1091
|
1 {-# OPTIONS --allow-unsolved-metas #-}
|
431
|
2 open import Level
|
|
3 open import Ordinals
|
|
4 module OD {n : Level } (O : Ordinals {n} ) where
|
|
5
|
1091
|
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
|
431
|
7 open import Relation.Binary.PropositionalEquality hiding ( [_] )
|
1091
|
8 open import Data.Nat.Properties
|
431
|
9 open import Data.Empty
|
1175
|
10 open import Data.Unit
|
431
|
11 open import Relation.Nullary
|
|
12 open import Relation.Binary hiding (_⇔_)
|
|
13 open import Relation.Binary.Core hiding (_⇔_)
|
|
14
|
|
15 open import logic
|
|
16 import OrdUtil
|
|
17 open import nat
|
|
18
|
|
19 open Ordinals.Ordinals O
|
1091
|
20 open Ordinals.IsOrdinals isOrdinal
|
|
21 open Ordinals.IsNext isNext
|
431
|
22 open OrdUtil O
|
|
23
|
|
24 -- Ordinal Definable Set
|
|
25
|
|
26 record OD : Set (suc n ) where
|
|
27 field
|
|
28 def : (x : Ordinal ) → Set n
|
|
29
|
|
30 open OD
|
|
31
|
|
32 open _∧_
|
|
33 open _∨_
|
|
34 open Bool
|
|
35
|
|
36 record _==_ ( a b : OD ) : Set n where
|
|
37 field
|
1091
|
38 eq→ : ∀ { x : Ordinal } → def a x → def b x
|
|
39 eq← : ∀ { x : Ordinal } → def b x → def a x
|
431
|
40
|
|
41 ==-refl : { x : OD } → x == x
|
|
42 ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x }
|
|
43
|
1091
|
44 open _==_
|
431
|
45
|
|
46 ==-trans : { x y z : OD } → x == y → y == z → x == z
|
|
47 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) }
|
|
48
|
1091
|
49 ==-sym : { x y : OD } → x == y → y == x
|
431
|
50 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
|
|
51
|
|
52
|
1091
|
53 ⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y
|
|
54 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
|
|
55 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
|
431
|
56
|
|
57 -- next assumptions are our axiom
|
|
58 --
|
|
59 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
|
|
60 -- correspondence to the OD then the OD looks like a ZF Set.
|
|
61 --
|
|
62 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
|
|
63 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
|
|
64 --
|
1091
|
65 -- In classical Set Theory, HOD is used, as a subset of OD,
|
431
|
66 -- HOD = { x | TC x ⊆ OD }
|
|
67 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
|
|
68 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
|
|
69 --
|
|
70 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
|
|
71 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
|
|
72 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
|
|
73 -- bound on each HOD.
|
|
74 --
|
|
75 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
|
1285
|
76 -- we need explict assumption on sup for unrestricted Replacement.
|
431
|
77 --
|
|
78 -- ==→o≡ is necessary to prove axiom of extensionality.
|
|
79
|
|
80 -- Ordinals in OD , the maximum
|
|
81 Ords : OD
|
1175
|
82 Ords = record { def = λ x → Lift n ⊤ }
|
431
|
83
|
|
84 record HOD : Set (suc n) where
|
|
85 field
|
|
86 od : OD
|
|
87 odmax : Ordinal
|
|
88 <odmax : {y : Ordinal} → def od y → y o< odmax
|
|
89
|
|
90 open HOD
|
|
91
|
1218
|
92 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
|
|
93
|
1091
|
94 record ODAxiom : Set (suc n) where
|
431
|
95 field
|
|
96 -- HOD is isomorphic to Ordinal (by means of Goedel number)
|
1091
|
97 & : HOD → Ordinal
|
|
98 * : Ordinal → HOD
|
431
|
99 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
|
|
100 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
|
|
101 *iso : {x : HOD } → * ( & x ) ≡ x
|
|
102 &iso : {x : Ordinal } → & ( * x ) ≡ x
|
|
103 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
|
1218
|
104 ∋-irr : {X : HOD} {x : Ordinal } → (a b : def (od X) x) → a ≅ b
|
1284
|
105
|
431
|
106 postulate odAxiom : ODAxiom
|
|
107 open ODAxiom odAxiom
|
|
108
|
1285
|
109 -- possible order restriction (required in the axiom of infinite )
|
|
110
|
|
111 record ODAxiom-ho< : Set (suc n) where
|
|
112 field
|
|
113 ho< : {x : HOD} → & x o< next (odmax x)
|
|
114
|
|
115 postulate odAxiom-ho< : ODAxiom-ho<
|
|
116 open ODAxiom-ho< odAxiom-ho<
|
|
117
|
431
|
118 -- odmax minimality
|
|
119 --
|
|
120 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
|
|
121 -- We can calculate the minimum using sup but it is tedius.
