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

annotate constructible-set.agda @ 15:497152f625ee

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fix
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 14 May 2019 03:52:42 +0900
parents e11e95d5ddee
children ac362cc8b10f
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
1 module constructible-set where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 open import Level
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
4 open import zf
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
8 open import Relation.Binary.PropositionalEquality
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
10 data Ordinal {n : Level } : (lv : Nat) → Set n where
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
11 Φ : {lv : Nat} → Ordinal {n} lv
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
12 T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
13 ℵ_ : (lv : Nat) → Ordinal (Suc lv)
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
15 data _o<_ {n : Level } : {lx ly : Nat} → Ordinal {n} lx → Ordinal {n} ly → Set n where
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
16 l< : {lx ly : Nat } → {x : Ordinal {n} lx } → {y : Ordinal {n} ly } → lx < ly → x o< y
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
17 Φ< : {lx : Nat} → {x : Ordinal {n} lx} → Φ {n} {lx} o< T-suc {n} {lx} x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
18 s< : {lx : Nat} → {x y : Ordinal {n} lx} → x o< y → T-suc {n} {lx} x o< T-suc {n} {lx} y
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
19 ℵΦ< : {lx : Nat} → {x : Ordinal {n} (Suc lx) } → Φ {n} {Suc lx} o< (ℵ lx)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
20 ℵ< : {lx : Nat} → {x : Ordinal {n} (Suc lx) } → T-suc {n} {Suc lx} x o< (ℵ lx)
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
22 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
23 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
24 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
25
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
26 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
27 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
28
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
29
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
30 nat< : { x y : Nat } → x ≡ y → x < y → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
31 nat< {Zero} {Zero} refl ()
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
32 nat< {Suc x} {.(Suc x)} refl (s≤s t) = nat< {x} {x} refl t
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
33
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
34 x≤x : { x : Nat } → x ≤ x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
35 x≤x {Zero} = z≤n
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
36 x≤x {Suc x} = s≤s ( x≤x )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
37
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
38 x<>y : { x y : Nat } → x > y → x < y → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
39 x<>y {.(Suc _)} {.(Suc _)} (s≤s lt) (s≤s lt1) = x<>y lt lt1
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
40
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
41 triO> : {n : Level } → {lx ly : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} ly } → ly < lx → x o< y → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
42 triO> {n} {lx} {ly} {x} {y} y<x xo<y with <-cmp lx ly
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
43 triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c = ¬c y<x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
44 triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c = ¬c y<x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
45 triO> {n} {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c = ¬a x₁
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
46 triO> {n} {lx} {ly} {Φ} {T-suc _} y<x Φ< | tri> ¬a ¬b c = ¬b refl
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
47 triO> {n} {lx} {ly} {T-suc px} {T-suc py} y<x (s< w) | tri> ¬a ¬b c = triO> y<x w
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
48 triO> {n} {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
49 triO> {n} {lx} {ly} {(T-suc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c = ¬b refl
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
50
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
51 trio! : {n : Level } → {lv : Nat} → {x : Ordinal {n} lv } → x o< x → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
52 trio! {n} {lx} {x} (l< y) = nat< refl y
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
53 trio! {n} {lx} {T-suc y} (s< t) = trio! t
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
55 trio<> : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → y o< x → x o< y → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
56 trio<> {n} {lx} {x} {y} (l< lt) _ = nat< refl lt
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
57 trio<> {n} {lx} {x} {y} _ (l< lt) = nat< refl lt
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
58 trio<> {n} {lx} {.(T-suc _)} {.(T-suc _)} (s< s) (s< t) =
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
59 trio<> s t
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
60
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
61 trio<≡ : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → x ≡ y → x o< y → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
62 trio<≡ refl = trio!
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
63
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
64 trio>≡ : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → x ≡ y → y o< x → ⊥
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
65 trio>≡ refl = trio!
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
66
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
67 triO : {n : Level } → {lx ly : Nat} → Ordinal {n} lx → Ordinal {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
68 triO {n} {lx} {ly} x y = <-cmp lx ly
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
69
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
70 triOonSameLevel : {n : Level } → {lx : Nat} → Trichotomous _≡_ ( _o<_ {n} {lx} {lx} )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
71 triOonSameLevel {n} {lv} Φ Φ = tri≈ trio! refl trio!
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
72 triOonSameLevel {n} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ trio! refl trio!
