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

annotate constructible-set.agda @ 16:ac362cc8b10f

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 14 May 2019 12:53:52 +0900
parents 497152f625ee
children 6a668c6086a5
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rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
2 module constructible-set (n : Level) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
4 open import zf
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
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
10 data OridinalD : (lv : Nat) → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
11 Φ : {lv : Nat} → OridinalD lv
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
12 OSuc : {lv : Nat} → OridinalD lv → OridinalD lv
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
13 ℵ_ : (lv : Nat) → OridinalD (Suc lv)
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
15 record Ordinal : Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
16 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
17 lv : Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
18 ord : OridinalD lv
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
19
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
20 data _o<_ : {lx ly : Nat} → OridinalD lx → OridinalD ly → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
21 l< : {lx ly : Nat } → {x : OridinalD lx } → {y : OridinalD ly } → lx < ly → x o< y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
22 Φ< : {lx : Nat} → {x : OridinalD lx} → Φ {lx} o< OSuc {lx} x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
23 s< : {lx : Nat} → {x y : OridinalD lx} → x o< y → OSuc {lx} x o< OSuc {lx} y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
24 ℵΦ< : {lx : Nat} → {x : OridinalD (Suc lx) } → Φ {Suc lx} o< (ℵ lx)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
25 ℵ< : {lx : Nat} → {x : OridinalD (Suc lx) } → OSuc {Suc lx} x o< (ℵ lx)
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
27 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
28 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
29 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
30
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
31 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
32 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
33
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
34
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
35 ≡→¬< : { x y : Nat } → x ≡ y → x < y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
36 ≡→¬< {Zero} {Zero} refl ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
37 ≡→¬< {Suc x} {.(Suc x)} refl (s≤s t) = ≡→¬< {x} {x} refl t
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
38
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
39 x≤x : { x : Nat } → x ≤ x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
40 x≤x {Zero} = z≤n
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
41 x≤x {Suc x} = s≤s ( x≤x )
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
42
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
43 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
44 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
45
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
46 triO> : {lx ly : Nat} {x : OridinalD lx } { y : OridinalD ly } → ly < lx → x o< y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
47 triO> {lx} {ly} {x} {y} y<x xo<y with <-cmp lx ly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
48 triO> {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c = ¬c y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
49 triO> {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c = ¬c y<x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
50 triO> {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c = ¬a x₁
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
51 triO> {lx} {ly} {Φ} {OSuc _} y<x Φ< | tri> ¬a ¬b c = ¬b refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
52 triO> {lx} {ly} {OSuc px} {OSuc py} y<x (s< w) | tri> ¬a ¬b c = triO> y<x w
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
53 triO> {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
54 triO> {lx} {ly} {(OSuc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c = ¬b refl
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
55
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
56 ≡→¬o< : {lv : Nat} → {x : OridinalD lv } → x o< x → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
57 ≡→¬o< {lx} {x} (l< y) = ≡→¬< refl y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
58 ≡→¬o< {lx} {OSuc y} (s< t) = ≡→¬o< t
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
60 trio<> : {lx : Nat} {x : OridinalD lx } { y : OridinalD lx } → y o< x → x o< y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
61 trio<> {lx} {x} {y} (l< lt) _ = ≡→¬< refl lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
62 trio<> {lx} {x} {y} _ (l< lt) = ≡→¬< refl lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
63 trio<> {lx} {.(OSuc _)} {.(OSuc _)} (s< s) (s< t) =
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
64 trio<> s t
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
65
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
66 trio<≡ : {lx : Nat} {x : OridinalD lx } { y : OridinalD lx } → x ≡ y → x o< y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
67 trio<≡ refl = ≡→¬o<
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
68
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
69 trio>≡ : {lx : Nat} {x : OridinalD lx } { y : OridinalD lx } → x ≡ y → y o< x → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
70 trio>≡ refl = ≡→¬o<
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
71
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
72 triO : {lx ly : Nat} → OridinalD lx → OridinalD ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
73 triO {lx} {ly} x y = <-cmp lx ly
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
74
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
