comparison OD.agda @ 297:be6670af87fa

maxod try
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 22 Jun 2020 16:43:31 +0900
parents ef93c56ad311
children 3795ffb127d0
comparison
equal deleted inserted replaced
296:42f89e5efb00 297:be6670af87fa
68 -- we need explict assumption on sup. 68 -- we need explict assumption on sup.
69 69
70 record ODAxiom : Set (suc n) where 70 record ODAxiom : Set (suc n) where
71 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) 71 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
72 field 72 field
73 maxod : Ordinal
73 od→ord : OD → Ordinal 74 od→ord : OD → Ordinal
74 ord→od : Ordinal → OD 75 ord→od : (x : Ordinal ) → x o< maxod → OD
76 o<max : {x : OD } → od→ord x o< maxod
75 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y 77 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
76 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x 78 oiso : {x : OD } → ord→od ( od→ord x ) o<max ≡ x
77 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x 79 diso : {x : Ordinal } → (lt : x o< maxod) → od→ord ( ord→od x lt ) ≡ x
78 ==→o≡ : { x y : OD } → (x == y) → x ≡ y 80 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
79 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) 81 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
80 sup-o : ( OD → Ordinal ) → Ordinal 82 sup-o : ( OD → Ordinal ) → Ordinal
81 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ 83 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ
84 sup-<od : { ψ : OD → OD } → ∀ {x : OD } → sup-o (λ x → od→ord (ψ x)) o< maxod
82 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use 85 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
83 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal 86 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
84 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) 87 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
85 88
86 postulate odAxiom : ODAxiom 89 postulate odAxiom : ODAxiom
91 94
92 -- Ordinals in OD , the maximum 95 -- Ordinals in OD , the maximum
93 Ords : OD 96 Ords : OD
94 Ords = record { def = λ x → One } 97 Ords = record { def = λ x → One }
95 98
96 maxod : {x : OD} → od→ord x o< od→ord Ords 99 -- maxod : {x : OD} → od→ord x o< od→ord Ords
97 maxod {x} = c<→o< OneObj 100 -- maxod {x} = c<→o< OneObj
98 101
99 -- Ordinal in OD ( and ZFSet ) Transitive Set 102 -- Ordinal in OD ( and ZFSet ) Transitive Set
100 Ord : ( a : Ordinal ) → OD 103 Ord : ( a : Ordinal ) → OD
101 Ord a = record { def = λ y → y o< a } 104 Ord a = record { def = λ y → y o< a }
102 105
103 od∅ : OD 106 od∅ : OD
104 od∅ = Ord o∅ 107 od∅ = Ord o∅
105 108
106 109
107 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) 110 -- o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y {!!} ) x ) → {x : OD } → x ≡ Ord (od→ord x)
108 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 111 -- o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
109 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y 112 -- lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y
110 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)) 113 -- 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))
111 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y 114 -- lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y
112 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) 115 -- lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt )
113 116
114 _∋_ : ( a x : OD ) → Set n 117 _∋_ : ( a x : OD ) → Set n
115 _∋_ a x = def a ( od→ord x ) 118 _∋_ a x = def a ( od→ord x )
116 119
117 _c<_ : ( x a : OD ) → Set n 120 _c<_ : ( x a : OD ) → Set n
127 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) ) 130 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )
128 131
129 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x )) 132 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
130 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )} 133 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )}
131 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where 134 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
132 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x)) 135 lemma : od→ord (ψ (ord→od (od→ord x) o<max )) o< sup-o (λ x → od→ord (ψ x))
133 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) ) 136 lemma = subst₂ (λ j k → j o< k ) refl (diso (sup-<od {ψ} {x}) ) (o<-subst (sup-o< ) refl (sym (diso sup-<od)))
134 137
135 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y 138 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
136 otrans x<a y<x = ordtrans y<x x<a 139 otrans x<a y<x = ordtrans y<x x<a
137 140
138 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X 141 -- def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X
139 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso 142 -- def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym (diso lemma)))) (diso lemma) (diso o<max) where
143 -- lemma : x o< maxod
144 -- lemma = subst (λ k → k o< maxod ) (diso {!!} ) (otrans o<max ( c<→o< lt ))
140 145
141 146
142 -- avoiding lv != Zero error 147 -- avoiding lv != Zero error
143 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y 148 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
144 orefl refl = refl 149 orefl refl = refl
145 150
146 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y 151 ==-iso : { x y : OD } → ord→od (od→ord x) o<max == ord→od (od→ord y) o<max → x == y
147 ==-iso {x} {y} eq = record { 152 ==-iso {x} {y} eq = record {
148 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; 153 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
149 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } 154 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
150 where 155 where
151 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z 156 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x) o<max ) z → def x z
152 lemma {x} {z} d = def-subst d oiso refl 157 lemma {x} {z} d = def-subst d oiso refl
