Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison OD.agda @ 334:ba3ebb9a16c6 release

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
HOD
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 05 Jul 2020 16:59:13 +0900
parents 0faa7120e4b5
children daafa2213dd2
comparison
equal deleted inserted replaced
289:9f926b2210bc 334:ba3ebb9a16c6
51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y 51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y
52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m 52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m 53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
54 54
55 -- next assumptions are our axiom 55 -- next assumptions are our axiom
56 -- it defines a subset of OD, which is called HOD, usually defined as 56 --
57 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
58 -- correspondence to the OD then the OD looks like a ZF Set.
59 --
60 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
61 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
62 --
63 -- In classical Set Theory, HOD is used, as a subset of OD,
57 -- HOD = { x | TC x ⊆ OD } 64 -- HOD = { x | TC x ⊆ OD }
58 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x 65 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
59 66 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
60 record ODAxiom : Set (suc n) where 67 --
61 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) 68 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
62 field 69 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
63 od→ord : OD → Ordinal 70 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
64 ord→od : Ordinal → OD 71 -- bound on each HOD.
65 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y 72 --
66 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x 73 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
67 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x 74 -- we need explict assumption on sup.
68 ==→o≡ : { x y : OD } → (x == y) → x ≡ y 75 --
69 -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) 76 -- ==→o≡ is necessary to prove axiom of extensionality.
70 sup-o : ( OD → Ordinal ) → Ordinal
71 sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ
72 -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
73 -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal
74 -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
75
76 postulate odAxiom : ODAxiom
77 open ODAxiom odAxiom
78 77
79 data One : Set n where 78 data One : Set n where
80 OneObj : One 79 OneObj : One
81 80
82 -- Ordinals in OD , the maximum 81 -- Ordinals in OD , the maximum
83 Ords : OD 82 Ords : OD
84 Ords = record { def = λ x → One } 83 Ords = record { def = λ x → One }
85 84
86 maxod : {x : OD} → od→ord x o< od→ord Ords 85 record HOD : Set (suc n) where
87 maxod {x} = c<→o< OneObj 86 field
87 od : OD
88 odmax : Ordinal
89 <odmax : {y : Ordinal} → def od y → y o< odmax
90
91 open HOD
92
93 record ODAxiom : Set (suc n) where
94 field
95 -- HOD is isomorphic to Ordinal (by means of Goedel number)
96 od→ord : HOD → Ordinal
97 ord→od : Ordinal → HOD
98 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
99 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
100 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
102 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
103 sup-o : (A : HOD) → (( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
104 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
105
106 postulate odAxiom : ODAxiom
107 open ODAxiom odAxiom
108
109 -- maxod : {x : OD} → od→ord x o< od→ord Ords
110 -- maxod {x} = c<→o< OneObj
111
112 -- we have not this contradiction
113 -- bad-bad : ⊥
114 -- bad-bad = osuc-< <-osuc (c<→o< { record { od = record { def = λ x → One }; <odmax = {!!} } } OneObj)
88 115
89 -- Ordinal in OD ( and ZFSet ) Transitive Set 116 -- Ordinal in OD ( and ZFSet ) Transitive Set
90 Ord : ( a : Ordinal ) → OD 117 Ord : ( a : Ordinal ) → HOD
91 Ord a = record { def = λ y → y o< a } 118 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
92 119 lemma : {x : Ordinal} → x o< a → x o< a
93 od∅ : OD 120 lemma {x} lt = lt
121
122 od∅ : HOD
94 od∅ = Ord o∅ 123 od∅ = Ord o∅
95 124
96 125 odef : HOD → Ordinal → Set n
97 o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) 126 odef A x = def ( od A ) x
98 o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where 127
99 lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y 128 o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x)
100 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)) 129 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
101 lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y 130 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y
102 lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) 131 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))
103 132 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y
104 _∋_ : ( a x : OD ) → Set n 133 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt )
105 _∋_ a x = def a ( od→ord x ) 134
106 135
107 _c<_ : ( x a : OD ) → Set n 136 _∋_ : ( a x : HOD ) → Set n
137 _∋_ a x = odef a ( od→ord x )
138
139 _c<_ : ( x a : HOD ) → Set n
108 x c< a = a ∋ x 140 x c< a = a ∋ x
109 141
110 cseq : {n : Level} → OD → OD 142 cseq : {n : Level} → HOD → HOD
111 cseq x = record { def = λ y → def x (osuc y) } where 143 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where
112 144 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x)
113 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x 145 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc )
114 def-subst df refl refl = df 146
115 147 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
116 sup-od : ( OD → OD ) → OD 148 odef-subst df refl refl = df
117 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) ) 149
118 150 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
119 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
