Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/Topology.agda @ 1198:6e0cc71097e0
fip <-> compact done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 01 Mar 2023 09:00:11 +0900 |
| parents | 0a88fcd5d1c9 |
| children | 1338b6c6a9b6 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module Topology {n : Level } (O : Ordinals {n}) where open import zf open import logic open _∧_ open _∨_ open Bool import OD open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import Relation.Binary.Definitions open import Relation.Binary.PropositionalEquality import BAlgebra open BAlgebra O open inOrdinal O open OD O open OD.OD open ODAxiom odAxiom import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal open Ordinals.IsNext isNext open OrdUtil O open ODUtil O import ODC open ODC O open import filter O open import OPair O record Topology ( L : HOD ) : Set (suc n) where field OS : HOD OS⊆PL : OS ⊆ Power L o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P OS∋od∅ : OS ∋ od∅ --- we may add -- OS∋L : OS ∋ L -- closed Set CS : HOD CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) os⊆L : {x : HOD} → OS ∋ x → x ⊆ L os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy ) cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) (sym *iso) xy ) CS∋L : CS ∋ L CS∋L = ⟪ subst (λ k → k ⊆ L) (sym *iso) (λ x → x) , subst (λ k → odef OS (& k)) (sym lem0) OS∋od∅ ⟫ where lem0 : L ＼ * (& L) ≡ od∅ lem0 = subst (λ k → L ＼ k ≡ od∅) (sym *iso) L＼L=0 CS⊆PL : CS ⊆ Power L CS⊆PL {x} Cx y xy = proj1 Cx xy P＼CS=OS : {cs : HOD} → CS ∋ cs → OS ∋ ( L ＼ cs ) P＼CS=OS {cs} ⟪ cs⊆L , olcs ⟫ = subst (λ k → odef OS k) (cong (λ k → & ( L ＼ k)) *iso) olcs P＼OS=CS : {cs : HOD} → OS ∋ cs → CS ∋ ( L ＼ cs ) P＼OS=CS {os} oos = ⟪ subst (λ k → k ⊆ L) (sym *iso) proj1 , subst (λ k → odef OS k) (cong (&) (trans (sym (L＼Lx=x (os⊆L oos))) (cong (λ k → L ＼ k) (sym *iso)) )) oos ⟫ open Topology -- Closure ( Intersection of Closed Set which include A ) Cl : {L : HOD} → (top : Topology L) → (A : HOD) → HOD Cl {L} top A = record { od = record { def = λ x → odef L x ∧ ( (c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x ) } ; odmax = & L ; <odmax = odef∧< } ClL : {L : HOD} → (top : Topology L) → Cl top L ≡ L ClL {L} top = ==→o≡ ( record { eq→ = λ {x} ic → subst (λ k → odef k x) *iso ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) (sym *iso) ( λ x → x))) ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } ) -- Closure is Closed Set CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → CS top ∋ Cl top A CS∋Cl {L} top A = subst (λ k → CS top ∋ k) (==→o≡ cc00) (P＼OS=CS top UOCl-is-OS) where OCl : HOD -- set of open set which it not contains A OCl = record { od = record { def = λ o → odef (OS top) o ∧ ( A ⊆ (L ＼ * o) ) } ; odmax = & (OS top) ; <odmax = odef∧< } OCl⊆OS : OCl ⊆ OS top OCl⊆OS ox = proj1 ox UOCl-is-OS : OS top ∋ Union OCl UOCl-is-OS = o∪ top OCl⊆OS cc00 : (L ＼ Union OCl) =h= Cl top A cc00 = record { eq→ = cc01 ; eq← = cc03 } where cc01 : {x : Ordinal} → odef (L ＼ Union OCl) x → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) cc01 {x} ⟪ Lx , nul ⟫ = ⟪ Lx , ( λ c cc ac → cc02 c cc ac nul ) ⟫ where cc02 : (c : Ordinal) → odef (CS top) c → A ⊆ * c → ¬ odef (Union OCl) x → odef (* c) x cc02 c cc ac nox with ODC.