annotate ordinal.agda @ 204:d4802eb159ff

Transfinite induction fixed
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 31 Jul 2019 15:29:51 +0900
parents 8edd2a13a7f3
children 22d435172d1a
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
34
c9ad0d97ce41 fix oridinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
1 {-# OPTIONS --allow-unsolved-metas #-}
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
2 open import Level
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
3 module ordinal where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
5 open import zf
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
8 open import Data.Empty
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
9 open import Relation.Binary.PropositionalEquality
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
11 data OrdinalD {n : Level} : (lv : Nat) → Set n where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
12 Φ : (lv : Nat) → OrdinalD lv
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
13 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
15 record Ordinal {n : Level} : Set n where
202
ed88384b5102 ε-induction like loop again
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 184
diff changeset
16 constructor ordinal
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
17 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
18 lv : Nat
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
19 ord : OrdinalD {n} lv
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
20
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
21 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
22 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
23 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
24
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
25 open Ordinal
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
26
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
27 _o<_ : {n : Level} ( x y : Ordinal ) → Set n
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
28 _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
30 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
31 s<refl {n} {lv} {Φ lv} = Φ<
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
32 s<refl {n} {lv} {OSuc lv x} = s< s<refl
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
33
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
34 trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
35 trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
36 trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< ()
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
37
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
38 d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
39 d<→lv Φ< = refl
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
40 d<→lv (s< lt) = refl
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
41
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
42 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
43 o<-subst df refl refl = df
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
44
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
45 open import Data.Nat.Properties
30
3b0fdb95618e problem on Ordinal ( OSuc ℵ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 29
diff changeset
46 open import Data.Unit using ( ⊤ )
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
47 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
48
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
49 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
50 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
51
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
52 o∅ : {n : Level} → Ordinal {n}
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
53 o∅ = record { lv = Zero ; ord = Φ Zero }
21
6d9fdd1dfa79 add transfinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 20
diff changeset
54
39
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
55 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
56
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
57 ordinal-cong : {n : Level} {x y : Ordinal {n}} →
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
58 lv x ≡ lv y → ord x ≅ ord y → x ≡ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
59 ordinal-cong refl refl = refl
21
6d9fdd1dfa79 add transfinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 20
diff changeset
60
46
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
61 ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
62 ordinal-lv refl = refl
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
63
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
64 ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
65 ordinal-d refl = refl
e584686a1307 == and ∅7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 43
diff changeset
66
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
67 ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
68 ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
69
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
70 trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
71 trio<≡ refl = ≡→¬d<
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
72
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
73 trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
74 trio>≡ refl = ≡→¬d<
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
75
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
76 triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
77 triO {n} {lx} {ly} x y = <-cmp lx ly
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
78
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
79 triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} )
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
80 triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d<
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
81 triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
82 triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
83 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
84 triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) )
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
85 triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d<
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
86 triOrdd {_} {lv} (OSuc lv x) (OSuc lv 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
87
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
88 osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n}
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
89 osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox }
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
90
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
91 <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
92 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ<
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
93 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl )
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
94
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
95 osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x )
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
96 osuc-lveq {n} = refl
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
97
113
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
98 osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
99 osucc {n} {ox} {oy} (case1 x) = case1 x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
100 osucc {n} {ox} {oy} (case2 x) with d<→lv x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
101 ... | refl = case2 (s< x)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
102
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
103 nat-<> : { x y : Nat } → x < y → y < x → ⊥
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
104 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
105
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
106 nat-<≡ : { x : Nat } → x < x → ⊥
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
107 nat-<≡ (s≤s lt) = nat-<≡ lt
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
108
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
109 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
110 nat-≡< refl lt = nat-<≡ lt
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
111
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
112 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
113 ¬a≤a (s≤s lt) = ¬a≤a lt
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
114
203
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
115 a<sa : {la : Nat} → la < Suc la
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
116 a<sa {Zero} = s≤s z≤n
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
117 a<sa {Suc la} = s≤s a<sa
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
118
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
119 =→¬< : {x : Nat } → ¬ ( x < x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
120 =→¬< {Zero} ()
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
121 =→¬< {Suc x} (s≤s lt) = =→¬< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
122
203
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
123 <-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) )
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
124 <-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
125 <-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n)
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
126 <-∨ {Suc x} {Zero} (s≤s ())
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
127 <-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
128 <-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq)