120 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )}
121 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
122 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x))
123 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) )
124
125 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y
126 otrans x<a y<x = ordtrans y<x x<a 151 otrans x<a y<x = ordtrans y<x x<a
127 152
128 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X 153 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X
129 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso 154 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso
130
131 155
132 -- avoiding lv != Zero error 156 -- avoiding lv != Zero error
133 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y 157 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y
134 orefl refl = refl 158 orefl refl = refl
135 159
136 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y 160 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y
137 ==-iso {x} {y} eq = record { 161 ==-iso {x} {y} eq = record {
138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; 162 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ;
139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } 163 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) }
140 where 164 where
141 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z 165 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z
142 lemma {x} {z} d = def-subst d oiso refl 166 lemma {x} {z} d = odef-subst d oiso refl
143 167
144 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) 168 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y)
145 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) 169 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso)
146 170
147 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y 171 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y
148 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where 172 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
149 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) 173 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy)
150 lemma ox ox refl = ==-refl 174 lemma ox ox refl = ==-refl
151 175
152 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y 176 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y)
153 o≡→== {x} {.x} refl = ==-refl 177 o≡→== {x} {.x} refl = ==-refl
154 178
155 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ 179 o∅≡od∅ : ord→od (o∅ ) ≡ od∅
156 o∅≡od∅ = ==→o≡ lemma where 180 o∅≡od∅ = ==→o≡ lemma where
157 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x 181 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x
158 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 182 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
159 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x 183 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x
160 lemma1 {x} lt = ⊥-elim (¬x<0 lt) 184 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
161 lemma : ord→od o∅ == od∅ 185 lemma : od (ord→od o∅) == od od∅
162 lemma = record { eq→ = lemma0 ; eq← = lemma1 } 186 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
163 187
164 ord-od∅ : od→ord (od∅ ) ≡ o∅ 188 ord-od∅ : od→ord (od∅ ) ≡ o∅
165 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) 189 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
166 190
167 ∅0 : record { def = λ x → Lift n ⊥ } == od∅ 191 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
168 eq→ ∅0 {w} (lift ()) 192 eq→ ∅0 {w} (lift ())
169 eq← ∅0 {w} lt = lift (¬x<0 lt) 193 eq← ∅0 {w} lt = lift (¬x<0 lt)
170 194
171 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) 195 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ )
172 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d 196 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
173 ∅< {x} {y} d eq | lift () 197 ∅< {x} {y} d eq | lift ()
174 198
175 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox 199 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
176 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) 200 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
177 201
178 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x 202 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
179 def-iso refl t = t 203 odef-iso refl t = t
180 204
181 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) 205 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
182 is-o∅ x with trio< x o∅ 206 is-o∅ x with trio< x o∅
183 is-o∅ x | tri< a ¬b ¬c = no ¬b 207 is-o∅ x | tri< a ¬b ¬c = no ¬b
184 is-o∅ x | tri≈ ¬a b ¬c = yes b 208 is-o∅ x | tri≈ ¬a b ¬c = yes b
185 is-o∅ x | tri> ¬a ¬b c = no ¬b 209 is-o∅ x | tri> ¬a ¬b c = no ¬b
186 210
187 _,_ : OD → OD → OD 211 _,_ : HOD → HOD → HOD
188 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) 212 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = omax (od→ord x) (od→ord y) ; <odmax = lemma } where
213 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y)
214 lemma {t} (case1 refl) = omax-x _ _
215 lemma {t} (case2 refl) = omax-y _ _
216
189 217
190 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 218 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
191 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) 219 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)
192 220
193 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD 221 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
194 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } 222 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
195 223
196 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) 224 -- Power Set of X ( or constructible by λ y → odef X (od→ord y )
197 225
198 ZFSubset : (A x : OD ) → OD 226 ZFSubset : (A x : HOD ) → HOD
199 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set 227 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; odmax = omin (odmax A) (odmax x) ; <odmax = lemma } where -- roughly x = A → Set
