Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/filter-util.agda @ 1476:32001d93755b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 28 Jun 2024 20:55:38 +0900 |
| parents | 5d69ed581269 |
| children | 88fdc41868f9 |
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{-# OPTIONS --cubical-compatible --safe #-} open import Level open import Ordinals open import logic open import Relation.Nullary open import Level open import Ordinals import HODBase import OD open import Relation.Nullary module filter-util {n : Level } (O : Ordinals {n} ) (HODAxiom : HODBase.ODAxiom O) (ho< : OD.ODAxiom-ho< O HODAxiom ) (AC : OD.AxiomOfChoice O HODAxiom ) where open import Relation.Binary.PropositionalEquality hiding ( [_] ) open import Data.Empty import OrdUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal import ODUtil open import logic open import nat open OrdUtil O open ODUtil O HODAxiom ho< open _∧_ open _∨_ open Bool open HODBase._==_ open HODBase.ODAxiom HODAxiom open OD O HODAxiom open HODBase.HOD open AxiomOfChoice AC open import ODC O HODAxiom AC as ODC open import filter O HODAxiom ho< AC open import ZProduct O HODAxiom ho< open Filter filter-⊆ : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → {x : HOD} → filter F ∋ x → { z : Ordinal } → odef x z → odef (ZFP P Q) z filter-⊆ {P} {Q} F {x} fx {z} xz = f⊆L F fx _ (subst (λ k → odef k z) ? xz ) rcp : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → RXCod (filter F) P (λ x fx → ZP-proj1 P Q x (filter-⊆ F fx)) rcp {P} {Q} F = record { ≤COD = λ {x} fx {z} ly → ZP1.aa ly } Filter-Proj1 : {P Q a : HOD } → ZFP P Q ∋ a → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → Filter {Power P} {P} (λ x → x) Filter-Proj1 {P} {Q} pqa F = record { filter = FP ; f⊆L = fp00 ; filter1 = f1 ; filter2 = f2 } where FP : HOD FP = Replace' (filter F) (λ x fx → ZP-proj1 P Q x (filter-⊆ F fx)) {P} (rcp F) isP→PxQ : {x : HOD} → (x⊆P : x ⊆ P ) → ZFP x Q ⊆ ZFP P Q isP→PxQ {x} x⊆P (ab-pair p q) = ab-pair (x⊆P p) q isQ→PxQ : {x : HOD} → (x⊆P : x ⊆ Q ) → ZFP P x ⊆ ZFP P Q isQ→PxQ {x} x⊆Q (ab-pair p q) = ab-pair p (x⊆Q q) fp00 : FP ⊆ Power P fp00 {x} record { z = z ; az = az ; x=ψz = x=ψz } w xw with subst (λ k → odef k w) (trans (cong (*) x=ψz) ? ) xw ... | t = ? -- record { b = b ; aa = aa ; bb = bb ; c∋ab = c∋ab } = aa f0 : {p q : HOD} → Power (ZFP P Q) ∋ q → filter F ∋ p → p ⊆ q → filter F ∋ q f0 {p} {q} PQq fp p⊆q = filter1 F PQq fp p⊆q f1 : {p q : HOD} → Power P ∋ q → FP ∋ p → p ⊆ q → FP ∋ q f1 {p} {q} Pq record { z = z ; az = az ; x=ψz = x=ψz } p⊆q = record { z = & (ZFP q Q) ; az = fp01 ty05 ty06 ; x=ψz = q=proj1 } where PQq : Power (ZFP P Q) ∋ ZFP q Q PQq z zpq = isP→PxQ {* (& q)} (Pq _) ( subst (λ k → odef k z ) (trans ? (cong (λ k → ZFP k Q) ?)) zpq ) q⊆P : q ⊆ P q⊆P {w} qw = Pq _ (subst (λ k → odef k w ) (sym ?) qw ) p⊆P : p ⊆ P p⊆P {w} pw = q⊆P (p⊆q pw) p=proj1 : & p ≡ & (ZP-proj1 P Q (* z) (filter-⊆ F (subst (odef (filter F)) (sym &iso) az))) p=proj1 = x=ψz p⊆ZP : (* z) ⊆ ZFP p Q p⊆ZP = subst (λ k → (* z) ⊆ ZFP k Q) (sym ?) ZP-proj1⊆ZFP ty05 : filter F ∋ ZFP p Q ty05 = filter1 F (λ z wz → isP→PxQ p⊆P (subst (λ k → odef k z) ? wz)) (subst (λ k → odef (filter F) k) (sym &iso) az) p⊆ZP ty06 : ZFP p Q ⊆ ZFP q Q ty06 (ab-pair wp wq ) = ab-pair (p⊆q wp) wq fp01 : filter F ∋ ZFP p Q → ZFP p Q ⊆ ZFP q Q → filter F ∋ ZFP q Q fp01 fzp zp⊆zq = filter1 F PQq fzp zp⊆zq q=proj1 : & q ≡ & (ZP-proj1 P Q (* (& (ZFP q Q))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (fp01 ty05 ty06)))) q=proj1 = ? -- cong (&) (ZP-proj1=rev (zp2 pqa) q⊆P *iso ) f2 : {p q : HOD} → FP ∋ p → FP ∋ q → Power P ∋ (p ∩ q) → FP ∋ (p ∩ q) f2 {p} {q} record { z = zp ; az = fzp ; x=ψz = x=ψzp } record { z = zq ; az = fzq ; x=ψz = x=ψzq } Ppq = record { z = _ ; az = ty50 ; x=ψz = pq=proj1 } where p⊆P : {zp : Ordinal} {p : HOD} (fzp : odef (filter F) zp) → ( & p ≡ & (ZP-proj1 P Q (* zp) (filter-⊆ F (subst (odef (filter F)) (sym &iso) fzp)))) → p ⊆ P p⊆P {zp} {p} fzp p=proj1 {x} px with subst (λ k → odef k x) ? px ... | t = ? -- record { b = b ; aa = aa ; bb = bb ; c∋ab = c∋ab } = aa x⊆pxq : {zp : Ordinal} {p : HOD} (fzp : odef (filter F) zp) → ( & p ≡ & (ZP-proj1 P Q (* zp) (filter-⊆ F (subst (odef (filter F)) (sym &iso) fzp)))) → * zp ⊆ ZFP p Q x⊆pxq {zp} {p} fzp p=proj1 = subst (λ k → (* zp) ⊆ ZFP k Q) (sym ?) ZP-proj1⊆ZFP ty54 : Power (ZFP P Q) ∋ (ZFP p Q ∩ ZFP q Q) ty54 z xz = subst (λ k → ZFProduct P Q k ) (zp-iso pqz) (ab-pair pqz1 pqz2 ) where pqz : odef (ZFP (p ∩ q) Q) z pqz = ? -- subst (λ k → odef k z ) (trans ? (sym (proj1 ZFP∩) )) xz pqz1 : odef P (zπ1 pqz) pqz1 = p⊆P fzp x=ψzp (proj1 (zp1 pqz)) pqz2 : odef Q (zπ2 pqz) pqz2 = zp2 pqz ty53 : filter F ∋ ZFP p Q ty53 = filter1 F (λ z wz → isP→PxQ (p⊆P fzp x=ψzp) ?) ? ? -- (subst (λ k → odef (filter F) k) (sym &iso) fzp ) (x⊆pxq fzp x=ψzp) ty52 : filter F ∋ ZFP q Q ty52 = filter1 F (λ z wz → isP→PxQ (p⊆P fzq x=ψzq) ?) ? (x⊆pxq fzq x=ψzq) ty51 : filter F ∋ ( ZFP p Q ∩ ZFP q Q ) ty51 = filter2 F ty53 ty52 ty54 ty50 : filter F ∋ ZFP (p ∩ q) Q ty50 = subst (λ k → filter F ∋ k ) ? ty51 pq=proj1 : & (p ∩ q) ≡ & (ZP-proj1 P Q (* (& (ZFP (p ∩ q) Q))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) ty50))) pq=proj1 = ? -- cong (&) (ZP-proj1=rev (zp2 pqa) (λ {x} pqx → Ppq _ (subst (λ k → odef k x) ? pqx)) *iso ) Filter-Proj1-UF : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) → ultra-filter (Filter-Proj1 pqa F) Filter-Proj1-UF {P} {Q} pqa F UF = record { proper = ty60 ; ultra = ty62 } where FP = Filter-Proj1 pqa F ty60 : ¬ (filter FP ∋ od∅) ty60 record { z = z ; az = az ; x=ψz = x=ψz } = ⊥-elim (ultra-filter.proper UF (filter1 F (λ x x<0 → ⊥-elim (¬x<0 (subst (λ k → odef k x) ? x<0))) (subst (λ k → odef (filter F) k ) (sym &iso) az) ty61 )) where ty61 : * z ⊆ od∅ ty61 {x} lt = ? -- ⊥-elim (¬x<0 (subst (λ k → odef k x) (trans (cong (*) (ZP-proj1-0 (sym ?))) *iso) lt )) ty62 : {p : HOD} → Power P ∋ p → Power P ∋ (P ＼ p) → (filter (Filter-Proj1 pqa F) ∋ p) ∨ (filter (Filter-Proj1 pqa F) ∋ (P ＼ p)) ty62 {p} Pp NEGP = uf05 where p⊆P : p ⊆ P p⊆P {z} px = Pp _ (subst (λ k → odef k z) ? px) mp : HOD mp = ZFP p Q uf03 : Power (ZFP P Q) ∋ mp uf03 x xz with subst (λ k → odef k x ) ? xz ... | t = ? -- ab-pair ax by = ab-pair (p⊆P ax) by uf04 : Power (ZFP P Q) ∋ (ZFP P Q ＼ mp) uf04 x xz = ? -- proj1 (subst (λ k → odef k x) *iso xz) uf02 : (filter F ∋ mp) ∨ (filter F ∋ (ZFP P Q ＼ mp)) uf02 = ultra-filter.ultra UF uf03 uf04 uf05 : (filter FP ∋ p) ∨ (filter FP ∋ (P ＼ p)) uf05 with uf02 ... | case1 fp = case1 record { z = _ ; az = fp ; x=ψz = ? } ... | case2 fnp = case2 record { z = _ ; az = fnp ; x=ψz = ? } rcq : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → RXCod (filter F) Q (λ x fx → ZP-proj2 P Q x (filter-⊆ F fx)) rcq {P} {Q} F = record { ≤COD = λ {x} fx {z} ly → ZP2.bb ly } -- -- Proj2 can be dervie from symmetry of ZFP (Product) -- but it makes Agda very slow -- so we copy Proj1 -- Filter-Proj2 : {P Q a : HOD } → ZFP P Q ∋ a → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → Filter {Power Q} {Q} (λ x → x) Filter-Proj2 {P} {Q} pqa F = record { filter = FQ ; f⊆L = fp00 ; filter1 = f1 ; filter2 = f2 } where FQ : HOD FQ = Replace' (filter F) (λ x fx → ZP-proj2 P Q x (filter-⊆ F fx)) {Q} (rcq