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comparison ordinal-definable.agda @ 104:d92411bed18c

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 16 Jun 2019 02:06:09 +0900
parents c8b79d303867
children ec6235ce0215
comparison
equal deleted inserted replaced
103:c8b79d303867 104:d92411bed18c
55 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} 55 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
56 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ 56 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ
57 -- a contra-position of minimality of supermum 57 -- a contra-position of minimality of supermum
58 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} 58 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
59 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) 59 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
60 -- sup-min : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → {z : Ordinal {n}} → ψ z o< z → sup-o ψ o< osuc z
61 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
62 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
63 minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) )
64
60 65
61 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n 66 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
62 _∋_ {n} a x = def a ( od→ord x ) 67 _∋_ {n} a x = def a ( od→ord x )
63 68
64 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n 69 Ord : { n : Level } → ( a : Ordinal {suc n} ) → OD {suc n}
65 x c< a = od→ord x o< od→ord a 70 Ord {n} a = record { def = λ y → y o< a }
66 71
67 postulate 72 _c<_ : { n : Level } → ( x a : Ordinal {n} ) → Set n
68 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y 73 x c< a = Ord a ∋ Ord x
74
75 c<→o< : { n : Level } → { x a : OD {n} } → record { def = λ y → y o< od→ord a } ∋ x → od→ord x o< od→ord a
76 c<→o< lt = lt
77
78 o<→c< : { n : Level } → { x a : OD {n} } → od→ord x o< od→ord a → record { def = λ y → y o< od→ord a } ∋ x
79 o<→c< lt = lt
80
81 ==→o≡' : {n : Level} → { x y : Ordinal {suc n} } → Ord x == Ord y → x ≡ y
82 ==→o≡' {n} {x} {y} eq with trio< {n} x y
83 ==→o≡' {n} {x} {y} eq | tri< a ¬b ¬c with eq← eq {x} a
84 ... | t = ⊥-elim ( o<¬≡ x x refl t )
85 ==→o≡' {n} {x} {y} eq | tri≈ ¬a refl ¬c = refl
86 ==→o≡' {n} {x} {y} eq | tri> ¬a ¬b c with eq→ eq {y} c
87 ... | t = ⊥-elim ( o<¬≡ y y refl t )
88
89 ∅∨ : { n : Level } → { x y : Ordinal {suc n} } → ( Ord {n} x == Ord y ) ∨ ( ¬ ( Ord x == Ord y ) )
90 ∅∨ {n} {x} {y} with trio< x y
91 ∅∨ {n} {x} {y} | tri< a ¬b ¬c = case2 ( λ eq → ¬b ( ==→o≡' eq ) )
92 ∅∨ {n} {x} {y} | tri≈ ¬a refl ¬c = case1 ( record { eq→ = id ; eq← = id } )
93 ∅∨ {n} {x} {y} | tri> ¬a ¬b c = case2 ( λ eq → ¬b ( ==→o≡' eq ) )
94
95 ¬x∋x' : { n : Level } → { x : Ordinal {n} } → ¬ ( record { def = λ y → y o< x } ∋ record { def = λ y → y o< x } )
96 ¬x∋x' {n} {record { lv = Zero ; ord = ord }} (case1 ())
97 ¬x∋x' {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = lx ; ord = Φ lx }} (case1 {!!})
98 ¬x∋x' {n} {record { lv = Suc lx ; ord = OSuc (Suc lx) ox }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = Suc lx ; ord = ox}} (case1 {!!})
99 ¬x∋x' {n} {record { lv = lv ; ord = Φ (lv) }} (case2 ())
100 ¬x∋x' {n} {record { lv = lv ; ord = OSuc (lv) ox }} (case2 x) =
101 ¬x∋x' {n} {record { lv = lv ; ord = ox }} (case2 {!!})
102
103 ¬x∋x : { n : Level } → { x : OD {n} } → ¬ x ∋ x
104 ¬x∋x = {!!}
105
106 oc-lemma : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → def (record { def = λ y → y o< oa }) oa → ⊥
107 oc-lemma {n} {x} {oa} lt = o<¬≡ oa oa refl lt
108
109 oc-lemma1 : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → od→ord (record { def = λ y → y o< oa }) o< oa → ⊥
110 oc-lemma1 {n} {x} {oa} lt = ¬x∋x' {n} lt -- lt : def (record { def = λ y → y o< oa }) (record { def = λ y → y o< oa })
111
112 oc-lemma2 : { n : Level } → { x a : OD {n} } → { oa : Ordinal {n} } → oa o< od→ord (record { def = λ y → y o< oa }) → ⊥
113 oc-lemma2 {n} {x} {oa} lt = {!!}
69 114
70 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) 115 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n)
71 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) 116 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
72 117
73 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x 118 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
74 def-subst df refl refl = df 119 def-subst df refl refl = df
75 120
76 -- sup-min : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → {z : Ordinal {n}} → ψ z o< z → sup-o ψ o< osuc z 121 o<-def : {n : Level } {x y : Ordinal {n} } → x o< y → def (record { def = λ x → x o< y }) x
122 o<-def x<y = x<y
123
124 def-o< : {n : Level } {x y : Ordinal {n} } → def (record { def = λ x → x o< y }) x → x o< y
125 def-o< x<y = x<y
77 126
78 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} 127 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n}
79 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) 128 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
80 129
81 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) 130 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
82 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )} 131 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )}
83 {!!} refl (cong ( λ k → od→ord (ψ k) ) oiso) 132 {!!} refl (cong ( λ k → od→ord (ψ k) ) oiso)
84 133
85 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) 134 od∅' : {n : Level} → OD {n}
86 ∅1 {n} x = {!!} 135 od∅' = record { def = λ x → x o< o∅ }
136
137 ∅0 : {n : Level} → od∅ {suc n} == record { def = λ x → x o< o∅ }
138 eq→ ∅0 {w} (lift ())
139 eq← ∅0 {w} (case1 ())
140 eq← ∅0 {w} (case2 ())
141
142 ∅1 : {n : Level} → ( x : Ordinal {n} ) → ¬ ( x c< o∅ {n} )
143 ∅1 {n} x lt = {!!}
87 144
88 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} 145 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
89 ∅3 {n} {x} = TransFinite {n} c2 c3 x where 146 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
90 c0 : Nat → Ordinal {n} → Set n 147 c0 : Nat → Ordinal {n} → Set n
91 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} 148 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n}
100 c3 lx (Φ .lx) d not | t | () 157 c3 lx (Φ .lx) d not | t | ()
101 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) 158 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } )
102 ... | t with t (case2 (s< s<refl ) ) 159 ... | t with t (case2 (s< s<refl ) )
103 c3 lx (OSuc .lx x₁) d not | t | () 160 c3 lx (OSuc .lx x₁) d not | t | ()
104 161
105 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x
106 transitive {n} {z} {y} {x} z∋y x∋y with ordtrans {!!} {!!}
107 ... | t = lemma0 (lemma t) where
108 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x)
109 lemma xo<z = {!!}
110 lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x)
111 lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl
112
113 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x 162 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
114 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) 163 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
115 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< 164 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
116 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) 165 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n)
117 166
164 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) 213 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso )))
165 214
166 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) 215 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } )
167 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where 216 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where
168 lemma : ord→od x == record { def = λ z → z o< x } 217 lemma : ord→od x == record { def = λ z → z o< x }
169 eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where 218 eq→ lemma {w} lt = {!!}
170 t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) 219 -- ?subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where
171 t = {!!} 220 --t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
172 eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} {!!} refl refl 221 --t = o<-subst lt ? ?
222 eq← lemma {w} lt = def-subst {suc n} {_} {_} {ord→od x} {w} {!!} refl refl
173 223
174 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } 224 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x }
175 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) 225 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} ))
176 226
177 ==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y 227 ==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y
191 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x 241 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x
192 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where 242 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where
193 t : def (ord→od (od→ord a)) (od→ord x) 243 t : def (ord→od (od→ord a)) (od→ord x)
194 t = {!!} 244 t = {!!}
195 245
196 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} 246 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅' {suc n}
197 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) 247 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅' {suc n} ))
198 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where 248 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
199 lemma : o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥ 249 lemma : o∅ {suc n } o< (od→ord (od∅' {suc n} )) → ⊥
200 lemma lt with def-subst {!!} oiso refl 250 lemma lt with def-subst {suc n} {_} {_} {_} {_} ( o<→c< ( o<-subst lt (sym diso) refl ) ) refl diso
201 lemma lt | t = {!!} 251 lemma lt | t = {!!}
202 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso 252 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
203 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) 253 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
204 254
205 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) 255 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y )
206 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt 256 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
207 257
208 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) 258 o<→¬c> : {n : Level} → { x y : Ordinal {n} } → x o< y → ¬ (y c< x )
209 o<→¬c> {n} {x} {y} olt clt = o<> olt {!!} where 259 o<→¬c> {n} {x} {y} olt clt = o<> olt {!!} where
210 260
211 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y 261 o≡→¬c< : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ¬ x c< y
212 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) lt 262 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ x y {!!} {!!}
213 263
214 tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) 264 tri-c< : {n : Level} → Trichotomous _≡_ (_c<_ {suc n})
215 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) 265 tri-c< {n} x y with trio< {n} x y
216 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst {!!} oiso refl) (o<→¬== a) ( o<→¬c> a ) 266 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst {!!} oiso refl) {!!} ( o<→¬c> a )
217 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) 267 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) {!!} (o≡→¬c< (sym b))
218 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst {!!} oiso refl) 268 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → {!!} ) (def-subst {!!} oiso refl)
219 269
220 c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ 270 c<> : {n : Level } { x y : Ordinal {suc n}} → x c< y → y c< x → ⊥
221 c<> {n} {x} {y} x<y y<x with tri-c< x y 271 c<> {n} {x} {y} x<y y<x with tri-c< x y
222 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x 272 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
223 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b x<y 273 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> {!!} {!!}
224 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y 274 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
225 275
226 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) 276 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} )
227 ∅< {n} {x} {y} d eq with eq→ eq d 277 ∅< {n} {x} {y} d eq with eq→ eq d
228 ∅< {n} {x} {y} d eq | lift () 278 ∅< {n} {x} {y} d eq | lift ()
229 279
230 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox 280 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox
231 ∅6 {n} {x} x∋x = c<> {n} {x} {x} {!!} {!!} 281 ∅6 {n} {x} x∋x = c<> {n} {{!!}} {{!!}} {!!} {!!}
232 282
233 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x 283 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x