200 228 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x)
201 Def : (A : OD ) → OD 229 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and))
202 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) 230
203 231 record _⊆_ ( A B : HOD ) : Set (suc n) where
204 -- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n
205 -- _⊆_ A B {x} = A ∋ x → B ∋ x
206
207 record _⊆_ ( A B : OD ) : Set (suc n) where
208 field 232 field
209 incl : { x : OD } → A ∋ x → B ∋ x 233 incl : { x : HOD } → A ∋ x → B ∋ x
210 234
211 open _⊆_ 235 open _⊆_
212
213 infixr 220 _⊆_ 236 infixr 220 _⊆_
214 237
215 subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) 238 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
216 subset-lemma {A} {x} = record { 239 subset-lemma {A} {x} = record {
217 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } 240 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
218 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } 241 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
219 } 242 }
220 243
244 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y)
245 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (subst (λ k → def (od x) k ) (sym diso) x>z )))
246
247 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x
248 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where
249 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y))
250 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y )
251
221 open import Data.Unit 252 open import Data.Unit
222 253
223 ε-induction : { ψ : OD → Set (suc n)} 254 ε-induction : { ψ : HOD → Set n}
224 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) 255 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
225 → (x : OD ) → ψ x 256 → (x : HOD ) → ψ x
226 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
227 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)
228 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 )))
229 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)
230 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
231 262
232 -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) 263 ε-induction1 : { ψ : HOD → Set (suc n)}
233 -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) 264 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
234 265 → (x : HOD ) → ψ x
235 OD→ZF : ZF 266 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where
236 OD→ZF = record { 267 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
237 ZFSet = OD 268 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
269 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
270 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy
271
272 HOD→ZF : ZF
273 HOD→ZF = record {
274 ZFSet = HOD
238 ; _∋_ = _∋_ 275 ; _∋_ = _∋_
239 ; _≈_ = _==_ 276 ; _≈_ = _=h=_
240 ; ∅ = od∅ 277 ; ∅ = od∅
241 ; _,_ = _,_ 278 ; _,_ = _,_
242 ; Union = Union 279 ; Union = Union
243 ; Power = Power 280 ; Power = Power
244 ; Select = Select 281 ; Select = Select
245 ; Replace = Replace 282 ; Replace = Replace
246 ; infinite = infinite 283 ; infinite = infinite
247 ; isZF = isZF 284 ; isZF = isZF
248 } where 285 } where
249 ZFSet = OD -- is less than Ords because of maxod 286 ZFSet = HOD -- is less than Ords because of maxod
250 Select : (X : OD ) → ((x : OD ) → Set n ) → OD 287 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
251 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } 288 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
252 Replace : OD → (OD → OD ) → OD 289 Replace : HOD → (HOD → HOD) → HOD
253 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } 290 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x }
291 ; odmax = rmax ; <odmax = rmax<} where
292 rmax : Ordinal
293 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))
294 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
295 rmax< lt = proj1 lt
254 _∩_ : ( A B : ZFSet ) → ZFSet 296 _∩_ : ( A B : ZFSet ) → ZFSet
255 A ∩ B = record { def = λ x → def A x ∧ def B x } 297 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
256 Union : OD → OD 298 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
257 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } 299 Union : HOD → HOD
300 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) }
301 ; odmax = osuc (od→ord U) ; <odmax = umax< } where
302 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U)
303 umax< {y} not = lemma (FExists _ lemma1 not ) where
304 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x
305 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y))
306 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U
307 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U))
308 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y)
309 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
310 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U)
311 lemma not with trio< y (od→ord U)
312 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
313 lemma not | tri≈ ¬a refl ¬c = <-osuc
314 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
258 _∈_ : ( A B : ZFSet ) → Set n 315 _∈_ : ( A B : ZFSet ) → Set n
259 A ∈ B = B ∋ A 316 A ∈ B = B ∋ A
260 Power : OD → OD 317
261 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) 318 OPwr : (A : HOD ) → HOD
319 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) )
320
321 Power : HOD → HOD
322 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
262 -- ｛_｝ : ZFSet → ZFSet 323 -- ｛_｝ : ZFSet → ZFSet
263 -- ｛ x ｝ = ( x , x ) -- it works but we don't use 324 -- ｛ x ｝ = ( x , x ) -- it works but we don't use
264 325
265 data infinite-d : ( x : Ordinal ) → Set n where 326 data infinite-d : ( x : Ordinal ) → Set n where
266 iφ : infinite-d o∅ 327 iφ : infinite-d o∅
267 isuc : {x : Ordinal } → infinite-d x → 328 isuc : {x : Ordinal } → infinite-d x →
268 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) 329 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
269 330