153 158
154 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) 159 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) o<max == y)
155 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) 160 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso)
156 161
162 <-irr : {x y z : Ordinal } → x ≡ y → (x o< z) ≡ (y o< z)
163 <-irr refl = refl
164
157 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y 165 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y
158 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where 166 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq) o<max o<max ) where
159 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) 167 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (x<m : ox o< maxod) (y<m : oy o< maxod) → (ord→od ox x<m ) == (ord→od oy y<m )
160 lemma ox ox refl = ==-refl 168 lemma ox ox refl x<m y<m = subst (λ k → ord→od ox x<m == ord→od ox k) {!!} ==-refl
161 169
162 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y 170 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x {!!} == ord→od y {!!}
163 o≡→== {x} {.x} refl = ==-refl 171 o≡→== {x} {.x} refl = ==-refl
164 172
165 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ 173 o∅≡od∅ : ord→od (o∅ ) {!!} ≡ od∅
166 o∅≡od∅ = ==→o≡ lemma where 174 o∅≡od∅ = ==→o≡ lemma where
167 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x 175 lemma0 : {x : Ordinal} → def (ord→od o∅ {!!} ) x → def od∅ x
168 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 176 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 {!!})
169 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x 177 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅ {!!} ) x
170 lemma1 {x} lt = ⊥-elim (¬x<0 lt) 178 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
171 lemma : ord→od o∅ == od∅ 179 lemma : ord→od o∅ {!!} == od∅
172 lemma = record { eq→ = lemma0 ; eq← = lemma1 } 180 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
173 181
174 ord-od∅ : od→ord (od∅ ) ≡ o∅ 182 ord-od∅ : od→ord (od∅ ) ≡ o∅
175 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) 183 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) (diso {!!}) (cong ( λ k → od→ord k ) o∅≡od∅ ) )
176 184
177 ∅0 : record { def = λ x → Lift n ⊥ } == od∅ 185 ∅0 : record { def = λ x → Lift n ⊥ } == od∅
178 eq→ ∅0 {w} (lift ()) 186 eq→ ∅0 {w} (lift ())
179 eq← ∅0 {w} lt = lift (¬x<0 lt) 187 eq← ∅0 {w} lt = lift (¬x<0 lt)
180 188
199 207
200 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 208 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
201 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) 209 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
202 210
203 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD 211 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
204 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } 212 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y {!!} ))))) }
205 213
206 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) 214 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
207 215
208 ZFSubset : (A x : OD ) → OD 216 ZFSubset : (A x : OD ) → OD
209 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set 217 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set
232 240
233 ε-induction : { ψ : OD → Set (suc n)} 241 ε-induction : { ψ : OD → Set (suc n)}
234 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) 242 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
235 → (x : OD ) → ψ x 243 → (x : OD ) → ψ x
236 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where 244 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
237 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) 245 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy {!!} )) → ψ (ord→od ox {!!} )
238 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) 246 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl (diso {!!}) )))
239 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) 247 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy {!!} )
240 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy 248 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy {!!} )} induction oy
241 249
242 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) 250 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
243 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) 251 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )
244 252
245 OD→ZF : ZF 253 OD→ZF : ZF
256 ; infinite = infinite 264 ; infinite = infinite
257 ; isZF = isZF 265 ; isZF = isZF
258 } where 266 } where
259 ZFSet = OD -- is less than Ords because of maxod 267 ZFSet = OD -- is less than Ords because of maxod
260 Select : (X : OD ) → ((x : OD ) → Set n ) → OD 268 Select : (X : OD ) → ((x : OD ) → Set n ) → OD
261 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } 269 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x {!!} )) }
262 Replace : OD → (OD → OD ) → OD 270 Replace : OD → (OD → OD ) → OD
263 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } 271 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x }
264 _∩_ : ( A B : ZFSet ) → ZFSet 272 _∩_ : ( A B : ZFSet ) → ZFSet
265 A ∩ B = record { def = λ x → def A x ∧ def B x } 273 A ∩ B = record { def = λ x → def A x ∧ def B x }
266 Union : OD → OD 274 Union : OD → OD
267 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } 275 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u {!!} ) x))) }
268 _∈_ : ( A B : ZFSet ) → Set n 276 _∈_ : ( A B : ZFSet ) → Set n
269 A ∈ B = B ∋ A 277 A ∈ B = B ∋ A
270 Power : OD → OD 278 Power : OD → OD
271 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) 279 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
272 -- {_} : ZFSet → ZFSet 280 -- {_} : ZFSet → ZFSet
273 -- { x } = ( x , x ) -- it works but we don't use 281 -- { x } = ( x , x ) -- it works but we don't use
274 282
275 data infinite-d : ( x : Ordinal ) → Set n where 283 data infinite-d : ( x : Ordinal ) → Set n where
276 iφ : infinite-d o∅ 284 iφ : infinite-d o∅
