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¬odmax based HOD
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 29 Jun 2020 17:56:06 +0900
parents 304c271b3d47
children 2c111bfcc89a
comparison
equal deleted inserted replaced
302:304c271b3d47 303:7963b76df6e1
65 -- ==→o≡ is necessary to prove axiom of extensionality. 65 -- ==→o≡ is necessary to prove axiom of extensionality.
66 -- 66 --
67 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic, 67 -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
68 -- we need explict assumption on sup. 68 -- we need explict assumption on sup.
69 69
70 record HOD (odmax : Ordinal) : Set (suc n) where 70 data One : Set n where
71 OneObj : One
72
73 -- Ordinals in OD , the maximum
74 Ords : OD
75 Ords = record { def = λ x → One }
76
77 record HOD : Set (suc n) where
71 field 78 field
72 hmax : {x : Ordinal } → x o< odmax 79 od : OD
73 hdef : Ordinal → Set n 80 ¬odmax : ¬ (od ≡ Ords)
74 81
75 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where 82 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
76 field 83 field
77 os→ : (x : Ordinal) → x o< maxordinal → Ordinal 84 os→ : (x : Ordinal) → x o< maxordinal → Ordinal
78 os← : Ordinal → Ordinal 85 os← : Ordinal → Ordinal
86 -- HOD→OD hod = record { def = hdef {!!} } 93 -- HOD→OD hod = record { def = hdef {!!} }
87 94
88 record ODAxiom : Set (suc n) where 95 record ODAxiom : Set (suc n) where
89 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) 96 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
90 field 97 field
91 od→ord : OD → Ordinal 98 od→ord : HOD → Ordinal
92 ord→od : Ordinal → OD 99 ord→od : Ordinal → HOD
93 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y 100 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
94 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x 101 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
95 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x 102 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
96 ==→o≡ : { x y : OD } → (x == y) → x ≡ y 103 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
97 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) 104 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
98 sup-o : ( OD → Ordinal ) → Ordinal 105 sup-od : ( HOD → HOD ) → HOD
99 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ 106 sup-c< : ( ψ : HOD → HOD ) → ∀ {x : HOD } → def (od ( sup-od ψ )) (od→ord ( ψ x ))
100 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use 107 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
101 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal 108 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
102 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) 109 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
103 110
104 record HODAxiom : Set (suc n) where
105 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
106 field
107 mod : Ordinal
108 mod-limit : ¬ ((y : Ordinal) → mod ≡ osuc y)
109 os : OrdinalSubset mod
110 od→ord : HOD mod → Ordinal
111 ord→od : Ordinal → HOD mod
112 c<→o< : {x y : HOD mod } → hdef y (od→ord x) → od→ord x o< od→ord y
113 oiso : {x : HOD mod } → ord→od ( od→ord x ) ≡ x
114 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
115 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
116 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
117 sup-o : ( HOD mod → Ordinal ) → Ordinal
118 sup-o< : { ψ : HOD mod → Ordinal } → ∀ {x : HOD mod } → ψ x o< sup-o ψ
119 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
120 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
121 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
122
123 postulate odAxiom : ODAxiom 111 postulate odAxiom : ODAxiom
124 open ODAxiom odAxiom 112 open ODAxiom odAxiom
125 113
126 data One : Set n where 114 -- maxod : {x : OD} → od→ord x o< od→ord Ords
127 OneObj : One 115 -- maxod {x} = c<→o< OneObj
128
129 -- Ordinals in OD , the maximum
130 Ords : OD
131 Ords = record { def = λ x → One }
132
133 maxod : {x : OD} → od→ord x o< od→ord Ords
134 maxod {x} = c<→o< OneObj
135 116
136 -- we have to avoid this contradiction 117 -- we have to avoid this contradiction
137 118
138 bad-bad : ⊥ 119 -- bad-bad : ⊥
139 bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj) 120 -- bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj)
140 121
141 -- Ordinal in OD ( and ZFSet ) Transitive Set 122 -- Ordinal in OD ( and ZFSet ) Transitive Set
142 Ord : ( a : Ordinal ) → OD 123 Ord : ( a : Ordinal ) → HOD
143 Ord a = record { def = λ y → y o< a } 124 Ord a = record { od = record { def = λ y → y o< a } ; ¬odmax = ? }
144 125
145 od∅ : OD 126 od∅ : HOD
146 od∅ = Ord o∅ 127 od∅ = Ord o∅
147 128
148 129 sup-o : ( HOD → Ordinal ) → Ordinal
149 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) 130 sup-o = ?