F) isP→PxQ : {x : HOD} → (x⊆P : x ⊆ P ) → ZFP x Q ⊆ ZFP P Q isP→PxQ {x} x⊆P (ab-pair p q) = ab-pair (x⊆P p) q isQ→PxQ : {x : HOD} → (x⊆P : x ⊆ Q ) → ZFP P x ⊆ ZFP P Q isQ→PxQ {x} x⊆Q (ab-pair p q) = ab-pair p (x⊆Q q) fp00 : FQ ⊆ Power Q fp00 {x} record { z = z ; az = az ; x=ψz = x=ψz } w xw with subst (λ k → odef k w) (trans (cong (*) x=ψz) ? ) xw ... | t = ? -- record { a = a ; aa = aa ; bb = bb ; c∋ab = c∋ab } = bb f0 : {p q : HOD} → Power (ZFP P Q) ∋ q → filter F ∋ p → p ⊆ q → filter F ∋ q f0 {p} {q} PQq fp p⊆q = filter1 F PQq fp p⊆q f1 : {p q : HOD} → Power Q ∋ q → FQ ∋ p → p ⊆ q → FQ ∋ q f1 {p} {q} Qq record { z = z ; az = az ; x=ψz = x=ψz } p⊆q = record { z = & (ZFP P q) ; az = fp01 ty05 ty06 ; x=ψz = q=proj2 } where PQq : Power (ZFP P Q) ∋ ZFP P q PQq z zpq = isQ→PxQ {* (& q)} (Qq _) ( subst (λ k → odef k z ) ? zpq ) q⊆P : q ⊆ Q q⊆P {w} qw = Qq _ (subst (λ k → odef k w ) ? qw ) p⊆P : p ⊆ Q p⊆P {w} pw = q⊆P (p⊆q pw) p=proj2 : & p ≡ & (ZP-proj2 P Q (* z) (filter-⊆ F (subst (odef (filter F)) (sym &iso) az))) p=proj2 = x=ψz p⊆ZP : (* z) ⊆ ZFP P p p⊆ZP = subst (λ k → (* z) ⊆ ZFP P k ) (sym ?) ZP-proj2⊆ZFP ty05 : filter F ∋ ZFP P p ty05 = filter1 F (λ z wz → isQ→PxQ p⊆P ?) (subst (λ k → odef (filter F) k) (sym &iso) az) p⊆ZP ty06 : ZFP P p ⊆ ZFP P q ty06 (ab-pair wp wq ) = ab-pair wp (p⊆q wq) fp01 : filter F ∋ ZFP P p → ZFP P p ⊆ ZFP P q → filter F ∋ ZFP P q fp01 fzp zp⊆zq = filter1 F PQq fzp zp⊆zq q=proj2 : & q ≡ & (ZP-proj2 P Q (* (& (ZFP P q))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (fp01 ty05 ty06)))) q=proj2 = ? -- cong (&) (ZP-proj2=rev (zp1 pqa) q⊆P *iso ) f2 : {p q : HOD} → FQ ∋ p → FQ ∋ q → Power Q ∋ (p ∩ q) → FQ ∋ (p ∩ q) f2 {p} {q} record { z = zp ; az = fzp ; x=ψz = x=ψzp } record { z = zq ; az = fzq ; x=ψz = x=ψzq } Ppq = record { z = _ ; az = ty50 ; x=ψz = pq=proj2 } where p⊆Q : {zp : Ordinal} {p : HOD} (fzp : odef (filter F) zp) → ( & p ≡ & (ZP-proj2 P Q (* zp) (filter-⊆ F (subst (odef (filter F)) (sym &iso) fzp)))) → p ⊆ Q p⊆Q {zp} {p} fzp p=proj2 {x} px with subst (λ k → odef k x) ? px ... | t = ? -- record { a = a ; aa = aa ; bb = bb ; c∋ab = c∋ab } = bb x⊆pxq : {zp : Ordinal} {p : HOD} (fzp : odef (filter F) zp) → ( & p ≡ & (ZP-proj2 P Q (* zp) (filter-⊆ F (subst (odef (filter F)) (sym &iso) fzp)))) → * zp ⊆ ZFP P p x⊆pxq {zp} {p} fzp p=proj2 = subst (λ k → (* zp) ⊆ ZFP P k ) (sym ?) ZP-proj2⊆ZFP ty54 : Power (ZFP P Q) ∋ (ZFP P p ∩ ZFP P q ) ty54 z xz = subst (λ k → ZFProduct P Q k ) (zp-iso pqz) (ab-pair pqz1 pqz2 ) where pqz : odef (ZFP P (p ∩ q) ) z pqz = ? --- subst (λ k → odef k z ) (trans *iso (sym (proj2 ZFP∩) )) xz pqz1 : odef P (zπ1 pqz) pqz1 = zp1 pqz pqz2 : odef Q (zπ2 pqz) pqz2 = p⊆Q fzp x=ψzp (proj1 (zp2 pqz)) ty53 : filter F ∋ ZFP P p ty53 = filter1 F (λ z wz → isQ→PxQ (p⊆Q fzp x=ψzp) ?) ? (x⊆pxq fzp x=ψzp) ty52 : filter F ∋ ZFP P q ty52 = filter1 F (λ z wz → isQ→PxQ (p⊆Q fzq x=ψzq) ?) ? (x⊆pxq fzq x=ψzq) ty51 : filter F ∋ ( ZFP P p ∩ ZFP P q ) ty51 = filter2 F ty53 ty52 ty54 ty50 : filter F ∋ ZFP P (p ∩ q) ty50 = subst (λ k → filter F ∋ k ) (sym ?) ty51 pq=proj2 : & (p ∩ q) ≡ & (ZP-proj2 P Q (* (& (ZFP P (p ∩ q) ))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) ty50))) pq=proj2 = ? -- cong (&) (ZP-proj2=rev (zp1 pqa) (λ {x} pqx → Ppq _ (subst (λ k → odef k x) ? pqx)) *iso ) Filter-Proj2-UF : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) → ultra-filter (Filter-Proj2 pqa F) Filter-Proj2-UF {P} {Q} pqa F UF = record { proper = ty60 ; ultra = ty62 } where FQ = Filter-Proj2 pqa F ty60 : ¬ (filter FQ ∋ od∅) ty60 record { z = z ; az = az ; x=ψz = x=ψz } = ⊥-elim (ultra-filter.proper UF (filter1 F (λ x x<0 → ⊥-elim (¬x<0 (subst (λ k → odef k x) ? x<0))) (subst (λ k → odef (filter F) k ) (sym &iso) az) ty61 )) where ty61 : * z ⊆ od∅ ty61 {x} lt = ⊥-elim (¬x<0 (subst (λ k → odef k x) (trans (cong (*) (ZP-proj2-0 (sym ?))) ?) lt )) ty62 : {p : HOD} → Power Q ∋ p → Power Q ∋ (Q ＼ p) → (filter (Filter-Proj2 pqa F) ∋ p) ∨ (filter (Filter-Proj2 pqa F) ∋ (Q ＼ p)) ty62 {p} Qp NEGQ = uf05 where p⊆Q : p ⊆ Q p⊆Q {z} px = Qp _ (subst (λ k → odef k z) ? px) mq : HOD mq = ZFP P p uf03 : Power (ZFP P Q) ∋ mq uf03 x xz with subst (λ k → odef k x ) ? xz ... | t = ? -- ab-pair ax by = ab-pair ax (p⊆Q by) uf04 : Power (ZFP P Q) ∋ (ZFP P Q ＼ mq) uf04 x xz = ? -- proj1 (subst (λ k → odef k x) *iso xz) uf02 : (filter F ∋ mq) ∨ (filter F ∋ (ZFP P Q ＼ mq)) uf02 = ultra-filter.ultra UF uf03 uf04 uf05 : (filter FQ ∋ p) ∨ (filter FQ ∋ (Q ＼ p)) uf05 with uf02 ... | case1 fp = case1 record { z = _ ; az = fp ; x=ψz = cong (&) ? } ... | case2 fnp = case2 record { z = _ ; az = fnp ; x=ψz = cong (&) ? } rcf : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → RXCod (filter F) (ZFP Q P) (λ x fx → ZPmirror P Q x (filter-⊆ F fx)) rcf {P} {Q} F = record { ≤COD = λ {x} fx {z} ly → ZPmirror⊆ZFPBA