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

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IChain?
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 11 Dec 2022 08:48:42 +0900
parents bd2a258f309c
children fc37082d2a8a
comparison
equal deleted inserted replaced
1059:bd2a258f309c 1060:a09f5e728f92
274 data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y ) 274 data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y )
275 (x : Ordinal) : (z : Ordinal) → Set n where 275 (x : Ordinal) : (z : Ordinal) → Set n where
276 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z 276 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z
277 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z 277 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z
278 278
279 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD 279 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD
280 UnionCF A f ay supf x 280 UnionCF A f ay supf x
281 = record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ; 281 = record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ;
282 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } 282 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
283 283
284 -- Union of all UChain under limit ordinal x
285
286 data IChain { A : HOD } { f : Ordinal → Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) {y : Ordinal } (ay : odef A y )
287 (x : Ordinal) : (z : Ordinal) → Set n where
288 ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain supf ay x z
289 ic-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u<x u ≡ u) ( fc : FClosure A f (supf u<x u) z ) → IChain supf ay x z
290
291 UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal) ( x : Ordinal ) → HOD
292 UnionIC A f ay supf x
293 = record { od = record { def = λ z → odef A z ∧ IChain {A} {f} supf ay x z } ;
294 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
284 295
285 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) 296 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y )
286 → supf x o< supf y → x o< y 297 → supf x o< supf y → x o< y
287 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y 298 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y
288 ... | tri< a ¬b ¬c = a 299 ... | tri< a ¬b ¬c = a
504 record Maximal ( A : HOD ) : Set (Level.suc n) where 515 record Maximal ( A : HOD ) : Set (Level.suc n) where
505 field 516 field
506 maximal : HOD 517 maximal : HOD
507 as : A ∋ maximal 518 as : A ∋ maximal
508 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative 519 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
509
510 record IChain (A : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) (z : Ordinal ) : Set n where
511 field
512 i : Ordinal
513 i<x : i o< x
514 fc : FClosure A f (supfz i<x) z
515 520
516 -- 521 --
517 -- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup 522 -- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup
518 -- 523 --
519 supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) 524 supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f)
1238 ; zo≤sz = ? ; supfmax = λ _ → refl ; is-minsup = ? ; cfcs = ? } where 1243 ; zo≤sz = ? ; supfmax = λ _ → refl ; is-minsup = ? ; cfcs = ? } where
1239 mf : ≤-monotonic-f A f 1244 mf : ≤-monotonic-f A f
1240 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where 1245 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
1241 mf00 : * x < * (f x) 1246 mf00 : * x < * (f x)
1242 mf00 = proj1 ( mf< x ax ) 1247 mf00 = proj1 ( mf< x ax )
1243 ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = ? ; supf-mono = supf-mono ; order = ? 1248 ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = asupf ; supf-mono = supf-mono ; order = ?
1244 ; zo≤sz = λ _ → ? ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs } where 1249 ; zo≤sz = ? ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs } where
1245 1250
1246 -- mf : ≤-monotonic-f A f 1251 -- mf : ≤-monotonic-f A f
1247 -- mf x ax = ? -- ⟪ case2 ( proj1 (mf< x ax)) , proj2 (mf< x ax ) ⟫ will result memory exhaust 1252 -- mf x ax = ? -- ⟪ case2 ( proj1 (mf< x ax)) , proj2 (mf< x ax ) ⟫ will result memory exhaust
1248 1253
1249 mf : ≤-monotonic-f A f 1254 mf : ≤-monotonic-f A f