150 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 131 sup-o< : { ψ : HOD → Ordinal } → ∀ {x : HOD } → ψ x o< sup-o ψ
151 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y 132 sup-o< = ?
152 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt)) 133
153 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y 134 odef : HOD → Ordinal → Set n
154 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) 135 odef A x = def ( od A ) x
155 136
156 _∋_ : ( a x : OD ) → Set n 137 o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x)
157 _∋_ a x = def a ( od→ord x ) 138 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
158 139 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y
159 _c<_ : ( x a : OD ) → Set n 140 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → odef x k ) (sym diso) lt))
141 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y
142 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt )
143
144 _∋_ : ( a x : HOD ) → Set n
145 _∋_ a x = odef a ( od→ord x )
146
147 _c<_ : ( x a : HOD ) → Set n
160 x c< a = a ∋ x 148 x c< a = a ∋ x
161 149
162 cseq : {n : Level} → OD → OD 150 cseq : {n : Level} → HOD → HOD
163 cseq x = record { def = λ y → def x (osuc y) } where 151 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; ¬odmax = ? } where
164 152
165 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x 153 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
166 def-subst df refl refl = df 154 odef-subst df refl refl = df
167 155
168 sup-od : ( OD → OD ) → OD 156 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
169 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )
170
171 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
172 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )}
173 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
174 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x))
175 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) )
176
177 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
178 otrans x<a y<x = ordtrans y<x x<a 157 otrans x<a y<x = ordtrans y<x x<a
179 158
180 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X 159 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X
181 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso 160 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso
182 161
183 162
184 -- avoiding lv != Zero error 163 -- avoiding lv != Zero error
185 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y 164 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
186 orefl refl = refl 165 orefl refl = refl
187 166
188 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y 167 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y
189 ==-iso {x} {y} eq = record { 168 ==-iso {x} {y} eq = record {
190 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; 169 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ;
191 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } 170 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) }
192 where 171 where
193 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z 172 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z
194 lemma {x} {z} d = def-subst d oiso refl 173 lemma {x} {z} d = odef-subst d oiso refl
195 174
196 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) 175 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y)
197 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) 176 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso)
198 177
199 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y 178 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y
200 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where 179 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
201 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) 180 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy)
202 lemma ox ox refl = ==-refl 181 lemma ox ox refl = ==-refl
203 182
204 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y 183 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y)
205 o≡→== {x} {.x} refl = ==-refl 184 o≡→== {x} {.x} refl = ==-refl
206 185
207 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ 186 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
208 o∅≡od∅ = ==→o≡ lemma where 187 o∅≡od∅ = ==→o≡ lemma where
209 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x 188 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x
210 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso 189 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (odef-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
211 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x 190 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x
212 lemma1 {x} lt = ⊥-elim (¬x<0 lt) 191 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
213 lemma : ord→od o∅ == od∅ 192 lemma : od (ord→od o∅) == od od∅
214 lemma = record { eq→ = lemma0 ; eq← = lemma1 } 193 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
215 194
216 ord-od∅ : od→ord (od∅ ) ≡ o∅ 195 ord-od∅ : od→ord (od∅ ) ≡ o∅
217 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) 196 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
218 197
219 ∅0 : record { def = λ x → Lift n ⊥ } == od∅ 198 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
220 eq→ ∅0 {w} (lift ()) 199 eq→ ∅0 {w} (lift ())
221 eq← ∅0 {w} lt = lift (¬x<0 lt) 200 eq← ∅0 {w} lt = lift (¬x<0 lt)
222 201
223 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) 202 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ )
224 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d 203 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
225 ∅< {x} {y} d eq | lift () 204 ∅< {x} {y} d eq | lift ()
226 205
227 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox 206 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
228 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) 207 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
229 208
230 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x 209 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
231 def-iso refl t = t 210 odef-iso refl t = t
232 211
233 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) 212 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
234 is-o∅ x with trio< x o∅ 213 is-o∅ x with trio< x o∅
235 is-o∅ x | tri< a ¬b ¬c = no ¬b 214 is-o∅ x | tri< a ¬b ¬c = no ¬b
236 is-o∅ x | tri≈ ¬a b ¬c = yes b 215 is-o∅ x | tri≈ ¬a b ¬c = yes b
237 is-o∅ x | tri> ¬a ¬b c = no ¬b 216 is-o∅ x | tri> ¬a ¬b c = no ¬b
238 217
239 _,_ : OD → OD → OD 218 _,_ : HOD → HOD → HOD
240 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) 219 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; ¬odmax = ? } -- Ord (omax (od→ord x) (od→ord y))