|
1091
|
122 -- Only Select has non minimum odmax.
|
431
|
123 -- We have the same problem on 'def' itself, but we leave it.
|
|
124
|
|
125 odmaxmin : Set (suc n)
|
|
126 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
|
|
127
|
|
128 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
|
|
129 ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
|
1175
|
130 ¬OD-order & * c<→o< = o≤> <-osuc (c<→o< {Ords} (lift tt) )
|
431
|
131
|
|
132 -- Ordinal in OD ( and ZFSet ) Transitive Set
|
1091
|
133 Ord : ( a : Ordinal ) → HOD
|
431
|
134 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
|
|
135 lemma : {x : Ordinal} → x o< a → x o< a
|
|
136 lemma {x} lt = lt
|
|
137
|
1091
|
138 od∅ : HOD
|
|
139 od∅ = Ord o∅
|
431
|
140
|
|
141 odef : HOD → Ordinal → Set n
|
|
142 odef A x = def ( od A ) x
|
|
143
|
|
144 _∋_ : ( a x : HOD ) → Set n
|
|
145 _∋_ a x = odef a ( & x )
|
|
146
|
|
147 -- _c<_ : ( x a : HOD ) → Set n
|
1091
|
148 -- x c< a = a ∋ x
|
431
|
149
|
|
150 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x)
|
|
151 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
|
|
152
|
|
153 -- odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
|
|
154 -- odef-subst df refl refl = df
|
|
155
|
|
156 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
|
|
157 otrans x<a y<x = ordtrans y<x x<a
|
|
158
|
|
159 -- If we have reverse of c<→o<, everything becomes Ordinal
|
|
160 ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x)
|
|
161 ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
|
|
162 lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
|
|
163 lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
|
|
164 lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
|
|
165 lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
|
|
166
|
|
167 -- avoiding lv != Zero error
|
|
168 orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y
|
|
169 orefl refl = refl
|
|
170
|
|
171 ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y
|
|
172 ==-iso {x} {y} eq = record {
|
1091
|
173 eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ;
|
|
174 eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) }
|
431
|
175 where
|
|
176 lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z
|
1091
|
177 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
|
431
|
178
|
|
179 =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y)
|
|
180 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
|
|
181
|
|
182 ord→== : { x y : HOD } → & x ≡ & y → od x == od y
|
|
183 ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
|
|
184 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy)
|
|
185 lemma ox ox refl = ==-refl
|
|
186
|
|
187 o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y)
|
|
188 o≡→== {x} {.x} refl = ==-refl
|
|
189
|
556
|
190 *≡*→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
|
|
191 *≡*→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) eq )
|
|
192
|
|
193 &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
|
|
194 &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq )
|
|
195
|
1091
|
196 o∅≡od∅ : * (o∅ ) ≡ od∅
|
431
|
197 o∅≡od∅ = ==→o≡ lemma where
|
|
198 lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x
|
|
199 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
|
|
200 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