P Q x (filter-⊆ F fx) ly } Filter-sym : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → Filter {Power (ZFP Q P)} {ZFP Q P} (λ x → x) Filter-sym {P} {Q} F = record { filter = fqp ; f⊆L = fqp<P ; filter1 = f1 ; filter2 = f2 } where fqp : HOD fqp = Replace' (filter F) (λ x fx → ZPmirror P Q x (filter-⊆ F fx)) {ZFP Q P} (rcf F) fqp<P : fqp ⊆ Power (ZFP Q P) fqp<P {z} record { z = x ; az = fx ; x=ψz = x=ψz } w xw = ZPmirror⊆ZFPBA P Q (* x) (filter-⊆ F (subst (λ k → odef (filter F) k) (sym &iso) fx )) (subst (λ k → odef k w) ? xw) f1 : {p q : HOD} → Power (ZFP Q P) ∋ q → fqp ∋ p → p ⊆ q → fqp ∋ q f1 {p} {q} QPq fqp p⊆q = record { z = _ ; az = fis00 {ZPmirror Q P p p⊆ZQP } {ZPmirror Q P q q⊆ZQP } fig01 fig03 fis04 ; x=ψz = fis05 } where fis00 : {p q : HOD} → Power (ZFP P Q) ∋ q → filter F ∋ p → p ⊆ q → filter F ∋ q fis00 = filter1 F q⊆ZQP : q ⊆ ZFP Q P q⊆ZQP {x} qx = QPq _ (subst (λ k → odef k x) ? qx) p⊆ZQP : p ⊆ ZFP Q P p⊆ZQP {z} px = q⊆ZQP (p⊆q px) fig06 : & p ≡ & (ZPmirror P Q (* (Replaced1.z fqp)) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (Replaced1.az fqp)))) fig06 = Replaced1.x=ψz fqp fig03 : filter F ∋ ZPmirror Q P p p⊆ZQP fig03 with Replaced1.az fqp ... | fz = subst (λ k → odef (filter F) k ) fig07 fz where fig07 : Replaced1.z fqp ≡ & (ZPmirror Q P p (λ {x} px → QPq x (subst (λ k → ? ) ? (p⊆q px)))) fig07 = trans (sym &iso) ( sym (cong (&) ?)) fig01 : Power (ZFP P Q) ∋ ZPmirror Q P q q⊆ZQP fig01 x xz = ZPmirror⊆ZFPBA Q P q q⊆ZQP (subst (λ k → odef k x) ? xz) fis04 : ZPmirror Q P p (λ z → q⊆ZQP (p⊆q z)) ⊆ ZPmirror Q P q q⊆ZQP fis04 = ZPmirror-⊆ p⊆q fis05 : & q ≡ & (ZPmirror P Q (* (& (ZPmirror Q P q q⊆ZQP))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (fis00 fig01 fig03 fis04)))) fis05 = cong (&) (sym ?) f2 : {p q : HOD} → fqp ∋ p → fqp ∋ q → Power (ZFP Q P) ∋ (p ∩ q) → fqp ∋ (p ∩ q) f2 {p} {q} fp fq QPpq = record { z = _ ; az = fis12 {ZPmirror Q P p p⊆ZQP} {ZPmirror Q P q q⊆ZQP} fig03 fig04 fig01 ; x=ψz = fis05 } where fis12 : {p q : HOD} → filter F ∋ p → filter F ∋ q → Power (ZFP P Q) ∋ (p ∩ q) → filter F ∋ (p ∩ q) fis12 {p} {q} fp fq PQpq = filter2 F fp fq PQpq p⊆ZQP : p ⊆ ZFP Q P p⊆ZQP {z} px = fqp<P fp _ (subst (λ k → odef k z) ? px) q⊆ZQP : q ⊆ ZFP Q P q⊆ZQP {z} qx = fqp<P fq _ (subst (λ k → odef k z) ? qx) pq⊆ZQP : (p ∩ q) ⊆ ZFP Q P pq⊆ZQP {z} pqx = QPpq _ (subst (λ k → odef k z) ? pqx) fig06 : & p ≡ & (ZPmirror P Q (* (Replaced1.z