241 220
242 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 221 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
243 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) 222 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
244 223
245 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD 224 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → HOD
246 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } 225 in-codomain X ψ = record { od = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } ; ¬odmax = ? }
247 226
248 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) 227 -- Power Set of X ( or constructible by λ y → odef X (od→ord y )
249 228
250 ZFSubset : (A x : OD ) → OD 229 ZFSubset : (A x : HOD ) → HOD
251 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set 230 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; ¬odmax = ? } -- roughly x = A → Set
252 231
253 OPwr : (A : OD ) → OD 232 OPwr : (A : HOD ) → HOD
254 OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) 233 OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) )
255 234
256 -- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n 235 -- _⊆_ : ( A B : HOD ) → ∀{ x : HOD } → Set n
257 -- _⊆_ A B {x} = A ∋ x → B ∋ x 236 -- _⊆_ A B {x} = A ∋ x → B ∋ x
258 237
259 record _⊆_ ( A B : OD ) : Set (suc n) where 238 record _⊆_ ( A B : HOD ) : Set (suc n) where
260 field 239 field
261 incl : { x : OD } → A ∋ x → B ∋ x 240 incl : { x : HOD } → A ∋ x → B ∋ x
262 241
263 open _⊆_ 242 open _⊆_
264 243
265 infixr 220 _⊆_ 244 infixr 220 _⊆_
266 245
267 subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) 246 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
268 subset-lemma {A} {x} = record { 247 subset-lemma {A} {x} = record {
269 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } 248 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
270 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } 249 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
271 } 250 }
272 251
273 open import Data.Unit 252 open import Data.Unit
274 253
275 ε-induction : { ψ : OD → Set (suc n)} 254 ε-induction : { ψ : HOD → Set (suc n)}
276 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) 255 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
277 → (x : OD ) → ψ x 256 → (x : HOD ) → ψ x
278 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where 257 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
279 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) 258 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
280 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) 259 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
281 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) 260 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
282 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy 261 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy
283 262
284 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) 263 -- minimal-2 : (x : HOD ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord y) )
285 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) 264 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )
286 265
287 OD→ZF : ZF 266 HOD→ZF : ZF
288 OD→ZF = record { 267 HOD→ZF = record {
289 ZFSet = OD 268 ZFSet = HOD
290 ; _∋_ = _∋_ 269 ; _∋_ = _∋_
291 ; _≈_ = _==_ 270 ; _≈_ = _=h=_
292 ; ∅ = od∅ 271 ; ∅ = od∅
293 ; _,_ = _,_ 272 ; _,_ = _,_
294 ; Union = Union 273 ; Union = Union
295 ; Power = Power 274 ; Power = Power
296 ; Select = Select 275 ; Select = Select
297 ; Replace = Replace 276 ; Replace = Replace
298 ; infinite = infinite 277 ; infinite = infinite
299 ; isZF = isZF 278 ; isZF = isZF
300 } where 279 } where
301 ZFSet = OD -- is less than Ords because of maxod 280 ZFSet = HOD -- is less than Ords because of maxod
302 Select : (X : OD ) → ((x : OD ) → Set n ) → OD 281 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
303 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } 282 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; ¬odmax = ? }
304 Replace : OD → (OD → OD ) → OD 283 Replace : HOD → (HOD → HOD ) → HOD
305 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } 284 Replace X ψ = record { od = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ odef (in-codomain X ψ) x } ; ¬odmax = ? }
306 _∩_ : ( A B : ZFSet ) → ZFSet 285 _∩_ : ( A B : ZFSet ) → ZFSet
307 A ∩ B = record { def = λ x → def A x ∧ def B x } 286 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ; ¬odmax = ? }
308 Union : OD → OD 287 Union : HOD → HOD
309 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } 288 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } ; ¬odmax = ? }
310 _∈_ : ( A B : ZFSet ) → Set n 289 _∈_ : ( A B : ZFSet ) → Set n
311 A ∈ B = B ∋ A 290 A ∈ B = B ∋ A
312 Power : OD → OD 291 Power : HOD → HOD
313 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) 292 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
314 -- ｛_｝ : ZFSet → ZFSet 293 -- ｛_｝ : ZFSet → ZFSet
315 -- ｛ x ｝ = ( x , x ) -- it works but we don't use 294 -- ｛ x ｝ = ( x , x ) -- it works but we don't use
316 295
317 data infinite-d : ( x : Ordinal ) → Set n where 296 data infinite-d : ( x : Ordinal ) → Set n where
318 iφ : infinite-d o∅ 297 iφ : infinite-d o∅
319 isuc : {x : Ordinal } → infinite-d x → 298 isuc : {x : Ordinal } → infinite-d x →
320 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) 299 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
321 300
322 infinite : OD 301 infinite : HOD
323 infinite = record { def = λ x → infinite-d x } 302 infinite = record { od = record { def = λ x → infinite-d x } ; ¬odmax = ? }
303
304 _=h=_ : (x y : HOD) → Set n
305 x =h= y = od x == od y
324 306
325 infixr 200 _∈_ 307 infixr 200 _∈_
326 -- infixr 230 _∩_ _∪_ 308 -- infixr 230 _∩_ _∪_
327 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite 309 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
328 isZF = record { 310 isZF = record {
329 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } 311 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
330 ; pair→ = pair→ 312 ; pair→ = pair→
331 ; pair← = pair← 313 ; pair← = pair←
332 ; union→ = union→ 314 ; union→ = union→
343 ; replacement→ = replacement→ 325 ; replacement→ = replacement→
344 -- ; choice-func = choice-func 326 -- ; choice-func = choice-func
345 -- ; choice = choice 327 -- ; choice = choice
346 } where 328 } where
347 329
348 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) 330 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