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
129 <-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
130
147
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
131 case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
132 case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
133 ... | refl = nat-≡< refl lt1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
134
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
135 case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
136 case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
137 ... | refl = nat-≡< refl lt1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
138
203
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
139 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥
111
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
140 o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 97
diff changeset
141 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
142
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
143 ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
144 ¬x<0 {n} {x} (case1 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
145 ¬x<0 {n} {x} (case2 ())
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
146
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
147 o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
148 o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
149 o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
150 o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
151 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ())
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
152 o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) =
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
153 o<> (case2 y<x) (case2 x<y)
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
154
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
155 orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
156 orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ<
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
157 orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
158
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
159 osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
160 osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt)
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
161 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
162 osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<)
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
163 osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ()))
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
164 osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
165 osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt )
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
166 ... | case1 refl = case1 refl
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
167 ... | case2 (case2 x) = case2 (case2( s< x) )
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
168 ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
169
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
170 osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
171 osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
172 osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁)
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
173 osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂)
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
174 osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
175 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
176 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
177 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x
75
714470702a8b Union done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 74
diff changeset
178
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
179 max : (x y : Nat) → Nat
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
180 max Zero Zero = Zero
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
181 max Zero (Suc x) = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
182 max (Suc x) Zero = (Suc x)
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
183 max (Suc x) (Suc y) = Suc ( max x y )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
184
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
185 maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
186 maxαd x y with triOrdd x y
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
187 maxαd x y | tri< a ¬b ¬c = y
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
188 maxαd x y | tri≈ ¬a b ¬c = x
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
189 maxαd x y | tri> ¬a ¬b c = x
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
190
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
191 minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
192 minαd x y with triOrdd x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
193 minαd x y | tri< a ¬b ¬c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
194 minαd x y | tri≈ ¬a b ¬c = y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
195 minαd x y | tri> ¬a ¬b c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
196
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
197 _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
198 a o≤ b = (a ≡ b) ∨ ( a o< b )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
199
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
200 ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
201 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ )
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
202 ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
203 ... | refl = case1 x₁
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
204 ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁
27
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
205 ... | refl = case1 x₂
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
206 ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
207 ... | refl | refl = case2 ( orddtrans x₁ x₂ )
bade0a35fdd9 OD, HOD, TC
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
208
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
209 trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
210 trio< a b with <-cmp (lv a) (lv b)
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
211 trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
212 lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
213 lemma1 (case1 x) = ¬c x
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
214 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ )
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
215 trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
216 lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
217 lemma1 (case1 x) = ¬a x
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
218 lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c )
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
219 trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b )
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
220 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
221 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
222 lemma1 refl = refl
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
223 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
224 lemma2 (case1 x) = ¬a x
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
225 lemma2 (case2 x) = trio<> x a
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
226 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
227 lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
228 lemma1 refl = refl
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
229 lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
230 lemma2 (case1 x) = ¬a x
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
231 lemma2 (case2 x) = trio<> x c
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
232 trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
233 lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
234 lemma1 (case1 x) = ¬a x
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
235 lemma1 (case2 x) = ≡→¬d< x
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
236
180
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
237 xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
238 xo<ab {n} {oa} {ob} a→b with trio< oa ob
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
239 xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
240 xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
241 xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) )
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
242
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
243 maxα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
244 maxα x y with trio< x y
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
245 maxα x y | tri< a ¬b ¬c = y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
246 maxα x y | tri> ¬a ¬b c = x
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
247 maxα x y | tri≈ ¬a refl ¬c = x
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
248
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
249 minα : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
250 minα {n} x y with trio< {n} x y
127
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
251 minα x y | tri< a ¬b ¬c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 113
diff changeset
252 minα x y | tri> ¬a ¬b c = y
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
253 minα x y | tri≈ ¬a refl ¬c = x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
254