|
|
201 lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x
|
|
202 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
|
|
203 lemma : od (* o∅) == od od∅
|
|
204 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
|
|
205
|
1091
|
206 ord-od∅ : & (od∅ ) ≡ o∅
|
431
|
207 ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
|
|
208
|
|
209 ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅
|
|
210 ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt))))
|
1091
|
211 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
|
431
|
212
|
1091
|
213 =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅
|
431
|
214 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
|
|
215
|
448
|
216 ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅
|
|
217 ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) )
|
|
218
|
1091
|
219 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
|
431
|
220 eq→ ∅0 {w} (lift ())
|
|
221 eq← ∅0 {w} lt = lift (¬x<0 lt)
|
|
222
|
|
223 ∅< : { x y : HOD } → odef x (& y ) → ¬ ( od x == od od∅ )
|
|
224 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
|
|
225 ∅< {x} {y} d eq | lift ()
|
450
|
226
|
1223
|
227 ¬x∋y→x≡od∅ : { x : HOD } → ({y : Ordinal } → ¬ odef x y ) → x ≡ od∅
|
|
228 ¬x∋y→x≡od∅ {x} nxy = ==→o≡ record { eq→ = λ {y} lt → ⊥-elim (nxy lt) ; eq← = λ {y} lt → ⊥-elim (¬x<0 lt) }
|
|
229
|
1148
|
230 0<P→ne : { x : HOD } → o∅ o< & x → ¬ ( od x == od od∅ )
|
|
231 0<P→ne {x} 0<x eq = ⊥-elim ( o<¬≡ (sym (=od∅→≡o∅ eq)) 0<x )
|
|
232
|
688
|
233 ∈∅< : { x : HOD } {y : Ordinal } → odef x y → o∅ o< (& x)
|
|
234 ∈∅< {x} {y} d with trio< o∅ (& x)
|
|
235 ... | tri< a ¬b ¬c = a
|
|
236 ... | tri≈ ¬a b ¬c = ⊥-elim ( ∅< {x} {* y} (subst (λ k → odef x k ) (sym &iso) d ) ( ≡o∅→=od∅ (sym b) ) )
|
|
237 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
|
|
238
|
431
|
239 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
|
|
240 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
|
|
241
|
|
242 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
|
|
243 odef-iso refl t = t
|
|
244
|
|
245 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
|
|
246 is-o∅ x with trio< x o∅
|
|
247 is-o∅ x | tri< a ¬b ¬c = no ¬b
|
|
248 is-o∅ x | tri≈ ¬a b ¬c = yes b
|
|
249 is-o∅ x | tri> ¬a ¬b c = no ¬b
|
|
250
|
1091
|
251 odef< : {b : Ordinal } { A : HOD } → odef A b → b o< & A
|
|
252 odef< {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
|
|
253
|
|
254 odef∧< : {A : HOD } {y : Ordinal} {n : Level } → {P : Set n} → odef A y ∧ P → y o< & A
|
|
255 odef∧< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
|
|
256
|
431
|
257 -- the pair
|
1091
|
258 _,_ : HOD → HOD → HOD
|
431
|
259 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where
|
|
260 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
|
|
261 lemma {t} (case1 refl) = omax-x _ _
|
|
262 lemma {t} (case2 refl) = omax-y _ _
|
|
263
|
1286
|
264 -- {x y : HOD} → & (x , y) ≡ omax (& x) (& y)
|
|
265
|
431
|
266 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
|
|
267 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
|
|
268 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
|
|
269 lemma (case1 refl) = y∋x
|
|
270 lemma (case2 refl) = y∋x
|
|
271
|
688
|
272 -- another possible restriction. We require no minimality on odmax, so it may arbitrary larger.