270 infinite : OD 331 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
271 infinite = record { def = λ x → infinite-d x } 332 -- We simply assumes nfinite-d y has a maximum.
333 --
334 -- This means that many of OD cannot be HODs because of the od→ord mapping divergence.
335 -- We should have some axioms to prevent this, but it may complicate thins.
336 --
337 postulate
338 ωmax : Ordinal
339 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
340
341 infinite : HOD
342 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
343
344 _=h=_ : (x y : HOD) → Set n
345 x =h= y = od x == od y
272 346
273 infixr 200 _∈_ 347 infixr 200 _∈_
274 -- infixr 230 _∩_ _∪_ 348 -- infixr 230 _∩_ _∪_
275 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite 349 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
276 isZF = record { 350 isZF = record {
277 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } 351 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
278 ; pair→ = pair→ 352 ; pair→ = pair→
279 ; pair← = pair← 353 ; pair← = pair←
280 ; union→ = union→ 354 ; union→ = union→
286 ; ε-induction = ε-induction 360 ; ε-induction = ε-induction
287 ; infinity∅ = infinity∅ 361 ; infinity∅ = infinity∅
288 ; infinity = infinity 362 ; infinity = infinity
289 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} 363 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
290 ; replacement← = replacement← 364 ; replacement← = replacement←
291 ; replacement→ = replacement→ 365 ; replacement→ = λ {ψ} → replacement→ {ψ}
292 -- ; choice-func = choice-func 366 -- ; choice-func = choice-func
293 -- ; choice = choice 367 -- ; choice = choice
294 } where 368 } where
295 369
296 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) 370 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
297 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) 371 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
298 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) 372 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y ))
299 373
300 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t 374 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
301 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) 375 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
302 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) 376 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))
303 377
304 empty : (x : OD ) → ¬ (od∅ ∋ x) 378 empty : (x : HOD ) → ¬ (od∅ ∋ x)
305 empty x = ¬x<0 379 empty x = ¬x<0
306 380
307 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) 381 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
308 o<→c< lt = record { incl = λ z → ordtrans z lt } 382 o<→c< lt = record { incl = λ z → ordtrans z lt }
309 383
312 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc 386 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
313 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc 387 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
314 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) 388 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl )
315 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) 389 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
316 390
317 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z 391 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
318 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx 392 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
319 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) 393 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
320 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) 394 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
321 union← X z UX∋z = FExists _ lemma UX∋z where 395 union← X z UX∋z = FExists _ lemma UX∋z where
322 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) 396 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
323 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 397 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
324 398
325 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y 399 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
326 ψiso {ψ} t refl = t 400 ψiso {ψ} t refl = t
327 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) 401 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
328 selection {ψ} {X} {y} = record { 402 selection {ψ} {X} {y} = record {
329 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } 403 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
330 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } 404 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
331 } 405 }
332 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x 406 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y))))
333 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where 407 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt )
408 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
409 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where
334 lemma : def (in-codomain X ψ) (od→ord (ψ x)) 410 lemma : def (in-codomain X ψ) (od→ord (ψ x))
335 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) 411 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
336 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) 412 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
337 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where 413 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
338 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) 414 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
339 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) 415 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)))
340 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where 416 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
341 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) 417 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