75 triOonSameLevel : {lx : Nat} → Trichotomous _≡_ ( _o<_ {lx} {lx} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
76 triOonSameLevel {lv} Φ Φ = tri≈ ≡→¬o< refl ≡→¬o<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
77 triOonSameLevel {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬o< refl ≡→¬o<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
78 triOonSameLevel {lv} Φ (OSuc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
79 triOonSameLevel {.(Suc lv)} Φ (ℵ lv) = tri< (ℵΦ< {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {lv} {Φ} )) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
80 triOonSameLevel {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ< {lv} {Φ} ) ) (λ ()) (ℵΦ< {lv} {Φ} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
81 triOonSameLevel {Suc lv} (ℵ lv) (OSuc y) = tri> ( λ lt → trio<> lt (ℵ< {lv} {y} ) ) (λ ()) (ℵ< {lv} {y} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
82 triOonSameLevel {lv} (OSuc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
83 triOonSameLevel {.(Suc lv)} (OSuc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
84 triOonSameLevel {lv} (OSuc x) (OSuc y) with triOonSameLevel x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
85 triOonSameLevel {lv} (OSuc x) (OSuc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
86 triOonSameLevel {lv} (OSuc x) (OSuc x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬o< refl ≡→¬o<
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
87 triOonSameLevel {lv} (OSuc x) (OSuc 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
88
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
89 <→≤ : {lx ly : Nat} → lx < ly → (Suc lx ≤ ly)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
90 <→≤ {Zero} {Suc ly} (s≤s lt) = s≤s z≤n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
91 <→≤ {Suc lx} {Zero} ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
92 <→≤ {Suc lx} {Suc ly} (s≤s lt) = s≤s (<→≤ lt)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
93
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
94 orddtrans : {lx ly lz : Nat} {x : OridinalD lx } { y : OridinalD ly } { z : OridinalD lz } → x o< y → y o< z → x o< z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
95 orddtrans {lx} {ly} {lz} x<y y<z with <-cmp lx ly | <-cmp ly lz
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
96 orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = l< ( <-trans a a₁ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
97 orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri≈ ¬a refl ¬c₁ = l< a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
98 orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = l< {!!} -- ⊥-elim ( ¬a c )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
99 orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = l< {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
100 orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri≈ ¬a₁ refl ¬c = l< {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
101 orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
102 orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = l< a
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
103 orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = l< {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
104 orddtrans {lx} {lx} {lx} x<y y<z | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = orddtrans1 x<y y<z where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
105 orddtrans1 : {lx : Nat} {x y z : OridinalD lx } → x o< y → y o< z → x o< z
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
106 orddtrans1 = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
107
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
108
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
109
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
110 max : (x y : Nat) → Nat
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
111 max Zero Zero = Zero
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
112 max Zero (Suc x) = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
113 max (Suc x) Zero = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
114 max (Suc x) (Suc y) = Suc ( max x y )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
116 -- use cannot use OridinalD (Data.Nat_⊔_ lx ly), I don't know why
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
117
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
118 maxα> : { lx ly : Nat } → OridinalD lx → OridinalD ly → lx > ly → OridinalD lx
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
119 maxα> x y _ = x
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
120
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
121 maxα= : { lx : Nat } → OridinalD lx → OridinalD lx → OridinalD lx
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
122 maxα= x y with triOonSameLevel x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
123 maxα= x y | tri< a ¬b ¬c = y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
124 maxα= x y | tri≈ ¬a b ¬c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
125 maxα= x y | tri> ¬a ¬b c = x
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
126
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
127 OrdTrans : Transitive (λ ( a b : Ordinal ) → (a ≡ b) ∨ (Ordinal.lv a < Ordinal.lv b) ∨ (Ordinal.ord a o< Ordinal.ord b) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
128 OrdTrans (case1 refl) (case1 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
129 OrdTrans (case1 refl) (case2 lt2) = case2 lt2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
130 OrdTrans (case2 lt1) (case1 refl) = case2 lt1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
131 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 ( case1 ( <-trans x y ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
132 OrdTrans (case2 (case1 x)) (case2 (case2 y)) = case2 {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
133 OrdTrans (case2 (case2 x)) (case2 (case1 y)) = case2 {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