fp)) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (Replaced1.az fp)))) fig06 = Replaced1.x=ψz fp fig09 : & q ≡ & (ZPmirror P Q (* (Replaced1.z fq)) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (Replaced1.az fq)))) fig09 = Replaced1.x=ψz fq fig03 : filter F ∋ ZPmirror Q P p p⊆ZQP fig03 = subst (λ k → odef (filter F) k ) fig07 ( Replaced1.az fp ) where fig07 : Replaced1.z fp ≡ & (ZPmirror Q P p p⊆ZQP ) fig07 = trans (sym &iso) ( sym (cong (&) ?)) fig04 : filter F ∋ ZPmirror Q P q q⊆ZQP fig04 = subst (λ k → odef (filter F) k ) fig08 ( Replaced1.az fq ) where fig08 : Replaced1.z fq ≡ & (ZPmirror Q P q q⊆ZQP ) fig08 = trans (sym &iso) ( sym (cong (&) ?)) fig01 : Power (ZFP P Q) ∋ ( ZPmirror Q P p p⊆ZQP ∩ ZPmirror Q P q q⊆ZQP ) fig01 x xz = ZPmirror⊆ZFPBA Q P q q⊆ZQP ? fis05 : & (p ∩ q) ≡ & (ZPmirror P Q (* (& (ZPmirror Q P p p⊆ZQP ∩ ZPmirror Q P q q⊆ZQP))) (filter-⊆ F (subst (odef (filter F)) (sym &iso) (fis12 fig03 fig04 fig01) ))) fis05 = ? -- cong (&) (sym ( ZPmirror-rev {Q} {P} {_} {_} {pq⊆ZQP} ? )) Filter-sym-UF : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) → ultra-filter (Filter-sym F) Filter-sym-UF {P} {Q} F UF = record { proper = uf00 ; ultra = uf01 } where FQP = Filter-sym F uf00 : ¬ (Replace' (filter F) (λ x fx → ZPmirror P Q x (filter-⊆ F fx)) {ZFP Q P} (rcf F) ∋ od∅) uf00 record { z = z ; az = az ; x=ψz = x=ψz } = ⊥-elim ( ultra-filter.proper UF (subst (λ k → odef (filter F) k) uf10 az )) where uf10 : z ≡ & od∅ uf10 = ZPmirror-0 (sym ?) uf01 : {p : HOD} → Power (ZFP Q P) ∋ p → Power (ZFP Q P) ∋ (ZFP Q P ＼ p) → (filter FQP ∋ p) ∨ (filter FQP ∋ (ZFP Q P ＼ p)) uf01 {p} QPp NEGP = uf05 where p⊆ZQP : p ⊆ ZFP Q P p⊆ZQP {z} px = QPp _ (subst (λ k → odef k z) ? px) mp : HOD mp = ZPmirror Q P p p⊆ZQP uf03 : Power (ZFP P Q) ∋ mp uf03 x xz = ZPmirror⊆ZFPBA Q P p p⊆ZQP ? uf04 : Power (ZFP P Q) ∋ (ZFP P Q ＼ mp) uf04 x xz = proj1 ? uf02 : (filter F ∋ mp) ∨ (filter F ∋ (ZFP P Q ＼ mp)) uf02 = ultra-filter.ultra UF uf03 uf04 uf05 : (filter FQP ∋ p) ∨ (filter FQP ∋ (ZFP Q P ＼ p)) uf05 with uf02 ... | case1 fp = case1 record { z = _ ; az = fp ; x=ψz = ? } ... | case2 fnp = case2 record { z = _ ; az = uf06 ; x=ψz = ? } where uf06 : odef (filter F) (& (ZPmirror Q P (ZFP Q P ＼ p) proj1 )) uf06 = subst (λ k → odef (filter F) k) ? fnp -- this makes check very slow -- Filter-Proj2 : {P Q a : HOD } → ZFP P Q ∋ a → -- (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) -- → Filter {Power Q} {Q} (λ x → x) -- Filter-Proj2 {P} {Q} {a} pqa F = Filter-Proj1 {Q} {P} qpa (Filter-sym F ) where -- qpa : ZFP Q P ∋ < * (zπ2 pqa) , * (zπ1 pqa) > -- qpa = ab-pair (zp2 pqa) (zp1 pqa) -- Filter-Proj2-UF : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) -- → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) -- → ultra-filter (Filter-Proj2 pqa F) -- Filter-Proj2-UF {P} {Q} {a} pqa F UF = Filter-Proj1-UF {Q} {P} qpa (Filter-sym F) (Filter-sym-UF F UF) where -- qpa : ZFP Q P ∋ < * (zπ2 pqa) , * (zπ1 pqa) > -- qpa = ab-pair (zp2 pqa) (zp1 pqa) FPSet⊆F : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → {x : Ordinal } → odef (filter (Filter-Proj1 {P} {Q} pqa F )) x → odef (filter F) (& (ZFP (* x) Q)) FPSet⊆F {P} {Q} {a} pqa F {x} record { z = z ; az = az ; x=ψz = x=ψz } = filter1 F uf09 (subst (λ k → odef (filter F) k) (sym &iso) az) uf08 where uf08 : * z ⊆ ZFP (* x) Q uf08 = subst (λ k → * z ⊆ ZFP k Q) ? ZP-proj1⊆ZFP uf09 : Power (ZFP P Q) ∋ ZFP (* x) Q uf09 z xqz = ? -- with subst (λ k → odef k z) *iso xqz -- ... | ab-pair {c} {d} xc by = ab-pair uf10 by where -- uf10 : odef P c -- uf10 with subst (λ k → odef k c) (sym (trans (sym *iso) (cong (*) (sym x=ψz)))) xc -- ... | record { b = b ; aa = aa ; bb = bb ; c∋ab = c∋ab } = aa FQSet⊆F : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) → {x : Ordinal } → odef (filter (Filter-Proj2 {P} {Q} pqa F )) x → odef (filter F) (& (ZFP P (* x) )) FQSet⊆F {P} {Q} {a} pqa F {x} record { z = z ; az = az ; x=ψz = x=ψz } = filter1 F uf09 (subst (λ k → odef (filter F) k) (sym &iso) az ) uf08 where uf08 : * z ⊆ ZFP P (* x) uf08 = subst (λ k → * z ⊆ ZFP P k ) ? ZP-proj2⊆ZFP uf09 : Power (ZFP P Q) ∋ ZFP P (* x) uf09 z xpz = ? -- with subst (λ k → odef k z) *iso xpz -- ... | ab-pair {c} {d} ax yc = ab-pair ax uf10 where -- uf10 : odef Q d -- uf10 with subst (λ k → odef k d) (sym (trans (sym *iso) (cong (*) (sym x=ψz)))) yc -- ... | record { a = a ; aa = aa ; bb = bb ; c∋ab = c∋ab } = bb -- FQSet⊆F : {P Q a : HOD } → (pqa : ZFP P Q ∋ a ) → -- (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) -- → {x : Ordinal } → odef (filter (Filter-Proj2 {P} {Q} pqa F )) x → odef (filter F) (& (ZFP P (* x) )) --FQSet⊆F {P} {Q} {a} pqa F {x} f2x = FPSet⊆F {P} {Q} {a} qpa (Filter-sym F) {x} ? where -- qpa : ZFP Q P ∋ < * (zπ2 pqa) , * (zπ1 pqa) > -- qpa = ab-pair (zp2 pqa) (zp1 pqa)