|
431
|
273 odmax<& : { x y : HOD } → x ∋ y → Set n
|
|
274 odmax<& {x} {y} x∋y = odmax x o< & x
|
|
275
|
1091
|
276 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
|
431
|
277 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) }
|
|
278
|
1091
|
279 _∩_ : ( A B : HOD ) → HOD
|
431
|
280 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
|
|
281 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
|
|
282
|
1096
|
283 _⊆_ : ( A B : HOD) → Set n
|
|
284 _⊆_ A B = { x : Ordinal } → odef A x → odef B x
|
431
|
285
|
|
286 infixr 220 _⊆_
|
|
287
|
|
288 -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o<
|
|
289 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
|
|
290 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
|
1091
|
291 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
|
431
|
292 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
|
|
293 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
|
|
294 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x )))
|
|
295 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
|
|
296 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
|
|
297 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
|
|
298 lemma (case1 refl) = refl
|
|
299 lemma (case2 refl) = refl
|
|
300 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
|
1091
|
301 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
|
431
|
302 lemma1 : osuc (& y) o< & (x , x)
|
1091
|
303 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
|
431
|
304
|
|
305 ε-induction : { ψ : HOD → Set (suc n)}
|
|
306 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
|
|
307 → (x : HOD ) → ψ x
|
|
308 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
|
|
309 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
|
1091
|
310 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
|
431
|
311 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
|
|
312 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
|
|
313
|
1109
|
314 ε-induction0 : { ψ : HOD → Set n}
|
|
315 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
|
|
316 → (x : HOD ) → ψ x
|
|
317 ε-induction0 {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
|
|
318 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
|
|
319 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
|
|
320 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
|
|
321 ε-induction-ord ox {oy} lt = inOrdinal.TransFinite0 O {λ oy → ψ (* oy)} induction oy
|
|
322
|
1091
|
323 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
|
|
324 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
|
|
325 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
|
|
326 next-ord : Ordinal → Ordinal
|
|
327 next-ord x = osuc x
|
|
328
|
|
329 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
|
431
|
330 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
|
|
331
|
1095
|
332 _=h=_ : (x y : HOD) → Set n
|
|
333 x =h= y = od x == od y
|
|
334
|
|
335 record Own (A : HOD) (x : Ordinal) : Set n where
|
|
336 field
|
|
337 owner : Ordinal
|
|
338 ao : odef A owner
|
|
339 ox : odef (* owner) x
|
|
340
|
|
341 Union : HOD → HOD
|
|
342 Union U = record { od = record { def = λ x → Own U x } ; odmax = osuc (& U) ; <odmax = umax } where
|
|
343 umax : {y : Ordinal} → Own U y → y o< osuc (& U)
|
|
344 umax {y} uy = o<→≤ ( ordtrans (odef< (Own.ox uy)) (subst (λ k → k o< & U) (sym &iso) umax1) ) where