234 def-iso refl t = t 284 def-iso refl t = t
235
236 is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y )
237 is-∋ {n} x y with tri-c< x y
238 is-∋ {n} x y | tri< a ¬b ¬c = no {!!}
239 is-∋ {n} x y | tri≈ ¬a b ¬c = no {!!}
240 is-∋ {n} x y | tri> ¬a ¬b c = yes {!!}
241 285
242 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) 286 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
243 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl 287 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
244 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) 288 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
245 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) 289 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
250 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where 294 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
251 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} 295 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
252 lemma ox ne with is-o∅ ox 296 lemma ox ne with is-o∅ ox
253 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where 297 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where
254 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ 298 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅
255 lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ 299 lemma1 = cong ( λ k → od→ord k ) {!!}
256 lemma o∅ ne | yes refl | () 300 lemma o∅ ne | yes refl | ()
257 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ {!!} 301 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) {!!} {!!}
258 302
259 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 303 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
260 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) 304 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
261 305
262 csuc : {n : Level} → OD {suc n} → OD {suc n} 306 csuc : {n : Level} → OD {suc n} → OD {suc n}
266 310
267 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} 311 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n}
268 ZFSubset A x = record { def = λ y → def A y ∧ def x y } 312 ZFSubset A x = record { def = λ y → def A y ∧ def x y }
269 313
270 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} 314 Def : {n : Level} → (A : OD {suc n}) → OD {suc n}
271 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) 315 Def {n} A = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )))) }
272 316
273 -- Constructible Set on α 317 -- Constructible Set on α
274 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} 318 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n}
275 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ 319 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅
276 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) 320 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) )
277 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) 321 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
278 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } 322 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) }
279 323
280 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} 324 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
281 OD→ZF {n} = record { 325 OD→ZF {n} = record {
282 ZFSet = OD {suc n} 326 ZFSet = OD {suc n}
283 ; _∋_ = _∋_ 327 ; _∋_ = _∋_
394 } 438 }
395 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x 439 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x
396 replacement {ψ} X x = sup-c< ψ {x} 440 replacement {ψ} X x = sup-c< ψ {x}
397 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 441 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅)
398 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq 442 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq
399 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
400 minimul x not = od∅
401 regularity : (x : OD) (not : ¬ (x == od∅)) → 443 regularity : (x : OD) (not : ¬ (x == od∅)) →
402 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) 444 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
403 proj1 (regularity x not ) = ¬∅=→∅∈ not 445 proj1 (regularity x not ) = x∋minimul x not
404 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where 446 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where
405 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y 447 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
406 reg {y} t with proj1 t 448 reg {y} t with minimul-1 x not (ord→od y) (proj2 t )
407 ... | x∈∅ = x∈∅ 449 ... | t1 = lift t1
408 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B 450 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
409 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d 451 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
410 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d 452 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
411 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } 453 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
412 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) 454 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x))
430 omega = record { lv = Suc Zero ; ord = Φ 1 } 472 omega = record { lv = Suc Zero ; ord = Φ 1 }
431 infinite : OD {suc n} 473 infinite : OD {suc n}
432 infinite = ord→od ( omega ) 474 infinite = ord→od ( omega )
433 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} 475 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
434 infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} 476 infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅}
435 {!!} refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) o∅≡od∅ )) 477 {!!} refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) {!!} ))
436 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega 478 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega
437 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where 479 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where
438 t : od→ord x o< od→ord (ord→od (omega)) 480 t : od→ord x o< od→ord (ord→od (omega))
439 t = ∋→o< {n} {infinite} {x} lt 481 t = ∋→o< {n} {infinite} {x} lt
440 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) 482 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x ))