∋-p O (* c) (* x) ... | yes y = subst (λ k → odef (* c) k) &iso y ... | no ncx = ⊥-elim ( nox record { owner = & ( L ＼ * c) ; ao = ⟪ proj2 cc , cc07 ⟫ ; ox = subst (λ k → odef k x) (sym *iso) cc06 } ) where cc06 : odef (L ＼ * c) x cc06 = ⟪ Lx , subst (λ k → ¬ odef (* c) k) &iso ncx ⟫ cc08 : * c ⊆ L cc08 = cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) cc ) cc07 : A ⊆ (L ＼ * (& (L ＼ * c))) cc07 {z} az = subst (λ k → odef k z ) ( begin * c ≡⟨ sym ( L＼Lx=x cc08 ) ⟩ L ＼ (L ＼ * c) ≡⟨ cong (λ k → L ＼ k ) (sym *iso) ⟩ L ＼ * (& (L ＼ * c)) ∎ ) ( ac az ) where open ≡-Reasoning cc03 : {x : Ordinal} → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) → odef (L ＼ Union OCl) x cc03 {x} ⟪ Lx , ccx ⟫ = ⟪ Lx , cc04 ⟫ where -- if x is in Cl A, it is in some c : CS, OCl says it is not , i.e. L ＼ o ∋ x, so it is in (L ＼ Union OCl) x cc04 : ¬ odef (Union OCl) x cc04 record { owner = o ; ao = ⟪ oo , A⊆L-o ⟫ ; ox = ox } = proj2 ( subst (λ k → odef k x) *iso cc05) ox where cc05 : odef (* (& (L ＼ * o))) x cc05 = ccx (& (L ＼ * o)) (P＼OS=CS top (subst (λ k → odef (OS top) k) (sym &iso) oo)) (subst (λ k → A ⊆ k) (sym *iso) A⊆L-o) -- Subbase P -- A set of countable intersection of P will be a base (x ix an element of the base) data Subbase (P : HOD) : Ordinal → Set n where gi : {x : Ordinal } → odef P x → Subbase P x g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) -- -- if y is in a Subbase, some element of P contains it sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal sbp P {x} (gi {y} px) = x sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) -- An open set generate from a base -- -- OS = { U ⊆ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊆ U } record Base (L P : HOD) (u x : Ordinal) : Set n where field b : Ordinal u⊆L : * u ⊆ L sb : Subbase P b b⊆u : * b ⊆ * u bx : odef (* b) x x⊆L : odef L x x⊆L = u⊆L (b⊆u bx) SO : (L P : HOD) → HOD SO L P = record { od = record { def = λ u → {x : Ordinal } → odef (* u) x → Base L P u x } ; odmax = osuc (& L) ; <odmax = tp00 } where tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) record IsSubBase (L P : HOD) : Set (suc n) where field P⊆PL : P ⊆ Power L -- we may need these if OS ∋ L is necessary -- p : {x : HOD} → L ∋ x → HOD -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x InducedTopology : (L P : HOD) → IsSubBase L P → Topology L InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x )) tp00 : SO L P ⊆ Power L tp00 {u} ou x ux with ou ux ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊆L (b⊆u bx) tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) tp01 {p} {q} op oq {x} ux = record { b = b ; u⊆L = subst (λ k → k ⊆ L) (sym *iso) ul ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where px : odef (* (& p)) x px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) qx : odef (* (& q)) x qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) b : Ordinal b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) tp08 : * b ⊆ * (& (p ∩ q) ) tp08 = subst₂ (λ j k → j ⊆ k ) (sym *iso) (sym *iso) (⊆∩-dist {(* (Base.b (op px)) ∩ * (Base.b (oq qx)))} {p} {q} tp09 tp10 ) where tp11 : * (Base.b (op px)) ⊆ * (& p ) tp11 = Base.b⊆u (op px) tp12 : * (Base.b (oq qx)) ⊆ * (& q ) tp12 = Base.b⊆u (oq qx) tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ ul : (p ∩ q) ⊆ L ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊆L (op px)) (pz pq) ) where pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) tp02 : { q : HOD} → q ⊆ SO L P → SO L P ∋ Union q tp02 {q} q⊆O {x} ux with subst (λ k → odef k x) *iso ux ... | record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = b ; u⊆L = subst (λ k → k ⊆ L) (sym *iso) tp04 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where tp05 : Union q ⊆ L tp05 {z} record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) tp04 : Union q ⊆ L tp04 = tp05 tp06 : * b ⊆ Union q tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } -- Product Topology open ZFProduct -- Product Topology is not -- ZFP (OS TP) (OS TQ) (box) record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where field p : Ordinal op : odef (OS TP) p prod : x ≡ & (ZFP (* p) Q ) record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where field q : Ordinal oq : odef (OS TQ) q prod : x ≡ & (ZFP P (* q )) pbase⊆PL : {P Q : HOD} → (TP : Topology P) → (TQ : Topology Q) → {x : Ordinal } → BaseP TP Q x ∨ BaseQ P TQ x → odef (Power (ZFP P Q)) x pbase⊆PL {P} {Q} TP TQ {z} (case1 record { p = p ; op = op ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) tp01 w wz with subst (λ k → odef k w ) *iso wz ... | ab-pair {a} {b} pa qb = ZFP→ (subst (λ k → odef P k ) (sym &iso) tp03 ) (subst (λ k → odef Q k ) (sym &iso) qb ) where tp03 : odef P a tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa pbase⊆PL {P} {Q} TP TQ {z} (case2 record { q = q ; oq = oq ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) tp01 w wz with subst (λ k → odef k w ) *iso wz ... | ab-pair {a} {b} pa qb = ZFP→ (subst (λ k → odef P k ) (sym &iso) pa ) (subst (λ k → odef Q k ) (sym &iso) tp03 ) where tp03 : odef Q b tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb pbase : {P Q : HOD} → Topology P → Topology Q → HOD pbase {P} {Q} TP TQ = record { od = record { def = λ x → BaseP TP Q x ∨ BaseQ P TQ x } ; odmax = & (Power (ZFP P Q)) ; <odmax = tp00 } where tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } -- covers ( q ⊆ Union P ) record _covers_ ( P q : HOD ) : Set n where field cover : {x : Ordinal } → odef q x → Ordinal P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x open _covers_ -- Finite Intersection Property record FIP {L : HOD} (top : Topology L) : Set n where field limit : {X : Ordinal } → * X ⊆ CS top → ( { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → Ordinal is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → {x : Ordinal } → odef (* X) x → odef L (limit CX fip) L∋limit {X} CX fip {x} xx = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX xx)) (is-limit CX fip xx) -- Compact data Finite-∪ (S : HOD) : Ordinal → Set n where fin-e : Finite-∪ S o∅ fin-i : {x : Ordinal } → odef S x → Finite-∪ S (& ( * x , * x )) fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) -- Finite-∪ S y → Union y ⊆ S record Compact {L : HOD} (top : Topology L) : Set n where field finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) -- FIP is Compact FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top FIP→Compact {L} top fip with trio< (& L) o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where -- L is empty fip02 : {x : Ordinal } → ¬ odef L x fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ fip00 {X} xo xcp = fin-e ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where -- set of coset of X CX : {X : Ordinal} → * X ⊆ OS top → Ordinal CX {X} ox = & ( Replace (* X) (λ z → L ＼ z )) CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top CCX {X} os {x} ox with subst (λ k → odef k x) *iso ox ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) fip07 : z ≡ & (L ＼ * x) fip07 = subst₂ (λ j k → j ≡ k) &iso (cong (λ k → & ( L ＼ k )) (cong (*) (sym x=ψz))) ( cong (&) ( ==→o≡ record { eq→ = fip09 ; eq← = fip08 } )) where fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) *iso not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) fip09 {w} zw = ⟪ os⊆L top (os (subst (λ k → odef (* X) k) (sym &iso) az)) zw , subst (λ k → ¬ odef k w) (sym *iso) fip10 ⟫ where fip10 : ¬ (odef (L ＼ * z) w) fip10 ⟪ Lw , nzw ⟫ = nzw zw fip06 : odef (OS top) (& (L ＼ * x)) fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) fip05 : * x ⊆ L fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso ) xw ) -- -- X covres L means Intersection of (CX X) contains nothing -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) -- it means there is a finite cover -- finCoverBase : {X : Ordinal } → * X ⊆ OS top → * X covers L → Subbase (Replace (* X) (λ z → L ＼ z)) o∅ finCoverBase {X} ox oc with ODC.p∨¬p O (Subbase (Replace (* X) (λ z → L ＼ z)) o∅) ... | case1 sb = sb ... | case2 ¬sb = ⊥-elim (fip09 fip25 fip20) where fip09 : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) fip09 {z} Lz nc = nc ( P∋cover oc Lz ) (subst (λ k → odef (* (cover oc Lz)) k) refl (isCover oc _ )) -- CX is finite intersection fip02 : {x : Ordinal} → Subbase (* (CX ox)) x → o∅ o< x fip02 {x} sc with trio< x o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri> ¬a ¬b c = c ... | tri≈ ¬a b ¬c = ⊥-elim (¬sb (subst₂ (λ j k → Subbase j k ) *iso b sc )) -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) ))) fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) )) )) (sym *iso) record { z = cover oc (x∋minimal L (0<P→ne 0<L)) ; az = P∋cover oc (x∋minimal L (0<P→ne 0<L)) ; x=ψz = refl } fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) fip20 {y} Xy yl = proj2 fip21 yl where fip22 : odef (* (CX ox)) (& ( L ＼ * y )) fip22 = subst (λ k → odef k (& ( L ＼ * y ))) (sym *iso) record { z = y ; az = Xy ; x=ψz = refl } fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) finCoverSet : {X : Ordinal } → (x : Ordinal) → Subbase (Replace (* X) (λ z → L ＼ z)) x → HOD finCoverSet {X} x (gi rx) = ( L ＼ * x ) , ( L ＼ * x ) finCoverSet {X} x∩y (g∩ {x} {y} sx sy) = finCoverSet {X} x sx ∪ finCoverSet {X} y sy -- -- this defines finite cover finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal finCover {X} ox oc = & ( finCoverSet o∅ (finCoverBase ox oc)) -- create Finite-∪ from cex isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) isFinite {X} xo xcp = fip60 o∅ (finCoverBase xo xcp) where fip60 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z)) x ) → Finite-∪ (* X) (& (finCoverSet {X} x sb)) fip60 x (gi rx) = subst (λ k → Finite-∪ (* X) k) fip62 (fin-i (fip61 rx)) where fip62 : & (* (& (L ＼ * x)) , * (& (L ＼ * x))) ≡ & ((L ＼ * x) , (L ＼ * x)) fip62 = cong₂ (λ j k → & (j , k )) *iso *iso fip61 : odef (Replace (* X) (_＼_ L)) x → odef (* X) ( & ((L ＼ * x ) )) fip61 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where fip34 : * z1 ⊆ L fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 fip33 : z1 ≡ & (L ＼ * x) fip33 = begin z1 ≡⟨ sym &iso ⟩ & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ & (L ＼ * x ) ∎ where open ≡-Reasoning fip60 x∩y (g∩ {x} {y} sx sy) = subst (λ k → Finite-∪ (* X) k) fip62 ( fin-∪ (fip60 x sx) (fip60 y sy) ) where fip62 : & (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡ & (finCoverSet x sx ∪ finCoverSet y sy) fip62 = cong (&) ( begin (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡⟨ cong₂ _∪_ *iso *iso ⟩ finCoverSet x sx ∪ finCoverSet y sy ∎ ) where open ≡-Reasoning -- is also a cover isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L isCover1 {X} xo xcp = subst₂ (λ j k → j covers k ) (sym *iso) (subst (λ k → L ＼ k ≡ L) (sym o∅≡od∅) L＼0=L) (fip70 o∅ (finCoverBase xo xcp)) where fip70 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z)) x ) → (finCoverSet {X} x sb) covers (L ＼ * x) fip70 x (gi rx) = record { cover = λ _ → & (L ＼ * x) ; P∋cover = λ _ → case1 refl ; isCover = λ {x} lt → subst (λ k → odef k x) (sym *iso) lt } fip70 x∩y (g∩ {x} {y} sx sy) = subst (λ k → finCoverSet (& (* x ∩ * y)) (g∩ sx sy) covers (L ＼ k)) (sym *iso) ( fip43 {_} {L} {* x} {* y} (fip71 (fip70 x sx)) (fip72 (fip70 y sy)) ) where fip71 : {a b c : HOD} → a covers c → (a ∪ b) covers c fip71 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case1 (P∋cover cov lt) ; isCover = isCover cov } fip72 : {a b c : HOD} → a covers c → (b ∪ a) covers c fip72 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case2 (P∋cover cov lt) ; isCover = isCover cov } fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) fip45 {L} {a} {b} {x} Lab with ODC.