Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison OD.agda @ 297:be6670af87fa
maxod try
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 22 Jun 2020 16:43:31 +0900 |
parents | ef93c56ad311 |
children | 3795ffb127d0 |
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296:42f89e5efb00 | 297:be6670af87fa |
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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 |