342 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) 418 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
343 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) 419 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
344 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) 420 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 ))
345 421
346 --- 422 ---
347 --- Power Set 423 --- Power Set
348 --- 424 ---
349 --- First consider ordinals in OD 425 --- First consider ordinals in HOD
350 --- 426 ---
351 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A 427 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A
352 -- 428 --
353 -- 429 --
354 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) 430 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
355 ∩-≡ {a} {b} inc = record { 431 ∩-≡ {a} {b} inc = record {
356 eq→ = λ {x} x<a → record { proj2 = x<a ; 432 eq→ = λ {x} x<a → record { proj2 = x<a ;
357 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; 433 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
358 eq← = λ {x} x<a∩b → proj2 x<a∩b } 434 eq← = λ {x} x<a∩b → proj2 x<a∩b }
359 -- 435 --
360 -- Transitive Set case 436 -- Transitive Set case
361 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t 437 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t
362 -- Def (Ord a) is a sup of ZFSubset (Ord a) t, so Def (Ord a) ∋ t 438 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t
363 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) 439 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
364 -- 440 --
365 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t 441 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
366 ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t} 442 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t}
367 lemma refl (lemma1 lemma-eq )where 443 lemma refl (lemma1 lemma-eq )where
368 lemma-eq : ZFSubset (Ord a) t == t 444 lemma-eq : ZFSubset (Ord a) t =h= t
369 eq→ lemma-eq {z} w = proj2 w 445 eq→ lemma-eq {z} w = proj2 w
370 eq← lemma-eq {z} w = record { proj2 = w ; 446 eq← lemma-eq {z} w = record { proj2 = w ;
371 proj1 = def-subst {_} {_} {(Ord a)} {z} 447 proj1 = odef-subst {_} {_} {(Ord a)} {z}
372 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } 448 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
373 lemma1 : {a : Ordinal } { t : OD } 449 lemma1 : {a : Ordinal } { t : HOD }
374 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t 450 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
375 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 )) 451 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 ))
376 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) 452 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a)))
377 lemma = sup-o< 453 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t)))
454 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord (ZFSubset (Ord a) (ord→od x)))
455 lemma = sup-o< _ lemma2
378 456
379 -- 457 --
380 -- Every set in OD is a subset of Ordinals, so make Def (Ord (od→ord A)) first 458 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
381 -- then replace of all elements of the Power set by A ∩ y 459 -- then replace of all elements of the Power set by A ∩ y
382 -- 460 --
383 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) 461 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )
384 462
385 -- we have oly double negation form because of the replacement axiom 463 -- we have oly double negation form because of the replacement axiom
386 -- 464 --
387 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) 465 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
388 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where 466 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
389 a = od→ord A 467 a = od→ord A
390 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) 468 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
391 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t 469 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t
392 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) 470 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
393 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) 471 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
394 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) 472 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
395 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) 473 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
396 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) 474 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x))
397 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not 475 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
398 476
399 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t 477 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
400 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 478 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
401 a = od→ord A 479 a = od→ord A
402 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x 480 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
403 lemma0 {x} t∋x = c<→o< (t→A t∋x) 481 lemma0 {x} t∋x = c<→o< (t→A t∋x)
404 lemma3 : Def (Ord a) ∋ t 482 lemma3 : OPwr (Ord a) ∋ t
405 lemma3 = ord-power← a t lemma0 483 lemma3 = ord-power← a t lemma0
406 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t 484 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
407 lemma4 = let open ≡-Reasoning in begin 485 lemma4 = let open ≡-Reasoning in begin
408 A ∩ ord→od (od→ord t) 486 A ∩ ord→od (od→ord t)
409 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ 487 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
410 A ∩ t 488 A ∩ t
411 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ 489 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
412 t 490 t
413 ∎ 491 ∎
414 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ x)) 492 sup1 : Ordinal