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
73 triOonSameLevel {n} {lv} Φ (T-suc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
74 triOonSameLevel {n} {.(Suc lv)} Φ (ℵ lv) = tri< (ℵΦ< {n} {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {n} {lv} {Φ} )) )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
75 triOonSameLevel {n} {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ< {n} {lv} {Φ} ) ) (λ ()) (ℵΦ< {n} {lv} {Φ} )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
76 triOonSameLevel {n} {Suc lv} (ℵ lv) (T-suc y) = tri> ( λ lt → trio<> lt (ℵ< {n} {lv} {y} ) ) (λ ()) (ℵ< {n} {lv} {y} )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
77 triOonSameLevel {n} {lv} (T-suc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
78 triOonSameLevel {n} {.(Suc lv)} (T-suc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
79 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) with triOonSameLevel x y
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
80 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
81 triOonSameLevel {n} {lv} (T-suc x) (T-suc x) | tri≈ ¬a refl ¬c = tri≈ trio! refl trio!
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
82 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
83
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
84
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
85 max : (x y : Nat) → Nat
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
86 max Zero Zero = Zero
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
87 max Zero (Suc x) = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
88 max (Suc x) Zero = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
89 max (Suc x) (Suc y) = Suc ( max x y )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
91 maxα> : {n : Level } → { lx ly : Nat } → Ordinal {n} lx → Ordinal {n} ly → lx > ly → Ordinal {n} lx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
92 maxα> x y _ = x
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
93
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
94 maxα= : {n : Level } → { lx : Nat } → Ordinal {n} lx → Ordinal {n} lx → Ordinal {n} lx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
95 maxα= x y with triOonSameLevel x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
96 maxα= x y | tri< a ¬b ¬c = y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
97 maxα= x y | tri≈ ¬a b ¬c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
98 maxα= x y | tri> ¬a ¬b c = x
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
99
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
100 -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
101
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
102 data Constructible {n : Level } {lv : Nat} ( α : Ordinal {n} lv ) : Set (suc n) where
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
103 fsub : ( ψ : Ordinal {n} lv → Set n ) → Constructible α
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
104 xself : Ordinal {n} lv → Constructible α
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
105
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
106 record ConstructibleSet {n : Level } : Set (suc n) where
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
107 field
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
108 level : Nat
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
109 α : Ordinal {n} level
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
110 constructible : Constructible α
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
111
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
112 open ConstructibleSet
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
113
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
114 data _c∋_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
115 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
116 c> : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' }
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
117 (ta : Constructible {n} {lv} α ) ( tx : Constructible {n} {lv'} α' ) → α' o< α → ta c∋ tx
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
118 xself-fsub : {lv : Nat} {α : Ordinal {n} lv }
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
119 (ta : Ordinal {n} lv ) ( ψ : Ordinal {n} lv → Set n ) → _c∋_ {n} {_} {_} {α} {α} (xself ta ) ( fsub ψ)
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
120 fsub-fsub : {lv lv' : Nat} {α : Ordinal {n} lv }
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
121 ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
122 ( ∀ ( x : Ordinal {n} lv ) → ψ x → ψ₁ x ) → _c∋_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
123
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
124 _∋_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
125 a ∋ x = constructible a c∋ constructible x
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
126
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
127 transitiveness : {n : Level} → (a b c : ConstructibleSet {n}) → a ∋ b → b ∋ c → a ∋ c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
128 transitiveness = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
129
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
130 data _c≈_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
131 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
132 crefl : {lv : Nat} {α : Ordinal {n} lv } → _c≈_ {n} {_} {_} {α} {α} (xself α ) (xself α )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
133 feq : {lv : Nat} {α : Ordinal {n} lv }
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
134 → ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
135 → (∀ ( x : Ordinal {n} lv ) → ψ x ⇔ ψ₁ x ) → _c≈_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
136
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
137 _≈_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
138 a ≈ x = constructible a c≈ constructible x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
139
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
140 ConstructibleSet→ZF : {n : Level } → ZF {suc n} {n}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
141 ConstructibleSet→ZF {n} = record {
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
142 ZFSet = ConstructibleSet
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
143 ; _∋_ = _∋_
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
144 ; _≈_ = _≈_
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
145 ; ∅ = record { level = Zero ; α = Φ ; constructible = xself Φ }
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
146 ; _×_ = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
147 ; Union = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
148 ; Power = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
149 ; Select = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
150 ; Replace = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
151 ; infinite = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
152 ; isZF = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
153 }