∋-p O b (* x) ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal fip44 {x} Lab with fip45 {L} {a} {b} Lab ... | case1 La = cover ca La ... | case2 Lb = cover cb Lb fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) fip46 {x} Lab with fip45 {L} {a} {b} Lab ... | case1 La = P∋cover ca La ... | case2 Lb = P∋cover cb Lb fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x fip47 {x} Lab with fip45 {L} {a} {b} Lab ... | case1 La = isCover ca La ... | case2 Lb = isCover cb Lb open _==_ Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top Compact→FIP {L} top compact with trio< (& L) o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a L=0 ¬c = record { limit = λ {X} CX fip → o∅ ; is-limit = λ {X} CX fip xx → ⊥-elim (fip000 CX fip xx) } where -- empty L case -- if 0 < X then 0 < x ∧ L ∋ xfrom fip -- if 0 ≡ X then ¬ odef X x fip000 : {X x : Ordinal} (CX : * X ⊆ CS top) → ({y : Ordinal} → Subbase (* X) y → o∅ o< y) → ¬ odef (* X) x fip000 {X} {x} CX fip xx with trio< o∅ X ... | tri< 0<X ¬b ¬c = ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) L=0)) o∅≡od∅ ) (sym &iso) ( cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) (CX xx)) Xe )) where 0<x : o∅ o< x 0<x = fip (gi xx ) e : HOD -- we have an element of x e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) Xe : odef (* x) (& e) Xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin * X ≡⟨ cong (*) (sym 0=X) ⟩ * o∅ ≡⟨ o∅≡od∅ ⟩ od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) -- (subst (λ k → odef (CS top) k) (sym &iso) ( CX xx ) ) )) ... | tri> ¬a ¬b 0<L = record { limit = limit ; is-limit = fip00 } where -- set of coset of X OX : {X : Ordinal} → * X ⊆ CS top → Ordinal OX {X} ox = & ( Replace (* X) (λ z → L ＼ z )) OOX : {X : Ordinal} → (cs : * X ⊆ CS top) → * (OX cs) ⊆ OS top OOX {X} cs {x} ox with subst (λ k → odef k x) *iso ox ... | record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef (OS top) k) (sym x=ψz) ( P＼CS=OS top (cs comp01)) where comp01 : odef (* X) (& (* z)) comp01 = subst (λ k → odef (* X) k) (sym &iso) az -- if all finite intersection of X contains something, -- there is no finite cover. From Compactness, (OX X) is not a cover of L ( contraposition of Compact) -- it means there is a limit has-intersection : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) → o∅ o< X → ¬ ( Intersection (* X) =h= od∅ ) has-intersection {X} CX fip 0<X i=0 = ⊥-elim ( ¬x<0 {NC.x not-covered} ( eq→ i=0 ⟪ fp06 , fp05 ⟫ )) where e : HOD -- we have an element of X e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) Xe : odef (* X) (& e) Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) no-cover : ¬ ( (* (OX CX)) covers L ) no-cover cov = ⊥-elim (no-finite-cover (Compact.isCover compact (OOX CX) cov)) where -- construct Subase from Finite-∪ fp01 : Ordinal fp01 = Compact.finCover compact (OOX CX) cov record SB (t : Ordinal) : Set n where field i : Ordinal sb : Subbase (* X) (& (L ＼ * i)) t⊆i : (L ＼ * i) ⊆ (L ＼ Union ( * t ) ) fp02 : (t : Ordinal) → Finite-∪ (* (OX CX)) t → SB t fp02 t fin-e = record { i = & ( L ＼ e) ; sb = gi (subst (λ k → odef (* X) k) fp21 