|
|
345 umax1 : Own.owner uy o< & U
|
|
346 umax1 = odef< (Own.ao uy)
|
|
347
|
|
348 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
349 union→ X z u xx = record { owner = & u ; ao = proj1 xx ; ox = subst (λ k → odef k (& z)) (sym *iso) (proj2 xx) }
|
|
350 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
351 union← X z UX∋z not = ⊥-elim ( not (* (Own.owner UX∋z)) ⟪ subst (λ k → odef X k) (sym &iso) ( Own.ao UX∋z) , Own.ox UX∋z ⟫ )
|
|
352
|
1285
|
353 record RCod (COD : HOD) (ψ : HOD → HOD) : Set (suc n) where
|
|
354 field
|
|
355 ≤COD : ∀ {x : HOD } → ψ x ⊆ COD
|
|
356
|
1095
|
357 record Replaced (A : HOD) (ψ : Ordinal → Ordinal ) (x : Ordinal ) : Set n where
|
|
358 field
|
|
359 z : Ordinal
|
|
360 az : odef A z
|
|
361 x=ψz : x ≡ ψ z
|
|
362
|
1285
|
363 Replace : (D : HOD) → (ψ : HOD → HOD) → {C : HOD} → RCod C ψ → HOD
|
|
364 Replace X ψ {C} rc = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = osuc (& C)
|
|
365 ; <odmax = rmax< } where
|
|
366 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< osuc (& C)
|
|
367 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RCod.≤COD rc) ) where
|
1095
|
368 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
|
|
369 r01 = sym (Replaced.x=ψz lt )
|
|
370
|
1285
|
371 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → {C : HOD} → (rc : RCod C ψ) → Replace X ψ rc ∋ ψ x
|
|
372 replacement← {ψ} X x lt {C} rc = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
|
|
373 replacement→ : {ψ : HOD → HOD} (X x : HOD) → {C : HOD} → (rc : RCod C ψ ) → (lt : Replace X ψ rc ∋ x)
|
|
374 → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
|
|
375 replacement→ {ψ} X x {C} rc lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
|
431
|
376
|
|
377 --
|
1091
|
378 -- If we have LEM, Replace' is equivalent to Replace
|
431
|
379 --
|
1095
|
380
|
1285
|
381 record RXCod (X COD : HOD) (ψ : (x : HOD) → X ∋ x → HOD) : Set (suc n) where
|
|
382 field
|
|
383 ≤COD : ∀ {x : HOD } → (lt : X ∋ x) → ψ x lt ⊆ COD
|
|
384
|
1095
|
385 record Replaced1 (A : HOD) (ψ : (x : Ordinal ) → odef A x → Ordinal ) (x : Ordinal ) : Set n where
|
|
386 field
|
|
387 z : Ordinal
|
|
388 az : odef A z
|
|
389 x=ψz : x ≡ ψ z az
|
431
|
390
|
1285
|
391 Replace' : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → RXCod X C ψ → HOD
|
|
392 Replace' X ψ {C} rc = record { od = record { def = λ x → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) x } ; odmax = osuc (& C) ; <odmax = rmax< } where
|
|
393 rmax< : {y : Ordinal} → Replaced1 X (λ z xz → & (ψ (* z) (subst (λ k → odef X k) (sym &iso) xz) )) y → y o< osuc (& C)
|
|
394 rmax< {y} lt = subst (λ k → k o< osuc (& C)) r01 ( ⊆→o≤ (RXCod.≤COD rc (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) ))) where
|
1095
|
395 r01 : & (ψ ( * (Replaced1.z lt ) ) (subst (λ k → odef X k) (sym &iso) (Replaced1.az lt) )) ≡ y
|
|
396 r01 = sym (Replaced1.x=ψz lt )
|
|
397
|
1285
|
398 cod-conv : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
|
|
399 → RXCod (* (& X)) C (λ y xy → ψ y (subst (λ k → k ∋ y) *iso xy))
|
|
400 cod-conv X ψ {C} rc = record { ≤COD = λ {x} lt → RXCod.≤COD rc (subst (λ k → odef k (& x)) *iso lt) }
|
1218
|
401
|
1285
|
402 Replace'-iso : (X : HOD) → (ψ : (x : HOD) → X ∋ x → HOD) → {C : HOD} → (rc : RXCod X C ψ )
|
|
403 → Replace' (* (& X)) (λ y xy → ψ y (subst (λ k → k ∋ y ) *iso xy) ) (cod-conv X ψ rc)
|
|
404 ≡ Replace' X ( λ y xy → ψ y xy ) rc
