Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/Topology.agda @ 1121:98af35c9711f
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 03 Jan 2023 14:17:53 +0900 |
| parents | e086a266c6b7 |
| children | 1c7474446754 |
line wrap: on
line source
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.PropositionalEquality import BAlgbra open BAlgbra 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 -- 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 ) open Topology -- 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)) -- 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 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00 ; o∪ = tp02 ; o∩ = tp01 } where 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 proj2 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 proj2 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 } -- covers 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 finite : {X : Ordinal } → * X ⊆ CS top → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal limit : {X : Ordinal } → (CX : * X ⊆ CS top ) → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → {x : Ordinal } → odef (* X) x → odef (* x) (finite CX fip) -- Compact data Finite-∪ (S : HOD) : Ordinal → Set n where fin-e : {x : Ordinal } → odef S x → Finite-∪ S x fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) 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 = 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 xz → L ＼ z )) CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top CCX {X} ox = ? -- CX has finite intersection CXfip : {X : Ordinal } → * X ⊆ OS top → Set n CXfip {X} ox = { x C : Ordinal } → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x Cex : {X : Ordinal } → * X ⊆ OS top → HOD Cex {X} ox = record { od = record { def = λ C → { x : Ordinal } → * C ⊆ * (CX ox) → Subbase (* C) o∅ } ; odmax = osuc ( & (Power L)) ; <odmax = ? } -- a counter example of fip , some CX has no finite intersection cex : {X : Ordinal } → * X ⊆ OS top → * X covers L → Ordinal cex {X} ox oc = & ( ODC.minimal O (Cex ox) fip00) where fip00 : ¬ ( Cex ox =h= od∅ ) fip00 cex=0 = fip03 ? ? where fip03 : {x z : Ordinal } → odef (* x) z → (¬ odef (* x) z) → ⊥ fip03 {x} {z} xz nxz = nxz xz fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x fip02 = ? fip01 : Ordinal fip01 = FIP.finite fip (CCX ox) fip02 ¬CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → * (cex ox oc) ⊆ * (CX ox) → Subbase (* (cex ox oc)) o∅ ¬CXfip {X} ox oc = ? where fip04 : odef (Cex ox) (cex ox oc) fip04 = ? -- this defines finite cover finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal finCover {X} ox oc = & ( Replace' (* (cex ox oc)) (λ z xz → L ＼ z )) -- create Finite-∪ from cex isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) isFinite = ? -- is also a cover isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L isCover1 = ? Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top Compact→FIP = {!!} -- 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 q : 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 p 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 ; q = q ; 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 { p = p ; 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 ) _Top⊗_ : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) _Top⊗_ {P} {Q} TP TQ = GeneratedTopogy (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } -- existence of Ultra Filter open Filter -- Ultra Filter has limit point record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) : Set (suc (suc n)) where field limit : Ordinal P∋limit : odef P limit is-limit : {o : Ordinal} → odef (OS TP) o → odef (* o) limit → (* o) ⊆ filter F -- FIP is UFL FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf FIP→UFLP {P} TP fip {L} LP F uf = record { limit = ? ; P∋limit = ? ; is-limit = ? } UFLP→FIP : {P : HOD} (TP : Topology P) → ( {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf ) → FIP TP UFLP→FIP {P} TP uflp = record { fip≠φ = ? } -- Product of UFL has limit point (Tychonoff) Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (TP Top⊗ TQ) Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (TP Top⊗ TQ) (UFLP→FIP (TP Top⊗ TQ) uflp ) where uflp : {L : HOD} (LPQ : L ⊆ Power (ZFP P Q)) (F : Filter LPQ) (uf : ultra-filter {L} {_} {LPQ} F) → UFLP (TP Top⊗ TQ) LPQ F uf uflp {L} LPQ F uf = record { limit = & < * ( UFLP.limit uflpp) , ? > ; P∋limit = ? ; is-limit = ? } where LP : (L : HOD ) (LPQ : L ⊆ Power (ZFP P Q)) → HOD LP L LPQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ1 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) ) LPP : (L : HOD) (LPQ : L ⊆ Power (ZFP P Q)) → LP L LPQ ⊆ Power P LPP L LPQ record { z = z ; az = az ; x=ψz = x=ψz } w xw = tp02 (subst (λ k → odef k w) (subst₂ (λ j k → j ≡ k) refl *iso (cong (*) x=ψz) ) xw) where tp02 : Replace' (* z) (λ z₁ xz → * (zπ1 (LPQ (subst (odef L) (sym &iso) az) (& z₁) (subst (λ k → odef k (& z₁)) (sym *iso) xz)))) ⊆ P tp02 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k ) (trans (sym &iso) (sym x=ψz1) ) (zp1 (LPQ (subst (λ k → odef L k) (sym &iso) az) _ (tp03 az1 ))) where tp03 : odef (* z) z1 → odef (* (& (* z))) (& (* z1)) tp03 lt = subst (λ k → odef k (& (* z1))) (sym *iso) (subst (odef (* z)) (sym &iso) lt) FP : Filter (LPP L LPQ) FP = record { filter = LP (filter F) (λ x → LPQ (f⊆L F x )) ; f⊆L = tp04 ; filter1 = ? ; filter2 = ? } where tp04 : LP (filter F) (λ x → LPQ (f⊆L F x )) ⊆ LP L LPQ tp04 record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = f⊆L F az ; x=ψz = ? } uFP : ultra-filter FP uFP = record { proper = ? ; ultra = ? } uflpp : UFLP {P} TP {LP L LPQ} (LPP L LPQ) FP uFP uflpp = FIP→UFLP TP (Compact→FIP TP CP) (LPP L LPQ) FP uFP LQ : HOD LQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ2 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) ) LQQ : LQ ⊆ Power Q LQQ = ?