349 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) 331 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
350 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) 332 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
351 333
352 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t 334 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
353 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) 335 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
354 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) 336 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
355 337
356 empty : (x : OD ) → ¬ (od∅ ∋ x) 338 empty : (x : HOD ) → ¬ (od∅ ∋ x)
357 empty x = ¬x<0 339 empty x = ¬x<0
358 340
359 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) 341 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
360 o<→c< lt = record { incl = λ z → ordtrans z lt } 342 o<→c< lt = record { incl = λ z → ordtrans z lt }
361 343
364 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc 346 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
365 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc 347 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
366 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) 348 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
367 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) 349 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
368 350
369 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z 351 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
370 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx 352 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
371 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) 353 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
372 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) 354 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
373 union← X z UX∋z = FExists _ lemma UX∋z where 355 union← X z UX∋z = FExists _ lemma UX∋z where
374 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) 356 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
375 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 357 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
376 358
377 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y 359 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
378 ψiso {ψ} t refl = t 360 ψiso {ψ} t refl = t
379 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) 361 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
380 selection {ψ} {X} {y} = record { 362 selection {ψ} {X} {y} = record {
381 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } 363 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
382 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } 364 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
383 } 365 }
384 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x 366 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
385 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where 367 replacement← {ψ} X x lt = record { proj1 = ? ; proj2 = lemma } where -- sup-c< ψ {x}
386 lemma : def (in-codomain X ψ) (od→ord (ψ x)) 368 lemma : odef (in-codomain X ψ) (od→ord (ψ x))
387 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) 369 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
388 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) 370 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
389 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where 371 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
390 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) 372 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
391 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) 373 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
392 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where 374 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
393 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) 375 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
394 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) 376 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
395 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) 377 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
396 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) 378 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
397 379
398 --- 380 ---
399 --- Power Set 381 --- Power Set
400 --- 382 ---
401 --- First consider ordinals in OD 383 --- First consider ordinals in HOD
402 --- 384 ---
403 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A 385 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A
404 -- 386 --
405 -- 387 --
406 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) 388 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
407 ∩-≡ {a} {b} inc = record { 389 ∩-≡ {a} {b} inc = record {
408 eq→ = λ {x} x<a → record { proj2 = x<a ; 390 eq→ = λ {x} x<a → record { proj2 = x<a ;
409 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; 391 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
410 eq← = λ {x} x<a∩b → proj2 x<a∩b } 392 eq← = λ {x} x<a∩b → proj2 x<a∩b }
411 -- 393 --
412 -- Transitive Set case 394 -- Transitive Set case
413 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t 395 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t
414 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t 396 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t
415 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) 397 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
416 -- 398 --
417 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t 399 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
418 ord-power← a t t→A = def-subst {_} {_} {OPwr (Ord a)} {od→ord t} 400 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
419 lemma refl (lemma1 lemma-eq )where 401 lemma refl (lemma1 lemma-eq )where
420 lemma-eq : ZFSubset (Ord a) t == t 402 lemma-eq : ZFSubset (Ord a) t =h= t
421 eq→ lemma-eq {z} w = proj2 w 403 eq→ lemma-eq {z} w = proj2 w
422 eq← lemma-eq {z} w = record { proj2 = w ; 404 eq← lemma-eq {z} w = record { proj2 = w ;
423 proj1 = def-subst {_} {_} {(Ord a)} {z} 405 proj1 = odef-subst {_} {_} {(Ord a)} {z}
424 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } 406 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
425 lemma1 : {a : Ordinal } { t : OD } 407 lemma1 : {a : Ordinal } { t : HOD }
426 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t 408 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
427 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) 409 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