277 isuc : {x : Ordinal } → infinite-d x → 285 isuc : {x : Ordinal } → infinite-d x →
278 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) 286 infinite-d (od→ord ( Union (ord→od x {!!} , (ord→od x {!!} , ord→od x {!!} ) ) ))
279 287
280 infinite : OD 288 infinite : OD
281 infinite = record { def = λ x → infinite-d x } 289 infinite = record { def = λ x → infinite-d x }
282 290
283 infixr 200 _∈_ 291 infixr 200 _∈_
319 327
320 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y 328 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
321 ⊆→o< {x} {y} lt with trio< x y 329 ⊆→o< {x} {y} lt with trio< x y
322 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc 330 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
323 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc 331 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
324 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) 332 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym (diso {!!})) refl )
325 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) 333 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt (diso {!!}) refl ))
326 334
327 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z 335 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
328 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx 336 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
329 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) 337 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
330 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) 338 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
331 union← X z UX∋z = FExists _ lemma UX∋z where 339 union← X z UX∋z = FExists _ lemma UX∋z where
332 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) 340 lemma : {y : Ordinal} → def X y ∧ def (ord→od y {!!} ) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
333 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 341 lemma {y} xx not = not (ord→od y {!!} ) record { proj1 = subst ( λ k → def X k ) (sym (diso {!!})) (proj1 xx ) ; proj2 = proj2 xx }
334 342
335 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y 343 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y
336 ψiso {ψ} t refl = t 344 ψiso {ψ} t refl = t
337 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) 345 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
338 selection {ψ} {X} {y} = record { 346 selection {ψ} {X} {y} = record {
343 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where 351 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
344 lemma : def (in-codomain X ψ) (od→ord (ψ x)) 352 lemma : def (in-codomain X ψ) (od→ord (ψ x))
345 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) 353 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
346 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) 354 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
347 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where 355 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
348 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) 356 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y {!!} ))))
349 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) 357 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} )))
350 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where 358 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
351 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) 359 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y {!!} ))) → (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} ))
352 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) 360 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) {!!} == k ) oiso (o≡→== eq )
353 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) 361 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} )) )
354 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) 362 lemma not y not2 = not (ord→od y {!!} ) (subst (λ k → k == ψ (ord→od y {!!} )) oiso ( proj2 not2 ))
355 363
356 --- 364 ---
357 --- Power Set 365 --- Power Set
358 --- 366 ---
359 --- First consider ordinals in OD 367 --- First consider ordinals in OD
362 -- 370 --
363 -- 371 --
364 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) 372 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
365 ∩-≡ {a} {b} inc = record { 373 ∩-≡ {a} {b} inc = record {
366 eq→ = λ {x} x<a → record { proj2 = x<a ; 374 eq→ = λ {x} x<a → record { proj2 = x<a ;
367 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; 375 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym (diso {!!})) )) refl (diso {!!}) } ;
368 eq← = λ {x} x<a∩b → proj2 x<a∩b } 376 eq← = λ {x} x<a∩b → proj2 x<a∩b }
369 -- 377 --
370 -- Transitive Set case 378 -- Transitive Set case
371 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t 379 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t
372 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t 380 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t
377 lemma refl (lemma1 lemma-eq )where 385 lemma refl (lemma1 lemma-eq )where
378 lemma-eq : ZFSubset (Ord a) t == t 386 lemma-eq : ZFSubset (Ord a) t == t
379 eq→ lemma-eq {z} w = proj2 w 387 eq→ lemma-eq {z} w = proj2 w
380 eq← lemma-eq {z} w = record { proj2 = w ; 388 eq← lemma-eq {z} w = record { proj2 = w ;
381 proj1 = def-subst {_} {_} {(Ord a)} {z} 389 proj1 = def-subst {_} {_} {(Ord a)} {z}
382 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } 390 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z {!!} )} w refl (sym (diso {!!})) )) refl (diso {!!}) }
383 lemma1 : {a : Ordinal } { t : OD } 391 lemma1 : {a : Ordinal } { t : OD }
384 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t 392 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max )) ≡ od→ord t
385 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 )) 393 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 ))
386 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) 394 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max ) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x))
