
74
819da8c08f05
ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
73
diff
changeset
|
398 union→ X y u xx | tri< a ¬b ¬c | () |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
399 union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where |
| 73 | 400 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX |
| 401 lemma refl lt = lt | |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
402 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 )) |
|
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
403 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) |
| 112 | 404 union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym diso) ; proj2 = union-lemma-u X∋z } |
| 54 | 405 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
| 406 ψiso {ψ} t refl = t | |
| 116 | 407 selection : {ψ : OD → Set (suc n)} {X y : OD} → (((y : OD) → X ∋ y → ψ y) ∧ (X ∋ y)) ⇔ (Select X ψ ∋ y) |
| 112 | 408 selection {ψ} {X} {y} = record { |
| 116 | 409 proj1 = λ cond → record { proj1 = λ y X>y → proj1 cond (ord→od y) (subst (λ k → def X k ) (sym diso) X>y) ; proj2 = proj2 cond } |
| 410 ; proj2 = λ select → record { proj1 = λ y X>y → subst (λ k → ψ k ) oiso (proj1 select (od→ord y) X>y ) ; proj2 = proj2 select } | |
| 411 } where | |
| 412 lemma : (cond : ((y : OD) → X ∋ y → ψ y ) ∧ (X ∋ y) ) → ψ y | |
| 413 lemma cond = (proj1 cond) y (proj2 cond) | |
| 93 | 414 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
| 415 replacement {ψ} X x = sup-c< ψ {x} | |
| 60 | 416 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
| 417 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
| 54 | 418 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
| 112 | 419 minimul x not = od∅ |
| 57 | 420 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 112 | 421 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
| 422 proj1 (regularity x not ) = ¬∅=→∅∈ not | |
| 423 proj2 (regularity x not ) = record { eq→ = reg ; eq← = lemma } where | |
| 424 lemma : {ox : Ordinal} → def od∅ ox → def (Select (minimul x not) (λ y → (minimul x not ∋ y) ∧ (x ∋ y))) ox | |
| 425 lemma (case1 ()) | |
| 426 lemma (case2 ()) | |
| 427 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
| 116 | 428 reg {y} t with proj1 t y (proj2 t) |
| 429 ... | x∈∅ = o<-subst (proj1 x∈∅) diso refl | |
| 76 | 430 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 431 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 432 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 89 | 433 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
| 112 | 434 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) |
| 89 | 435 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} |
| 112 | 436 xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where |
| 91 | 437 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) |
|
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
438 lemma1 {x} = c<→o< ( proj1 (pair x x ) ) |
| 91 | 439 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) |
| 112 | 440 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) |
| 89 | 441 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
| 91 | 442 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
| 90 | 443 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
| 112 | 444 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where |
| 90 | 445 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) |
| 112 | 446 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) |
| 91 | 447 uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } |
| 448 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt | |
| 449 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
| 450 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) | |
| 112 | 451 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) |
| 452 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
453 infinite : OD {suc n} |
| 112 | 454 infinite = ord→od ( omega ) |
| 455 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} | |
| 456 infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} | |
| 457 (o<→c< ( case1 (s≤s z≤n ))) refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) o∅≡od∅ )) | |
| 458 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega | |
| 459 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where | |
| 460 t : od→ord x o< od→ord (ord→od (omega)) | |
| 461 t = ∋→o< {n} {infinite} {x} lt | |
| 462 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) | |
| 463 infinite∋uxxx x lt = o<∋→ t where | |
| 464 t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) | |
| 465 t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt ) | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
466 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 112 | 467 infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where |
| 91 | 468 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega |
| 469 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 470 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 471 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
| 472 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
| 473 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
| 474 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
475 |