428 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) 410 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x))
429 lemma = sup-o< 411 lemma = sup-o<
430 412
431 -- 413 --
432 -- Every set in OD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first 414 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
433 -- then replace of all elements of the Power set by A ∩ y 415 -- then replace of all elements of the Power set by A ∩ y
434 -- 416 --
435 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) 417 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
436 418
437 -- we have oly double negation form because of the replacement axiom 419 -- we have oly double negation form because of the replacement axiom
438 -- 420 --
439 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) 421 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
440 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where 422 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
441 a = od→ord A 423 a = od→ord A
442 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) 424 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
443 lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t 425 lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t
444 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) 426 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
445 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) 427 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
446 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) 428 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
447 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) 429 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
448 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) 430 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
449 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not 431 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
450 432
451 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t 433 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
452 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 434 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
453 a = od→ord A 435 a = od→ord A
454 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x 436 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
455 lemma0 {x} t∋x = c<→o< (t→A t∋x) 437 lemma0 {x} t∋x = c<→o< (t→A t∋x)
456 lemma3 : OPwr (Ord a) ∋ t 438 lemma3 : OPwr (Ord a) ∋ t
457 lemma3 = ord-power← a t lemma0 439 lemma3 = ord-power← a t lemma0
458 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t 440 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
459 lemma4 = let open ≡-Reasoning in begin 441 lemma4 = let open ≡-Reasoning in begin
464 t 446 t
465 ∎ 447 ∎
466 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) 448 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x))
467 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) 449 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x)))
468 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) 450 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} )
469 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) 451 lemma2 : odef (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
470 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where 452 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
471 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 453 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
472 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) 454 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ t→A )))
473 455
474 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) 456 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
475 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where 457 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
476 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y 458 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
477 lemma lt y<x with osuc-≡< lt 459 lemma lt y<x with osuc-≡< lt
478 lemma lt y<x | case1 refl = c<→o< y<x 460 lemma lt y<x | case1 refl = c<→o< y<x
479 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a 461 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
480 462
481 continuum-hyphotheis : (a : Ordinal) → Set (suc n) 463 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
482 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) 464 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
483 465
484 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B 466 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
485 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d 467 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
486 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d 468 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
487 469
488 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) 470 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
489 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d 471 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
490 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d 472 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
491 473
492 infinity∅ : infinite ∋ od∅ 474 infinity∅ : infinite ∋ od∅
493 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where 475 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
494 lemma : o∅ ≡ od→ord od∅ 476 lemma : o∅ ≡ od→ord od∅
495 lemma = let open ≡-Reasoning in begin 477 lemma = let open ≡-Reasoning in begin
496 o∅ 478 o∅
497 ≡⟨ sym diso ⟩ 479 ≡⟨ sym diso ⟩
498 od→ord ( ord→od o∅ ) 480 od→ord ( ord→od o∅ )
499 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ 481 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
500 od→ord od∅ 482 od→ord od∅
501 ∎ 483 ∎
502 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) 484 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
503 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where 485 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
504 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) 486 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
505 ≡ od→ord (Union (x , (x , x))) 487 ≡ od→ord (Union (x , (x , x)))
506 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso 488 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
507 489
508 490
509 Union = ZF.Union OD→ZF 491 Union = ZF.Union HOD→ZF
510 Power = ZF.Power OD→ZF 492 Power = ZF.Power HOD→ZF
511 Select = ZF.Select OD→ZF 493 Select = ZF.Select HOD→ZF
512 Replace = ZF.Replace OD→ZF 494 Replace = ZF.Replace HOD→ZF
513 isZF = ZF.isZF OD→ZF 495 isZF = ZF.isZF HOD→ZF