annotate Ordinals.agda @ 326:feeba7fd499a

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 05 Jul 2020 03:45:41 +0900
parents fbabb20f222e
children 5544f4921a44
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
220
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
2 module Ordinals 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
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
6 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
7 open import Data.Empty
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
8 open import Relation.Binary.PropositionalEquality
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 204
diff changeset
9 open import logic
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 204
diff changeset
10 open import nat
220
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
11 open import Data.Unit using ( ⊤ )
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
12 open import Relation.Nullary
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
13 open import Relation.Binary
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
14 open import Relation.Binary.Core
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15
320
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
16 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord ) (osuc : ord → ord ) (_o<_ : ord → ord → Set n) (next : ord → ord ) : Set (suc (suc n)) where
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
17 field
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
18 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
19 OTri : Trichotomous {n} _≡_ _o<_
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
20 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
21 <-osuc : { x : ord } → x o< osuc x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
22 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
320
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
23 is-limit : ( x : ord ) → Dec ( ¬ ((y : ord) → x ≡ osuc y) )
321
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 320
diff changeset
24 next-limit : { y : ord } → (y o< next y ) ∧ ((x : ord) → x o< next y → osuc x o< next y )
324
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 321
diff changeset
25 TransFinite : { ψ : ord → Set n }
222
59771eb07df0 TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 221
diff changeset
26 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
59771eb07df0 TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 221
diff changeset
27 → ∀ (x : ord) → ψ x
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
28
222
59771eb07df0 TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 221
diff changeset
29
59771eb07df0 TransFinite induction fixed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 221
diff changeset
30 record Ordinals {n : Level} : Set (suc (suc n)) where
220
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
31 field
95a26d1698f4 try to separate Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 218
diff changeset
32 ord : Set n
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
33 o∅ : ord
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
34 osuc : ord → ord
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
35 _o<_ : ord → ord → Set n
320
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
36 next : ord → ord
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
37 isOrdinal : IsOrdinals ord o∅ osuc _o<_ next
17
6a668c6086a5 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 16
diff changeset
38
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
39 module inOrdinal {n : Level} (O : Ordinals {n} ) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
41 Ordinal : Set n
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
42 Ordinal = Ordinals.ord O
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
43
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
44 _o<_ : Ordinal → Ordinal → Set n
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
45 _o<_ = Ordinals._o<_ O
218
eee983e4b402 try func
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 213
diff changeset
46
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
47 osuc : Ordinal → Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
48 osuc = Ordinals.osuc O
218
eee983e4b402 try func
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 213
diff changeset
49
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
50 o∅ : Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
51 o∅ = Ordinals.o∅ O
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
52
320
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
53 next : Ordinal → Ordinal
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
54 next = Ordinals.next O
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
55
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
56 ¬x<0 = IsOrdinals.¬x<0 (Ordinals.isOrdinal O)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
57 osuc-≡< = IsOrdinals.osuc-≡< (Ordinals.isOrdinal O)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
58 <-osuc = IsOrdinals.<-osuc (Ordinals.isOrdinal O)
235
846e0926bb89 fix cardinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 222
diff changeset
59 TransFinite = IsOrdinals.TransFinite (Ordinals.isOrdinal O)
320
21203fe0342f infinite ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 309
diff changeset
60 next-limit = IsOrdinals.next-limit (Ordinals.isOrdinal O)
321
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 320
diff changeset
61
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
62 o<-dom : { x y : Ordinal } → x o< y → Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
63 o<-dom {x} _ = x
74
819da8c08f05 ordinal atomical successor?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 73
diff changeset
64
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
65 o<-cod : { x y : Ordinal } → x o< y → Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
66 o<-cod {_} {y} _ = y
147
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 144
diff changeset
67
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
68 o<-subst : {Z X z x : Ordinal } → Z o< X → Z ≡ z → X ≡ x → z o< x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
69 o<-subst df refl refl = df
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
70
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
71 ordtrans : {x y z : Ordinal } → x o< y → y o< z → x o< z