|
|
405 Replace'-iso X ψ rc = ==→o≡ record { eq→ = ri0 ; eq← = ri1 } where
|
1218
|
406 ri2 : {z : Ordinal } (a b : X ∋ (* z)) → & (ψ (* z) a) ≡ & (ψ (* z) b)
|
|
407 ri2 {z} a b = cong (λ k → & (ψ (* z) k)) ( HE.≅-to-≡ ( ∋-irr {X} {& (* z)} a b ) )
|
|
408 ri0 : {x : Ordinal}
|
|
409 → Replaced1 (* (& X)) (λ z xz → & (ψ (* z) (subst (λ k → k ∋ * z) *iso (subst (odef (* (& X))) (sym &iso) xz)))) x
|
|
410 → Replaced1 X (λ z xz → & (ψ (* z) (subst (odef X) (sym &iso) xz))) x
|
|
411 ri0 {x} record { z = z ; az = az ; x=ψz = refl } = record { z = z ; az = subst (λ k → odef k z) *iso az
|
|
412 ; x=ψz = ri2 (subst (λ k → k ∋ * z) *iso (subst (odef (* (& X))) (sym &iso) az))
|
|
413 (subst (odef X) (sym &iso) (subst (λ k → odef k z) *iso az) ) }
|
|
414 ri1 : {x : Ordinal}
|
|
415 → Replaced1 X (λ z xz → & (ψ (* z) (subst (odef X) (sym &iso) xz))) x
|
|
416 → Replaced1 (* (& X)) (λ z xz → & (ψ (* z) (subst (λ k → k ∋ * z) *iso (subst (odef (* (& X))) (sym &iso) xz)))) x
|
|
417 ri1 {x} record { z = z ; az = az ; x=ψz = refl } = record { z = z ; az = subst (λ k → odef k z) (sym *iso) az
|
|
418 ; x=ψz = ri2 (subst (λ k → odef X k) (sym &iso) az ) -- brain damaged below
|
|
419 (subst (λ k → k ∋ * z) *iso (subst (odef (* (& X))) (sym &iso) (subst (λ k → odef k z) (sym *iso) az))) }
|
|
420
|
|
421
|
1095
|
422 -- replacement←1 : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace1 X ψ ∋ ψ x
|
|
423 -- replacement←1 {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
|
|
424 -- replacement→1 : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace1 X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
|
|
425 -- replacement→1 {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
|
|
426
|
431
|
427 _∈_ : ( A B : HOD ) → Set n
|
|
428 A ∈ B = B ∋ A
|
|
429
|
1095
|
430 Power : HOD → HOD
|
|
431 Power A = record { od = record { def = λ x → ( ( z : Ordinal) → odef (* x) z → odef A z ) } ; odmax = osuc (& A)
|
|
432 ; <odmax = p00 } where
|
|
433 p00 : {y : Ordinal} → ((z : Ordinal) → odef (* y) z → odef A z) → y o< osuc (& A)
|
|
434 p00 {y} y⊆A = p01 where
|
|
435 p01 : y o≤ & A
|
|
436 p01 = subst (λ k → k o≤ & A) &iso ( ⊆→o≤ (λ {x} yx → y⊆A x yx ))
|
431
|
437
|
1095
|
438 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
|
|
439 power→ A t P∋t {x} t∋x = P∋t (& x) (subst (λ k → odef k (& x) ) (sym *iso) t∋x )
|
|
440
|
|
441 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
442 power← A t t⊆A z xz = subst (λ k → odef A k ) &iso ( t⊆A (subst₂ (λ j k → odef j k) *iso (sym &iso) xz ))
|
|
443
|
1180
|
444 Intersection : (X : HOD ) → HOD -- ∩ X
|
1186
|
445 Intersection X = record { od = record { def = λ x → (x o≤ & X ) ∧ ( {y : Ordinal} → odef X y → odef (* y) x )} ; odmax = osuc (& X) ; <odmax = λ lt → proj1 lt }
|
1180
|
446
|
|
447
|
431
|
448 -- {_} : ZFSet → ZFSet
|
|
449 -- { x } = ( x , x ) -- better to use (x , x) directly
|
|
450
|
|
451
|
|
452 data infinite-d : ( x : Ordinal ) → Set n where
|
|
453 iφ : infinite-d o∅
|
|
454 isuc : {x : Ordinal } → infinite-d x →
|
|
455 infinite-d (& ( Union (* x , (* x , * x ) ) ))
|
|
456
|
|
457 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
|
|
458 -- We simply assumes infinite-d y has a maximum.
|
1091
|
459 --
|
431
|
460 -- This means that many of OD may not be HODs because of the & mapping divergence.
|
|
461 -- We should have some axioms to prevent this such as & x o< next (odmax x).