134 OrdTrans (case2 (case2 x)) (case2 (case2 y)) = case2 {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
135
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
136 OrdPreorder : Preorder n n n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
137 OrdPreorder = record { Carrier = Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
138 ; _≈_ = _≡_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
139 ; _∼_ = λ a b → (a ≡ b) ∨ (Ordinal.lv a < Ordinal.lv b) ∨ (Ordinal.ord a o< Ordinal.ord b )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
140 ; isPreorder = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
141 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
142 ; reflexive = case1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
143 ; trans = OrdTrans
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
144 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
145 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
146
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
147 -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
148
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
149 data Constructible {lv : Nat} ( α : OridinalD lv ) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
150 fsub : ( ψ : OridinalD lv → Set n ) → Constructible α
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
151 xself : OridinalD lv → Constructible α
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
152
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
153 record ConstructibleSet : Set (suc n) where
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
154 field
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
155 level : Nat
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
156 α : OridinalD level
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
157 constructible : Constructible α
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
158
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
159 open ConstructibleSet
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
160
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
161 data _c∋_ : {lv lv' : Nat} {α : OridinalD lv } {α' : OridinalD lv' } →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
162 Constructible {lv} α → Constructible {lv'} α' → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
163 c> : {lv lv' : Nat} {α : OridinalD lv } {α' : OridinalD lv' }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
164 (ta : Constructible {lv} α ) ( tx : Constructible {lv'} α' ) → α' o< α → ta c∋ tx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
165 xself-fsub : {lv : Nat} {α : OridinalD lv }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
166 (ta : OridinalD lv ) ( ψ : OridinalD lv → Set n ) → _c∋_ {_} {_} {α} {α} (xself ta ) ( fsub ψ)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
167 fsub-fsub : {lv lv' : Nat} {α : OridinalD lv }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
168 ( ψ : OridinalD lv → Set n ) ( ψ₁ : OridinalD lv → Set n ) →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
169 ( ∀ ( x : OridinalD lv ) → ψ x → ψ₁ x ) → _c∋_ {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
170
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
171 _∋_ : (ConstructibleSet ) → (ConstructibleSet ) → Set n
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
172 a ∋ x = constructible a c∋ constructible x
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
173
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
174 transitiveness : (a b c : ConstructibleSet ) → a ∋ b → b ∋ c → a ∋ c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
175 transitiveness a b c a∋b b∋c with constructible a c∋ constructible b | constructible b c∋ constructible c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
176 ... | t1 | t2 = {!!}
15
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
177
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
178 data _c≈_ : {lv lv' : Nat} {α : OridinalD lv } {α' : OridinalD lv' } →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
179 Constructible {lv} α → Constructible {lv'} α' → Set n where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
180 crefl : {lv : Nat} {α : OridinalD lv } → _c≈_ {_} {_} {α} {α} (xself α ) (xself α )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
181 feq : {lv : Nat} {α : OridinalD lv }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
182 → ( ψ : OridinalD lv → Set n ) ( ψ₁ : OridinalD lv → Set n )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
183 → (∀ ( x : OridinalD lv ) → ψ x ⇔ ψ₁ x ) → _c≈_ {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
11
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 10
diff changeset
184
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
185 _≈_ : (ConstructibleSet ) → (ConstructibleSet ) → Set n
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
186 a ≈ x = constructible a c≈ constructible x
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
187
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
188 ConstructibleSet→ZF : ZF {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
189 ConstructibleSet→ZF = record {
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
190 ZFSet = ConstructibleSet
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
191 ; _∋_ = _∋_
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
192 ; _≈_ = _≈_
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
193 ; ∅ = record { level = Zero ; α = Φ ; constructible = xself Φ }
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
194 ; _×_ = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
195 ; Union = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
196 ; Power = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
197 ; Select = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
198 ; Replace = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
199 ; infinite = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
200 ; isZF = {!!}
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
201 }