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
72 ordtrans = IsOrdinals.Otrans (Ordinals.isOrdinal O)
94
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 91
diff changeset
73
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
74 trio< : Trichotomous _≡_ _o<_
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
75 trio< = IsOrdinals.OTri (Ordinals.isOrdinal O)
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
76
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
77 o<¬≡ : { ox oy : Ordinal } → ox ≡ oy → ox o< oy → ⊥
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
78 o<¬≡ {ox} {oy} eq lt with trio< ox oy
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
79 o<¬≡ {ox} {oy} eq lt | tri< a ¬b ¬c = ¬b eq
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
80 o<¬≡ {ox} {oy} eq lt | tri≈ ¬a b ¬c = ¬a lt
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
81 o<¬≡ {ox} {oy} eq lt | tri> ¬a ¬b c = ¬b eq
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
82
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
83 o<> : {x y : Ordinal } → y o< x → x o< y → ⊥
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
84 o<> {ox} {oy} lt tl with trio< ox oy
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
85 o<> {ox} {oy} lt tl | tri< a ¬b ¬c = ¬c lt
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
86 o<> {ox} {oy} lt tl | tri≈ ¬a b ¬c = ¬a tl
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
87 o<> {ox} {oy} lt tl | tri> ¬a ¬b c = ¬a tl
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
88
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
89 osuc-< : { x y : Ordinal } → y o< osuc x → x o< y → ⊥
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
90 osuc-< {x} {y} y<ox x<y with osuc-≡< y<ox
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
91 osuc-< {x} {y} y<ox x<y | case1 refl = o<¬≡ refl x<y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
92 osuc-< {x} {y} y<ox x<y | case2 y<x = o<> x<y y<x
180
11490a3170d4 new ordinal-definable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 167
diff changeset
93
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
94 osucc : {ox oy : Ordinal } → oy o< ox → osuc oy o< osuc ox
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
95 ---- y < osuc y < x < osuc x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
96 ---- y < osuc y = x < osuc x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
97 ---- y < osuc y > x < osuc x -> y = x ∨ x < y → ⊥
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
98 osucc {ox} {oy} oy<ox with trio< (osuc oy) ox
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
99 osucc {ox} {oy} oy<ox | tri< a ¬b ¬c = ordtrans a <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
100 osucc {ox} {oy} oy<ox | tri≈ ¬a refl ¬c = <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
101 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c with osuc-≡< c
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
102 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case1 eq = ⊥-elim (o<¬≡ (sym eq) oy<ox)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
103 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case2 lt = ⊥-elim (o<> lt oy<ox)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
104
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
105 open _∧_
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
106
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
107 osuc2 : ( x y : Ordinal ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
108 proj2 (osuc2 x y) lt = osucc lt
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
109 proj1 (osuc2 x y) ox<ooy with osuc-≡< ox<ooy
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
110 proj1 (osuc2 x y) ox<ooy | case1 ox=oy = o<-subst <-osuc refl ox=oy
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
111 proj1 (osuc2 x y) ox<ooy | case2 ox<oy = ordtrans <-osuc ox<oy
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
112
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
113 _o≤_ : Ordinal → Ordinal → Set n
326
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
114 a o≤ b = a o< osuc b -- (a ≡ b) ∨ ( a o< b )
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
115
129
2a5519dcc167 ord power set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 127
diff changeset
116
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
117 xo<ab : {oa ob : Ordinal } → ( {ox : Ordinal } → ox o< oa → ox o< ob ) → oa o< osuc ob
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
118 xo<ab {oa} {ob} a→b with trio< oa ob
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
119 xo<ab {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
120 xo<ab {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
121 xo<ab {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) )
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
122
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
123 maxα : Ordinal → Ordinal → Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
124 maxα x y with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
125 maxα x y | tri< a ¬b ¬c = y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
126 maxα x y | tri> ¬a ¬b c = x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
127 maxα x y | tri≈ ¬a refl ¬c = x
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
128
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
129 omin : Ordinal → Ordinal → Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
130 omin x y with trio< x y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
131 omin x y | tri< a ¬b ¬c = x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
132 omin x y | tri> ¬a ¬b c = y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
133 omin x y | tri≈ ¬a refl ¬c = x
88
975e72ae0541 osuc work around done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 87
diff changeset
134
308
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 302
diff changeset
135 min1 : {x y z : Ordinal } → z o< x → z o< y → z o< omin x y
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
136 min1 {x} {y} {z} z<x z<y with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
137 min1 {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
138 min1 {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
139 min1 {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y
84
4b2aff372b7c omax ..