|
1091
|
462 --
|
1097
|
463 -- Since we have Ord (next o∅), we don't need this, but ZF axiom requires this and ho<
|
431
|
464
|
1286
|
465 infinite-od : OD
|
|
466 infinite-od = record { def = λ x → infinite-d x }
|
|
467
|
|
468 postulate
|
|
469 infinite-odmax : Ordinal
|
|
470 infinite-odmax< : {z : Ordinal } → def infinite-od z → z o< infinite-odmax
|
|
471
|
1091
|
472 infinite : HOD
|
1286
|
473 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = infinite-odmax ; <odmax = infinite-odmax< }
|
|
474
|
|
475 -- where
|
|
476 -- u : (y : Ordinal ) → HOD
|
|
477 -- u y = Union (* y , (* y , * y))
|
|
478 -- -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
|
|
479 -- lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y))
|
|
480 -- lemma8 = ho<
|
|
481 -- --- (x,y) < next (omax x y) < next (osuc y) = next y
|
|
482 -- lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y)
|
|
483 -- lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
|
|
484 -- lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y))
|
|
485 -- lemma81 {y} = nexto=n (subst (λ k → & (* y , * y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
|
|
486 -- lemma9 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y , * y))
|
|
487 -- lemma9 = lemmaa (c<→o< (case1 refl))
|
|
488 -- lemma71 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y))
|
|
489 -- lemma71 = ? -- next< lemma81 lemma9
|
|
490 -- lemma1 : {y : Ordinal} → & (u y) o< next (osuc (& (* y , (* y , * y))))
|
|
491 -- lemma1 = ho<
|
|
492 -- --- main recursion
|
|
493 -- lemma : {y : Ordinal} → infinite-d y → y o< next o∅
|
|
494 -- lemma {o∅} iφ = x<nx
|
|
495 -- lemma (isuc {y} x) = ? -- next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1))
|
431
|
496
|
|
497 empty : (x : HOD ) → ¬ (od∅ ∋ x)
|
1091
|
498 empty x = ¬x<0
|
431
|
499
|
1091
|
500 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
|
431
|
501 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
|
|
502 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
|
|
503
|
1091
|
504 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
|
431
|
505 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
|
|
506 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
|
|
507
|
1091
|
508 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
|
1096
|
509 o<→c< lt {z} ox = ordtrans ox lt
|
431
|
510
|
|
511 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
|
1091
|
512 ⊆→o< {x} {y} lt with trio< x y
|
431
|
513 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
|
|
514 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
|
1096
|
515 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (o<-subst c (sym &iso) refl )
|
431
|
516 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
|
|
517
|
|
518 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
|
|
519 ψiso {ψ} t refl = t
|
|
520 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
|
|
521 selection {ψ} {X} {y} = ⟪
|
|
522 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ )
|
|
523 , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ )
|
|
524 ⟫
|
|
525
|
1091
|
526 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
|
431
|
527 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
|
|
528
|
|
529 ---
|
|
530 --- Power Set
|
|
531 ---
|
|
532 --- First consider ordinals in HOD
|
|
533 ---
|
|
534 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
|
|
535 --
|
|
536 --
|
|
537 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
|
|
538 ∩-≡ {a} {b} inc = record {
|
|
539 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ;
|
|
540 eq← = λ {x} x<a∩b → proj2 x<a∩b }
|
|
541
|
|
542 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
|
1091
|
543 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d
|
|
544 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d
|
431
|
545
|
|
546 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
|
|
547 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
|
1091
|
548 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
|
431
|
549
|
1091
|
550 infinity∅ : infinite ∋ od∅
|
|
551 infinity∅ = subst (λ k → odef infinite k ) lemma iφ where
|
431
|
552 lemma : o∅ ≡ & od∅
|
|
553 lemma = let open ≡-Reasoning in begin
|
|
554 o∅
|
|
555 ≡⟨ sym &iso ⟩
|
|
556 & ( * o∅ )
|
|
557 ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
|
|
558 & od∅
|
|
559 ∎
|
|
560 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
|
|
561 infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where
|
|
562 lemma : & (Union (* (& x) , (* (& x) , * (& x))))
|
|
563 ≡ & (Union (x , (x , x)))
|
1091
|
564 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
|
431
|
565
|
1284
|
566 open import zf
|
|
567
|
1285
|
568 record ODAxiom-sup : Set (suc n) where
|
|
569 field
|
|
570 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal -- required in Replace
|
|
571 sup-o≤ : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal }
|
|
572 → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤ sup-o A ψ
|
|
573 sup-c≤ : (ψ : HOD → HOD) → {X x : HOD} → def (od X) (& x) → & (ψ x) o≤ (sup-o X (λ y X∋y → & (ψ (* y))))
|
|
574 sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o≤ X lt )
|
|
575
|
|
576 -- sup-o may contradict
|
|
577 -- If we have open monotonic function in Ordinal, there is no sup-o.
