Mercurial > hg > Members > kono > Proof > ZF-in-agda
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 16 Jan 2023 02:22:03 +0900 |
parents | 73b256c5474b |
children | 1966127fc14f |
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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∅ -- 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 --- we may add -- OS∋L : OS ∋ L open Topology Cl : {L : HOD} → (top : Topology L) → (A : HOD) → A ⊆ L → HOD Cl {L} top A A⊆L = 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) → {f : L ⊆ L } → Cl top L f ≡ L ClL {L} top {f} = ==→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) ⟫ } ) -- 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 ; 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 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 } -- 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 ) ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) ProductTopology {P} {Q} TP TQ = GeneratedTopogy (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } -- 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 limit : {X : Ordinal } → * X ⊆ CS top → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal is-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) (limit CX fip) L∋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 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 : {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} 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 ) -- 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 -- -- X covres L means Intersection of (CX X) contains nothing -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP ) -- it means there is a finite cover -- record CFIP (x : Ordinal) : Set n where field is-CS : * x ⊆ CS top y : Ordinal sy : Subbase (* y) o∅ Cex : HOD Cex = record { od = record { def = λ x → CFIP x } ; odmax = osuc (& (CS top)) ; <odmax = λ {x} lt → subst₂ (λ j k → j o≤ k ) &iso refl ( ⊆→o≤ (CFIP.is-CS lt )) } cex : {X : Ordinal } → * X ⊆ OS top → * X covers L → Ordinal cex {X} ox oc = & ( ODC.minimal O Cex fip00) where fip00 : ¬ ( Cex =h= od∅ ) fip00 cex=0 = ⊥-elim (fip09 fip25 fip20) 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 {C} {x} C<CX sc with trio< x o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri> ¬a ¬b c = c ... | tri≈ ¬a b ¬c = ⊥-elim (¬x<0 ( _==_.eq→ cex=0 record { is-CS = fip10 ; y = C ; sy = subst (λ k → Subbase (* C) k) b sc } )) where fip10 : * C ⊆ CS top fip10 {w} cw = CCX ox ( C<CX cw ) fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) fip25 = FIP.L∋limit fip (CCX ox) fip02 ? 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 ) 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 _ )) ¬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 (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 = {!!} -- 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 {L} {P} LP ) (FL : filter F ∋ P) (ultra : ultra-filter 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 UFLP→FIP : {P : HOD} (TP : Topology P) → {L : HOD} (LP : L ⊆ Power P ) → ( (F : Filter {L} {P} LP ) (FP : filter F ∋ P) (UF : ultra-filter F ) → UFLP TP LP F FP UF ) → FIP TP UFLP→FIP {P} TP {L} LP uflp = record { limit = uf00 ; is-limit = {!!} } where fip : {X : Ordinal} → * X ⊆ CS TP → Set n fip {X} CSX = {C x : Ordinal} → * C ⊆ * X → Subbase (* C) x → o∅ o< x F : Filter {L} {P} LP F = ? uf00 : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal uf00 {X} CSX fip = UFLP.limit ( uflp F ? (F→ultra LP ? ? F ? ? ? ) ) FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (FP : filter F ∋ P) (UF : ultra-filter F ) → UFLP {P} TP {L} LP F FP UF FIP→UFLP {P} TP fip {L} LP F FP UF = ? -- product topology of compact topology is compact Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) LPQ uflp ) where L = pbase TP TQ LPQ = pbase⊆PL TP TQ -- Product of UFL has limit point uflp : (F : Filter {pbase TP TQ} LPQ) (LF : filter F ∋ ZFP P Q) (UF : ultra-filter F) → UFLP (ProductTopology TP TQ) LPQ F LF UF uflp F LF UF = record { limit = & < ? , {!!} > ; P∋limit = {!!} ; is-limit = {!!} }