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 81
diff changeset
140
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
141 --
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
142 -- max ( osuc x , osuc y )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
143 --
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
144
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
145 omax : ( x y : Ordinal ) → Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
146 omax x y with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
147 omax x y | tri< a ¬b ¬c = osuc y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
148 omax x y | tri> ¬a ¬b c = osuc x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
149 omax x y | tri≈ ¬a refl ¬c = osuc x
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
150
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
151 omax< : ( x y : Ordinal ) → x o< y → osuc y ≡ omax x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
152 omax< x y lt with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
153 omax< x y lt | tri< a ¬b ¬c = refl
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
154 omax< x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
155 omax< x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
86
08be6351927e internal error
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 85
diff changeset
156
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
157 omax≡ : ( x y : Ordinal ) → x ≡ y → osuc y ≡ omax x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
158 omax≡ x y eq with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
159 omax≡ x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
160 omax≡ x y eq | tri≈ ¬a refl ¬c = refl
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
161 omax≡ x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
91
b4742cf4ef97 infinity axiom done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
162
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
163 omax-x : ( x y : Ordinal ) → x o< omax x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
164 omax-x x y with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
165 omax-x x y | tri< a ¬b ¬c = ordtrans a <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
166 omax-x x y | tri> ¬a ¬b c = <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
167 omax-x x y | tri≈ ¬a refl ¬c = <-osuc
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
168
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
169 omax-y : ( x y : Ordinal ) → y o< omax x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
170 omax-y x y with trio< x y
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
171 omax-y x y | tri< a ¬b ¬c = <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
172 omax-y x y | tri> ¬a ¬b c = ordtrans c <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
173 omax-y x y | tri≈ ¬a refl ¬c = <-osuc
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
174
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
175 omxx : ( x : Ordinal ) → omax x x ≡ osuc x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
176 omxx x with trio< x x
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
177 omxx x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
178 omxx x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
179 omxx x | tri≈ ¬a refl ¬c = refl
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
180
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
181 omxxx : ( x : Ordinal ) → omax x (omax x x ) ≡ osuc (osuc x)
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
182 omxxx x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
183
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
184 open _∧_
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
185
326
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
186 o≤-refl : { i j : Ordinal } → i ≡ j → i o≤ j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
187 o≤-refl {i} {j} eq = subst (λ k → i o< osuc k ) eq <-osuc
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
188 OrdTrans : Transitive _o≤_
326
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
189 OrdTrans a≤b b≤c with osuc-≡< a≤b | osuc-≡< b≤c
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
190 OrdTrans a≤b b≤c | case1 refl | case1 refl = <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
191 OrdTrans a≤b b≤c | case1 refl | case2 a≤c = ordtrans a≤c <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
192 OrdTrans a≤b b≤c | case2 a≤c | case1 refl = ordtrans a≤c <-osuc
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
193 OrdTrans a≤b b≤c | case2 a<b | case2 b<c = ordtrans (ordtrans a<b b<c) <-osuc
97
f2b579106770 power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 94
diff changeset
194
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
195 OrdPreorder : Preorder n n n
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
196 OrdPreorder = record { Carrier = Ordinal
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
197 ; _≈_ = _≡_
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
198 ; _∼_ = _o≤_
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
199 ; isPreorder = record {
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
200 isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
326
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 324
diff changeset
201 ; reflexive = o≤-refl
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
202 ; trans = OrdTrans
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
203 }
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
204 }
165
d16b8bf29f4f minor fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 158
diff changeset
205
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
206 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
207 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
208 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
209 → ¬ p
258
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 235
diff changeset
210 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
221
1709c80b7275 fix Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 220
diff changeset
211
309
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
212 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
213 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
214 os→ : (x : Ordinal) → x o< maxordinal → Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
215 os← : Ordinal → Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
216 os←limit : (x : Ordinal) → os← x o< maxordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
217 os-iso← : {x : Ordinal} → os→ (os← x) (os←limit x) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
218 os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) → os← (os→ x lt) ≡ x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 308
diff changeset
219