415 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ x))) 493 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord (ZFSubset (Ord (od→ord A)) (ord→od x)))
416 lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) 494 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A)))
417 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) 495 lemma9 = <-osuc
496 lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1
497 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9
498 lemmad : Ord (osuc (od→ord A)) ∋ t
499 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt)))
500 lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A)
501 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
502 lemmaf : {x : Ordinal} → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x
503 lemmaf {x} lt = proj1 lt
504 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x
505 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt }
506 lemmae : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A))
507 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac)
508 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t)
509 lemma7 with osuc-≡< lemmad
510 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
511 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where
512 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x
513 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t))
514 diso
515 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt )))
516 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
517 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t
518 lemmai = let open ≡-Reasoning in begin
519 od→ord (Ord (od→ord A))
520 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩
521 od→ord (Ord (od→ord t))
522 ≡⟨ sym diso ⟩
523 od→ord (ord→od (od→ord (Ord (od→ord t))))
524 ≡⟨ sym eq1 ⟩
525 od→ord (ord→od (od→ord t))
526 ≡⟨ diso ⟩
527 od→ord t
528 ∎
529 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
530 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A))
531 lemmak = let open ≡-Reasoning in begin
532 od→ord (ord→od (od→ord (Ord (od→ord t))))
533 ≡⟨ diso ⟩
534 od→ord (Ord (od→ord t))
535 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩
536 od→ord (Ord (od→ord A))
537 ∎
538 lemmaj : od→ord t o< od→ord (Ord (od→ord A))
539 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt
540 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))
541 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))))
542 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 )
543 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
418 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where 544 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
419 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 545 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
420 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) 546 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A )))
547
421 548
422 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) 549 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
423 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where 550 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where
424 lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y 551 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y
425 lemma lt y<x with osuc-≡< lt 552 lemma lt y<x with osuc-≡< lt
426 lemma lt y<x | case1 refl = c<→o< y<x 553 lemma lt y<x | case1 refl = c<→o< y<x
427 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a 554 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a
428 555
429 continuum-hyphotheis : (a : Ordinal) → Set (suc n) 556 continuum-hyphotheis : (a : Ordinal) → Set (suc n)
430 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) 557 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)
431 558
432 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B 559 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
433 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d 560 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
434 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d 561 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
435 562
436 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) 563 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
437 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d 564 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
438 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d 565 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
439 566
440 infinity∅ : infinite ∋ od∅ 567 infinity∅ : infinite ∋ od∅
441 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where 568 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
442 lemma : o∅ ≡ od→ord od∅ 569 lemma : o∅ ≡ od→ord od∅
443 lemma = let open ≡-Reasoning in begin 570 lemma = let open ≡-Reasoning in begin
444 o∅ 571 o∅
445 ≡⟨ sym diso ⟩ 572 ≡⟨ sym diso ⟩
446 od→ord ( ord→od o∅ ) 573 od→ord ( ord→od o∅ )
447 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ 574 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
448 od→ord od∅ 575 od→ord od∅
449 ∎ 576 ∎
450 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) 577 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
451 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where 578 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
452 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) 579 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
453 ≡ od→ord (Union (x , (x , x))) 580 ≡ od→ord (Union (x , (x , x)))
454 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso 581 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
455 582
456 583
457 Union = ZF.Union OD→ZF 584 Union = ZF.Union HOD→ZF
458 Power = ZF.Power OD→ZF 585 Power = ZF.Power HOD→ZF
459 Select = ZF.Select OD→ZF 586 Select = ZF.Select HOD→ZF
460 Replace = ZF.Replace OD→ZF 587 Replace = ZF.Replace HOD→ZF
461 isZF = ZF.isZF OD→ZF 588 isZF = ZF.isZF HOD→ZF