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
255 min1 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
256 min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
257 min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
258 min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
259 min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
260
85
7494ae6b83c6 omax-induction does not work
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
261 --
7494ae6b83c6 omax-induction does not work
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
262 -- max ( osuc x , osuc y )
7494ae6b83c6 omax-induction does not work
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 84
diff changeset
263 --
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
264
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
265 omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n}
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
266 omax {n} x y with trio< x y
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
267 omax {n} x y | tri< a ¬b ¬c = osuc y
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
268 omax {n} x y | tri> ¬a ¬b c = osuc x
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
269 omax {n} x y | tri≈ ¬a refl ¬c = osuc x
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
270
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
271 omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
272 omax< {n} x y lt with trio< x y
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
273 omax< {n} x y lt | tri< a ¬b ¬c = refl
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
274 omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
275 omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
276
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
277 omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
278 omax≡ {n} x y eq with trio< x y
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
279 omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
280 omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
281 omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
282
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
283 omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
284 omax-x {n} x y with trio< x y
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
285 omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
286 omax-x {n} x y | tri> ¬a ¬b c = <-osuc
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
287 omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
288
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
289 omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
290 omax-y {n} x y with trio< x y
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
291 omax-y {n} x y | tri< a ¬b ¬c = <-osuc
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
292 omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
293 omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
294
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
295 omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
296 omxx {n} x with trio< x x
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
297 omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
298 omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
299 omxx {n} x | tri≈ ¬a refl ¬c = refl
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
300
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
301 omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x)
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
302 omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
303
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
304 open _∧_
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
305
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
306 osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
307 proj1 (osuc2 {n} x y) (case1 lt) = case1 lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
308 proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
309 proj2 (osuc2 {n} x y) (case1 lt) = case1 lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
310 proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
311 ... | refl = case2 (s< lt)
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
312
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
313 OrdTrans : {n : Level} → Transitive {suc n} _o≤_
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
314 OrdTrans (case1 refl) (case1 refl) = case1 refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
315 OrdTrans (case1 refl) (case2 lt2) = case2 lt2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
316 OrdTrans (case2 lt1) (case1 refl) = case2 lt1
81
96c932d0145d simpler ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 75
diff changeset
317 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y)
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
318
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
319 OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n)
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
320 OrdPreorder {n} = record { Carrier = Ordinal
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
321 ; _≈_ = _≡_
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
322 ; _∼_ = _o≤_
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
323 ; isPreorder = record {
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
324 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
325 ; reflexive = case1
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
326 ; trans = OrdTrans
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
327 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
328 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
329
203
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
330 TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m }
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
331 → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
24
3186bbee99dd separte level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
diff changeset
332 → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
22
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 21
diff changeset
333 → ∀ (x : Ordinal) → ψ x
204
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
334 TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
335 TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) )
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
336 TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where
203
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
337 lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
338 lemma x (case1 ())
8edd2a13a7f3 fixing transfinte induction...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
339 lemma x (case2 ())
204
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
340 lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
341 lemma1 x (case1 ())
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
342 lemma1 x (case2 ())
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
343 TransFinite1 (Suc lx) (Φ (Suc lx)) = record { proj1 = caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) ; proj2 = (λ x → lemma (lv x) (ord x)) } where
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
344 lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy)
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
345 lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
346 lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy)
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
347 lemma lx1 ox1 (case1 lt) with <-∨ lt
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
348 lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) )
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
349 lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa))
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
350 lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0 lx (Φ lx) (case1 (s≤s lt1))
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
351 lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa )))
d4802eb159ff Transfinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
352 TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )}
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
353
184
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 183
diff changeset
354 -- we cannot prove this in intutionistic logic
142
c30bc9f5bd0d Power Set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 129
diff changeset
355 -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
356 -- postulate
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
357 -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m )
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
358 -- → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
359 -- → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → p )
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
360 -- → p
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
361 --
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
362 -- Instead we prove
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
363 --
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
364 TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m )
165
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
365 → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → ¬ p )
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
366 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
367 → ¬ p
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 165
diff changeset
368 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
165
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
369