387 lemma = sup-o< 395 lemma = sup-o<
388 396
389 -- 397 --
390 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first 398 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first
391 -- then replace of all elements of the Power set by A ∩ y 399 -- then replace of all elements of the Power set by A ∩ y
399 a = od→ord A 407 a = od→ord A
400 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) 408 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y)))
401 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t 409 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
402 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) 410 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
403 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) 411 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
404 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) 412 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ (ord→od y {!!} ))))
405 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) 413 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
406 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) 414 lemma5 : {y : Ordinal} → t == (A ∩ (ord→od y {!!})) → ¬ ¬ (def A (od→ord x))
407 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not 415 lemma5 {y} eq not = (lemma3 (ord→od y {!!} ) eq) not
408 416
409 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t 417 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
410 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 418 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
411 a = od→ord A 419 a = od→ord A
412 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x 420 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
413 lemma0 {x} t∋x = c<→o< (t→A t∋x) 421 lemma0 {x} t∋x = c<→o< (t→A t∋x)
414 lemma3 : Def (Ord a) ∋ t 422 lemma3 : Def (Ord a) ∋ t
415 lemma3 = ord-power← a t lemma0 423 lemma3 = ord-power← a t lemma0
416 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t 424 lemma4 : (A ∩ ord→od (od→ord t) {!!} ) ≡ t
417 lemma4 = let open ≡-Reasoning in begin 425 lemma4 = let open ≡-Reasoning in begin
418 A ∩ ord→od (od→ord t) 426 A ∩ ord→od (od→ord t) {!!}
419 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ 427 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
420 A ∩ t 428 A ∩ t
421 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ 429 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
422 t 430 t
423 431
424 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) 432 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x))
425 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) 433 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x)))
426 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) 434 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} )
427 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) 435 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
428 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where 436 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
429 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 437 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t) {!!} )
430 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) 438 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A )))
431 439
432 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) 440 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
433 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where 441 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
434 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y 442 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
438 446
439 continuum-hyphotheis : (a : Ordinal) → Set (suc n) 447 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
440 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) 448 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
441 449
442 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B 450 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
443 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d 451 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym (diso {!!})) (proj1 (eq (ord→od x {!!} ))) d
444 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d 452 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym (diso {!!})) (proj2 (eq (ord→od x {!!} ))) d
445 453
446 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) 454 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
447 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d 455 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
448 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d 456 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
449 457
450 infinity∅ : infinite ∋ od∅ 458 infinity∅ : infinite ∋ od∅
451 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where 459 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
452 lemma : o∅ ≡ od→ord od∅ 460 lemma : o∅ ≡ od→ord od∅
453 lemma = let open ≡-Reasoning in begin 461 lemma = let open ≡-Reasoning in begin
454 o∅ 462 o∅
455 ≡⟨ sym diso ⟩ 463 ≡⟨ sym (diso {!!}) ⟩
456 od→ord ( ord→od o∅ ) 464 od→ord ( ord→od o∅ {!!} )
457 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ 465 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
458 od→ord od∅ 466 od→ord od∅
459 467
460 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) 468 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
461 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where 469 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
462 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) 470 lemma : od→ord (Union (ord→od (od→ord x) {!!} , (ord→od (od→ord x) {!!} , ord→od (od→ord x) {!!} )))
463 ≡ od→ord (Union (x , (x , x))) 471 ≡ od→ord (Union (x , (x , x)))
464 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso 472 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
465 473
466 474
467 Union = ZF.Union OD→ZF 475 Union = ZF.Union OD→ZF