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

comparison ordinal.agda @ 349:adc3c3a37308

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 14 Jul 2020 09:00:24 +0900
parents feb0fcc430a9
children 7389120cd6c0
comparison
equal deleted inserted replaced
348:08d94fec239c 349:adc3c3a37308
218 ; ¬x<0 = ¬x<0 218 ; ¬x<0 = ¬x<0
219 ; <-osuc = <-osuc 219 ; <-osuc = <-osuc
220 ; osuc-≡< = osuc-≡< 220 ; osuc-≡< = osuc-≡<
221 ; TransFinite = TransFinite1 221 ; TransFinite = TransFinite1
222 ; TransFinite1 = TransFinite2 222 ; TransFinite1 = TransFinite2
223 ; not-limit = not-limit 223 ; not-limit-p = not-limit
224 ; next-limit = next-limit 224 } ;
225 isNext = record {
226 x<nx = x<nx
227 ; osuc<nx = λ {x} {y} → osuc<nx {x} {y}
228 ; ¬nx<nx = ¬nx<nx
225 } 229 }
226 } where 230 } where
227 next : Ordinal {suc n} → Ordinal {suc n} 231 next : Ordinal {suc n} → Ordinal {suc n}
228 next (ordinal lv ord) = ordinal (Suc lv) (Φ (Suc lv)) 232 next (ordinal lv ord) = ordinal (Suc lv) (Φ (Suc lv))
229 next-limit : {y : Ordinal} → (y o< next y) ∧ ((x : Ordinal) → x o< next y → osuc x o< next y) ∧ 233 x<nx : { y : Ordinal } → (y o< next y )
230 ( (x : Ordinal) → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ (x ≡ osuc z) )) 234 x<nx = case1 a<sa
231 next-limit {y} = record { proj1 = case1 a<sa ; proj2 = record { proj1 = lemma ; proj2 = lemma2 } } where 235 osuc<nx : { x y : Ordinal } → x o< next y → osuc x o< next y
232 lemma : (x : Ordinal) → x o< next y → osuc x o< next y 236 osuc<nx (case1 lt) = case1 lt
233 lemma x (case1 lt) = case1 lt 237 ¬nx<nx : {x y : Ordinal} → y o< x → x o< osuc (next y) → ¬ ((z : Ordinal) → ¬ (x ≡ osuc z))
234 lemma2 : (x : Ordinal) → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ x ≡ osuc z) 238 ¬nx<nx {x} {y} = lemma2 x where
239 lemma2 : (x : Ordinal) → y o< x → x o< osuc (next y) → ¬ ((z : Ordinal) → ¬ x ≡ osuc z)
235 lemma2 (ordinal Zero (Φ 0)) (case2 ()) (case1 (s≤s z≤n)) not 240 lemma2 (ordinal Zero (Φ 0)) (case2 ()) (case1 (s≤s z≤n)) not
236 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 Φ<) (case1 (s≤s z≤n)) not = not _ refl 241 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 Φ<) (case1 (s≤s z≤n)) not = not _ refl
237 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 (s< x)) (case1 (s≤s z≤n)) not = not _ refl 242 lemma2 (ordinal Zero (OSuc 0 dx)) (case2 (s< x)) (case1 (s≤s z≤n)) not = not _ refl
238 lemma2 (ordinal (Suc lx) (OSuc (Suc lx) ox)) y<x (case1 (s≤s (s≤s lt))) not = not _ refl 243 lemma2 (ordinal (Suc lx) (OSuc (Suc lx) ox)) y<x (case1 (s≤s (s≤s lt))) not = not _ refl
239 lemma2 (ordinal (Suc lx) (Φ (Suc lx))) (case1 x) (case1 (s≤s (s≤s lt))) not = lemma3 x lt where 244 lemma2 (ordinal (Suc lx) (Φ (Suc lx))) (case1 x) (case1 (s≤s (s≤s lt))) not = lemma3 x lt where
240 lemma3 : {n l : Nat} → (Suc (Suc n) ≤ Suc l) → l ≤ n → ⊥ 245 lemma3 : {n l : Nat} → (Suc (Suc n) ≤ Suc l) → l ≤ n → ⊥
241 lemma3 (s≤s sn≤l) (s≤s l≤n) = lemma3 sn≤l l≤n 246 lemma3 (s≤s sn≤l) (s≤s l≤n) = lemma3 sn≤l l≤n
247 lemma2 (ordinal (Suc lx) (Φ (Suc lx))) (case1 lt) (case2 lt2) not = {!!}
248 lemma2 (ordinal (Suc lx) (OSuc (Suc lx) os)) lt (case2 lt2) not = {!!}
249
242 not-limit : (x : Ordinal) → Dec (¬ ((y : Ordinal) → ¬ (x ≡ osuc y))) 250 not-limit : (x : Ordinal) → Dec (¬ ((y : Ordinal) → ¬ (x ≡ osuc y)))
243 not-limit (ordinal lv (Φ lv)) = no (λ not → not (λ y () )) 251 not-limit (ordinal lv (Φ lv)) = no (λ not → not (λ y () ))
244 not-limit (ordinal lv (OSuc lv ox)) = yes (λ not → not (ordinal lv ox) refl ) 252 not-limit (ordinal lv (OSuc lv ox)) = yes (λ not → not (ordinal lv ox) refl )
245 ord1 : Set (suc n) 253 ord1 : Set (suc n)
246 ord1 = Ordinal {suc n} 254 ord1 = Ordinal {suc n}