Xe) ; t⊆i = fp23 } where -- t ≡ o∅, no cover. Any subst of L is ok and we have e ⊆ L fp22 : e ⊆ L fp22 {x} lt = cs⊆L top (CX Xe) lt fp21 : & e ≡ & (L ＼ * (& (L ＼ e))) fp21 = cong (&) (trans (sym (L＼Lx=x fp22)) (cong (λ k → L ＼ k) (sym *iso))) fp23 : (L ＼ * (& (L ＼ e))) ⊆ (L ＼ Union (* o∅)) fp23 {x} ⟪ Lx , _ ⟫ = ⟪ Lx , ( λ lt → ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) o∅≡od∅ (sym &iso) (Own.ao lt )))) ⟫ fp02 t (fin-i {x} tx ) = record { i = x ; sb = gi fp03 ; t⊆i = fp24 } where -- we have a single cover x, L ＼ * x is single finite intersection fp24 : (L ＼ * x) ⊆ (L ＼ Union (* (& (* x , * x)))) fp24 {y} ⟪ Lx , not ⟫ = ⟪ Lx , subst (λ k → ¬ odef (Union k) y) (sym *iso) fp25 ⟫ where fp25 : ¬ odef (Union (* x , * x)) y fp25 record { owner = .(& (* x)) ; ao = (case1 refl) ; ox = ox } = not (subst (λ k → odef k y) *iso ox ) fp25 record { owner = .(& (* x)) ; ao = (case2 refl) ; ox = ox } = not (subst (λ k → odef k y) *iso ox ) fp03 : odef (* X) (& (L ＼ * x)) -- becase x is an element of Replace (* X) (λ z → L ＼ z ) fp03 with subst (λ k → odef k x ) *iso tx ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where fip34 : * z1 ⊆ L fip34 {w} wz1 = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX az1) ) wz1 fip33 : z1 ≡ & (L ＼ * x) fip33 = begin z1 ≡⟨ sym &iso ⟩ & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ & (L ＼ * x ) ∎ where open ≡-Reasoning fp02 t (fin-∪ {tx} {ty} ux uy ) = record { i = & (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))) ; sb = fp11 ; t⊆i = fp35 } where fp35 : (L ＼ * (& (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))) ⊆ (L ＼ Union (* (& (* tx ∪ * ty)))) fp35 = subst₂ (λ j k → (L ＼ j ) ⊆ (L ＼ Union k)) (sym *iso) (sym *iso) fp36 where fp40 : {z tz : Ordinal } → Finite-∪ (* (OX CX)) tz → odef (Union (* tz )) z → odef L z fp40 {z} {.(Ordinals.o∅ O)} fin-e record { owner = owner ; ao = ao ; ox = ox } = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) o∅≡od∅ (sym &iso) ao )) fp40 {z} {.(& (* _ , * _))} (fin-i {w} x) uz = fp41 x (subst (λ k → odef (Union k) z) *iso uz) where fp41 : (x : odef (* (OX CX)) w) → (uz : odef (Union (* w , * w)) z ) → odef L z fp41 x record { owner = .(& (* w)) ; ao = (case1 refl) ; ox = ox } = os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) *iso ox ) fp41 x record { owner = .(& (* w)) ; ao = (case2 refl) ; ox = ox } = os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) *iso ox ) fp40 {z} {.(& (* _ ∪ * _))} (fin-∪ {x1} {y1} ftx fty) uz with subst (λ k → odef (Union k) z ) *iso uz ... | record { owner = o ; ao = case1 x1o ; ox = oz } = fp40 ftx record { owner = o ; ao = x1o ; ox = oz } ... | record { owner = o ; ao = case2 y1o ; ox = oz } = fp40 fty record { owner = o ; ao = y1o ; ox = oz } fp36 : (L ＼ (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* tx ∪ * ty)) fp36 {z} ⟪ Lz , not ⟫ = ⟪ Lz , fp37 ⟫ where fp37 : ¬ odef (Union (* tx ∪ * ty)) z fp37 record { owner = owner ; ao = (case1 ax) ; ox = ox } = not (case1 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where fp38 : (L ＼ (* (SB.i (fp02 tx ux)))) ⊆ (L ＼ Union (* tx)) fp38 = SB.t⊆i (fp02 tx ux) fp39 : Union (* tx) ⊆ (* (SB.i (fp02 tx ux))) fp39 {w} txw with ∨L＼X {L} {* (SB.i (fp02 tx ux))} (fp40 ux txw) ... | case1 sb = sb ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) txw ) fp37 record { owner = owner ; ao = (case2 ax) ; ox = ox } = not (case2 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where fp38 : (L ＼ (* (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* ty)) fp38 = SB.t⊆i (fp02 ty uy) fp39 : Union (* ty) ⊆ (* (SB.i (fp02 ty uy))) fp39 {w} tyw with ∨L＼X {L} {* (SB.i (fp02 ty uy))} (fp40 uy tyw) ... | case1 sb = sb ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) tyw ) fp04 : {tx ty : Ordinal} → & (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) ≡ & (L ＼ * (& (* tx ∪ * ty))) fp04 {tx} {ty} = cong (&) ( ==→o≡ record { eq→ = fp05 ; eq← = fp09 } ) where fp05 : {x : Ordinal} → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x → odef (L ＼ * (& (* tx ∪ * ty))) x fp05 {x} lt with subst₂ (λ j k → odef (j ∩ k) x ) *iso *iso lt ... | ⟪ ⟪ Lx , ¬tx ⟫ , ⟪ Ly , ¬ty ⟫ ⟫ = subst (λ k → odef (L ＼ k) x) (sym *iso) ⟪ Lx , fp06 ⟫ where fp06 : ¬ odef (* tx ∪ * ty) x fp06 (case1 tx) = ¬tx tx fp06 (case2 ty) = ¬ty ty fp09 : {x : Ordinal} → odef (L ＼ * (& (* tx ∪ * ty))) x → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x fp09 {x} lt with subst (λ k → odef (L ＼ k) x) (*iso) lt ... | ⟪ Lx , ¬tx∨ty ⟫ = subst₂ (λ j k → odef (j ∩ k) x ) (sym *iso) (sym *iso) ⟪ ⟪ Lx , ( λ tx → ¬tx∨ty (case1 tx)) ⟫ , ⟪ Lx , ( λ ty → ¬tx∨ty (case2 ty)) ⟫ ⟫ fp11 : Subbase (* X) (& (L ＼ * (& ((* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))))) fp11 = subst (λ k → Subbase (* X) k ) fp04 ( g∩ (SB.sb (fp02 tx ux)) (SB.sb (fp02 ty uy )) ) -- -- becase of fip, finite cover cannot be a cover -- fcov : Finite-∪ (* (OX CX)) (Compact.finCover compact (OOX CX) cov) fcov = Compact.isFinite compact (OOX CX) cov 0<sb : {i : Ordinal } → (sb : Subbase (* X) (& (L ＼ * i))) → o∅ o< & (L ＼ * i) 0<sb {i} sb = fip sb sb : SB (Compact.finCover compact (OOX CX) cov) sb = fp02 fp01 (Compact.isFinite compact (OOX CX) cov) no-finite-cover : ¬ ( (* (Compact.finCover compact (OOX CX) cov)) covers L ) no-finite-cover fcovers = ⊥-elim ( o<¬≡ (cong (&) (sym (==→o≡ f22))) f25 ) where f23 : (L ＼ * (SB.i sb)) ⊆ ( L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) f23 = SB.t⊆i sb f22 : (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) =h= od∅ f22 = record { eq→ = λ lt → ⊥-elim ( f24 lt) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where f24 : {x : Ordinal } → ¬ ( odef (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) x ) f24 {x} ⟪ Lx , not ⟫ = not record { owner = cover fcovers Lx ; ao = P∋cover fcovers Lx ; ox = isCover fcovers Lx } f25 : & od∅ o< (& (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ) f25 = ordtrans<-≤ (subst (λ k → k o< & (L ＼ * (SB.i sb))) (sym ord-od∅) (0<sb (SB.sb sb) ) ) ( begin & (L ＼ * (SB.i sb)) ≤⟨ ⊆→o≤ f23 ⟩ & (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ∎ ) where open o≤-Reasoning O record NC : Set n where -- find an element xi, which is not covered (which is a limit point) field x : Ordinal yx : {y : Ordinal} (Xy : odef (* X) y) → odef (* y) x not-covered : NC not-covered with ODC.p∨¬p O NC ... | case1 nc = nc ... | case2 ¬nc = ⊥-elim ( no-cover record { cover = λ Lx → & (L ＼ coverf Lx) ; P∋cover = fp22 ; isCover = fp23 } ) where coverSet : {x : Ordinal} → odef L x → HOD coverSet {x} Lx = record { od = record { def = λ y → odef (* X) y ∧ odef (L ＼ * y) x } ; odmax = X ; <odmax = λ {x} lt → subst (λ k → x o< k) &iso ( odef< (proj1 lt)) } fp17 : {x : Ordinal} → (Lx : odef L x ) → ¬ ( coverSet Lx =h= od∅ ) fp17 {x} Lx eq = ⊥-elim (¬nc record { x = x ; yx = fp19 } ) where fp19 : {y : Ordinal} → odef (* X) y → odef (* y) x fp19 {y} Xy with ∨L＼X {L} {* y} {x} Lx ... | case1 yx = yx ... | case2 lyx = ⊥-elim ( ¬x<0 {y} ( eq→ eq fp20 )) where fp20 : odef (* X) y ∧ odef (L ＼ * y) x fp20 = ⟪ Xy , lyx ⟫ coverf : {x : Ordinal} → (Lx : odef L x ) → HOD coverf Lx = ODC.minimal O (coverSet Lx) (fp17 Lx) fp22 : {x : Ordinal} (lt : odef L x) → odef (* (OX CX)) (& (L ＼ coverf lt)) fp22 {x} Lx = subst (λ k → odef k (& (L ＼ coverf Lx ))) (sym *iso) record { z = _ ; az = fp25 ; x=ψz = fp24 } where fp24 : & (L ＼ coverf Lx) ≡ & (L ＼ * (& (coverf Lx))) fp24 = cong (λ k → & ( L ＼ k )) (sym *iso) fp25 : odef (* X) (& (coverf Lx)) fp25 = proj1 ( ODC.x∋minimal O (coverSet Lx) (fp17 Lx) ) fp23 : {x : Ordinal} (lt : odef L x) → odef (* (& (L ＼ coverf lt))) x fp23 {x} Lx = subst (λ k → odef k x) (sym *iso) ⟪ Lx , fp26 ⟫ where fp26 : ¬ odef (coverf Lx) x fp26 = subst (λ k → ¬ odef k x ) *iso (proj2 (proj2 ( ODC.x∋minimal O (coverSet Lx) (fp17 Lx) )) ) fp05 : {y : Ordinal } → (Xy : odef (* X) y ) → odef ( * y) (NC.x not-covered ) fp05 {y} Xy = NC.yx not-covered Xy fp06 : NC.x not-covered o≤ & (* X) fp06 = begin NC.x not-covered ≡⟨ sym &iso ⟩ & (* (NC.x not-covered)) <⟨ c<→o< (subst₂ (λ j k → odef j k ) *iso (sym &iso) (NC.yx not-covered Xe)) ⟩ & e <⟨ c<→o< Xe ⟩ & (* X) ∎ where open o≤-Reasoning O limit : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) → Ordinal limit {X} CX fip with trio< X o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a b ¬c = o∅ ... | tri> ¬a ¬b c = & (ODC.minimal O (Intersection (* X)) ( has-intersection CX fip c)) fip00 : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) {x : Ordinal} → odef (* X) x → odef (* x) (limit CX fip ) fip00 {X} CX fip {x} Xx with trio< X o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) (subst (λ k → o∅ o< k) &iso (∈∅< Xx) ) ) ... | tri> ¬a ¬b c with ODC.x∋minimal O (Intersection (* X)) ( has-intersection CX fip c ) ... | ⟪ 0<m , intersect ⟫ = intersect Xx open Filter -- Ultra Filter has limit point record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where field u : Ordinal ou : odef (OS TP) u ux : odef (* u) x v⊆P : * v ⊆ P u⊆v : * u ⊆ * v -- -- Neighbor on x is a Filter (on Power P) -- NeighborF : {P : HOD} (TP : Topology P) (x : Ordinal ) → Filter {Power P} {P} (λ x → x) NeighborF {P} TP x = record { filter = NF ; f⊆L = NF⊆PP ; filter1 = f1 ; filter2 = f2 } where nf00 : {v : Ordinal } → Neighbor TP x v → odef (Power P) v nf00 {v} nei y vy = Neighbor.v⊆P nei vy NF : HOD NF = record { od = record { def = λ v → Neighbor TP x v } ; odmax = & (Power P) ; <odmax = λ lt → odef< (nf00 lt) } NF⊆PP : NF ⊆ Power P NF⊆PP = nf00 f1 : {p q : HOD} → Power P ∋ q → NF ∋ p → p ⊆ q → NF ∋ q f1 {p} {q} Pq Np p⊆q = record { u = Neighbor.u Np ; ou = Neighbor.ou Np ; ux = Neighbor.ux Np ; v⊆P = Pq _ ; u⊆v = f11 } where f11 : * (Neighbor.u Np) ⊆ * (& q) f11 {x} ux = subst (λ k → odef k x ) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso (Neighbor.u⊆v Np ux)) ) f2 : {p q : HOD} → NF ∋ p → NF ∋ q → Power P ∋ (p ∩ q) → NF ∋ (p ∩ q) f2 {p} {q} Np Nq Ppq = record { u = upq ; ou = ou ; ux = ux ; v⊆P = Ppq _ ; u⊆v = f20 } where upq : Ordinal upq = & ( * (Neighbor.u Np) ∩ * (Neighbor.u Nq) ) ou : odef (OS TP) upq ou = o∩ TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou Np)) (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou Nq)) ux : odef (* upq) x ux = subst ( λ k → odef k x ) (sym *iso) ⟪ Neighbor.ux Np , Neighbor.ux Nq ⟫ f20 : * upq ⊆ * (& (p ∩ q)) f20 = subst₂ (λ j k → j ⊆ k ) (sym *iso) (sym *iso) ( λ {x} pq → ⟪ subst (λ k → odef k x) *iso (Neighbor.u⊆v Np (proj1 pq)) , subst (λ k → odef k x) *iso (Neighbor.u⊆v Nq (proj2 pq)) ⟫ ) CAP : (P : HOD) {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) CAP P {p} {q} Pp Pq x pqx with subst (λ k → odef k x ) *iso pqx ... | ⟪ px , qx ⟫ = Pp _ (subst (λ k → odef k x) (sym *iso) px ) NEG : (P : HOD) {p : HOD } → Power P ∋ p → Power P ∋ (P ＼ p ) NEG P {p} Pp x vx with subst (λ k → odef k x) *iso vx ... | ⟪ Px , npx ⟫ = Px