|
|
578 -- for example, if we may have countable sequence of Ordinal, which contains some ordinal larger than any given Ordinal.
|
|
579 -- This happens when we have a coutable model. In this case, we have to have codomain restriction in Replacement axiom.
|
|
580 -- that is, Replacement axiom does not create new ZF set.
|
|
581
|
|
582 open ODAxiom-sup
|
|
583
|
|
584 ZFReplace : ODAxiom-sup → HOD → (HOD → HOD) → HOD
|
|
585 ZFReplace os X ψ = record { od = record { def = λ x → Replaced X (λ z → & (ψ (* z))) x } ; odmax = rmax ; <odmax = rmax< } where
|
|
586 rmax : Ordinal
|
|
587 rmax = osuc ( sup-o os X (λ y X∋y → & (ψ (* y) )) )
|
|
588 rmax< : {y : Ordinal} → Replaced X (λ z → & (ψ (* z))) y → y o< rmax
|
|
589 rmax< {y} lt = subst (λ k → k o< rmax) r01 ( sup-o≤ os X (Replaced.az lt) ) where
|
|
590 r01 : & (ψ ( * (Replaced.z lt ) )) ≡ y
|
|
591 r01 = sym (Replaced.x=ψz lt )
|
|
592
|
|
593 zf-replacement← : (os : ODAxiom-sup) → {ψ : HOD → HOD} (X x : HOD) → X ∋ x → ZFReplace os X ψ ∋ ψ x
|
|
594 zf-replacement← os {ψ} X x lt = record { z = & x ; az = lt ; x=ψz = cong (λ k → & (ψ k)) (sym *iso) }
|
|
595 zf-replacement→ : (os : ODAxiom-sup ) → {ψ : HOD → HOD} (X x : HOD) → (lt : ZFReplace os X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
|
|
596 zf-replacement→ os {ψ} X x lt eq = eq (* (Replaced.z lt)) (ord→== (Replaced.x=ψz lt))
|
|
597
|
|
598 isZF : (os : ODAxiom-sup) → IsZF HOD _∋_ _=h=_ od∅ _,_ Union Power Select (ZFReplace os) infinite
|
|
599 isZF os = record {
|
431
|
600 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
|
|
601 ; pair→ = pair→
|
|
602 ; pair← = pair←
|
|
603 ; union→ = union→
|
|
604 ; union← = union←
|
|
605 ; empty = empty
|
1091
|
606 ; power→ = power→
|
|
607 ; power← = power←
|
|
608 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
|
431
|
609 ; ε-induction = ε-induction
|
|
610 ; infinity∅ = infinity∅
|
|
611 ; infinity = infinity
|
|
612 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
|
1285
|
613 ; replacement← = zf-replacement← os
|
|
614 ; replacement→ = λ {ψ} → zf-replacement→ os {ψ}
|
1091
|
615 }
|
431
|
616
|
1285
|
617 HOD→ZF : ODAxiom-sup → ZF
|
|
618 HOD→ZF os = record {
|
1091
|
619 ZFSet = HOD
|
|
620 ; _∋_ = _∋_
|
|
621 ; _≈_ = _=h=_
|
431
|
622 ; ∅ = od∅
|
|
623 ; _,_ = _,_
|
|
624 ; Union = Union
|
|
625 ; Power = Power
|
|
626 ; Select = Select
|
1285
|
627 ; Replace = ZFReplace os
|
431
|
628 ; infinite = infinite
|
1285
|
629 ; isZF = isZF os
|
1091
|
630 }
|
431
|
631
|
1091
|
632
|