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 | Tue, 10 Jan 2023 03:00:04 +0900 |
parents | edef8810a023 |
children | c4f4868a8cdd |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module filter {n : Level } (O : Ordinals {n}) where open import zf open import logic import OD open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core 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 _∧_ open _∨_ open Bool -- Kunen p.76 and p.53, we use ⊆ record Filter { L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field filter : HOD f⊆L : filter ⊆ L filter1 : { p q : HOD } → L ∋ q → filter ∋ p → p ⊆ q → filter ∋ q filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → L ∋ (p ∩ q) → filter ∋ (p ∩ q) open Filter record prime-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) prime : {p q : HOD } → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) record ultra-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) ultra : {p : HOD } → L ∋ p → L ∋ ( P \ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P \ p) ) ∈-filter : {L P p : HOD} → {LP : L ⊆ Power P} → (F : Filter {L} {P} LP ) → filter F ∋ p → L ∋ p ∈-filter {L} {p} {LP} F lt = ( f⊆L F) lt ⊆-filter : {L P p q : HOD } → {LP : L ⊆ Power P } → (F : Filter {L} {P} LP) → L ∋ q → q ⊆ P ⊆-filter {L} {P} {p} {q} {LP} F lt = power→⊆ P q ( LP lt ) ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L ∪-lemma1 {L} {p} {q} lt p∋x = lt (case1 p∋x) ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L ∪-lemma2 {L} {p} {q} lt p∋x = lt (case2 p∋x) q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q q∩q⊆q lt = proj1 lt open HOD ----- -- -- ultra filter is prime -- filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ (P \ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) → (F : Filter {L} {P} LP) → ultra-filter F → prime-filter F filter-lemma1 {P} {L} LP NG IN F u = record { proper = ultra-filter.proper u ; prime = lemma3 } where lemma3 : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) lemma3 {p} {q} Lp Lq _ lt with ultra-filter.ultra u Lp (NG Lp) ... | case1 p∈P = case1 p∈P ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P \ p)} Lq lemma7 lemma8) where lemma5 : ((p ∪ q ) ∩ (P \ p)) =h= (q ∩ (P \ p)) lemma5 = record { eq→ = λ {x} lt → ⟪ lemma4 x lt , proj2 lt ⟫ ; eq← = λ {x} lt → ⟪ case2 (proj1 lt) , proj2 lt ⟫ } where lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (P \ p)) x → odef q x lemma4 x lt with proj1 lt lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) lemma4 x lt | case2 qx = qx lemma9 : L ∋ ((p ∪ q ) ∩ (P \ p)) lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (IN Lq (NG Lp)) lemma6 : filter F ∋ ((p ∪ q ) ∩ (P \ p)) lemma6 = filter2 F lt ¬p∈P lemma9 lemma7 : filter F ∋ (q ∩ (P \ p)) lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 lemma8 : (q ∩ (P \ p)) ⊆ q lemma8 lt = proj1 lt ----- -- -- if Filter {L} {P} contains L, prime filter is ultra -- filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P \ p)) → (F : Filter {L} {P} LP) → filter F ∋ P → prime-filter F → ultra-filter F filter-lemma2 {P} {L} LP Lm F f∋P prime = record { proper = prime-filter.proper prime ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (Lm L∋p) (lemma10 L∋p ((f⊆L F) f∋P) ) (lemma p (p⊆L L∋p )) } where open _==_ p⊆L : {p : HOD} → L ∋ p → p ⊆ P p⊆L {p} lt = power→⊆ P p ( LP lt ) p+1-p=1 : {p : HOD} → p ⊆ P → P =h= (p ∪ (P \ p)) eq→ (p+1-p=1 {p} p⊆P) {x} lt with ODC.decp O (odef p x) eq→ (p+1-p=1 {p} p⊆P) {x} lt | yes p∋x = case1 p∋x eq→ (p+1-p=1 {p} p⊆P) {x} lt | no ¬p = case2 ⟪ lt , ¬p ⟫ eq← (p+1-p=1 {p} p⊆P) {x} ( case1 p∋x ) = subst (λ k → odef P k ) &iso (p⊆P ( subst (λ k → odef p k) (sym &iso) p∋x )) eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P \ p)) lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P lemma10 : {p : HOD} → L ∋ p → L ∋ P → L ∋ (p ∪ (P \ p)) lemma10 {p} L∋p lt = subst (λ k → L ∋ k ) (==→o≡ (p+1-p=1 {p} (p⊆L L∋p))) lt ----- -- -- if there is a filter , there is a ultra filter under the axiom of choise -- Zorn Lemma -- filter→ultra : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P \ p)) → (F : Filter {L} {P} LP) → ultra-filter F -- filter→ultra {P} {L} LP Lm F = {!!} record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field dense : HOD d⊆P : dense ⊆ L dense-f : {p : HOD} → L ∋ p → HOD dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p record Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field ideal : HOD i⊆L : ideal ⊆ L ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) open Ideal proper-ideal : {L P : HOD} → (LP : L ⊆ Power P) → (P : Ideal {L} {P} LP ) → {p : HOD} → Set n proper-ideal {L} {P} LP I = ideal I ∋ od∅ prime-ideal : {L P : HOD} → (LP : L ⊆ Power P) → Ideal {L} {P} LP → ∀ {p q : HOD } → Set n prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where field genf : Filter {L} {P} LP generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) : Set (suc n) where field mf : Filter {L} {P} LP proper : ¬ (filter mf ∋ od∅) is-maximum : ( f : Filter {L} {P} LP ) → ¬ (filter f ∋ od∅) → ¬ ( filter mf ⊂ filter f ) record Fp {L P : HOD} (LP : L ⊆ Power P) (mx : MaximumFilter {L} {P} LP ) (p x : Ordinal ) : Set n where field y : Ordinal mfy : odef (filter (MaximumFilter.mf mx)) y x=y∪p : x ≡ & ( * y ∪ * p ) max→ultra : {L P : HOD} (LP : L ⊆ Power P) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p \ q)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q)) → (mx : MaximumFilter {L} {P} LP ) → ultra-filter ( MaximumFilter.mf mx ) max→ultra {L} {P} LP NEG CUP mx = record { proper = MaximumFilter.proper mx ; ultra = ultra } where mf = MaximumFilter.mf mx ultra : {p : HOD} → L ∋ p → L ∋ (P \ p) → (filter mf ∋ p) ∨ (filter mf ∋ (P \ p)) ultra {p} Lp lnp with ∋-p (filter mf) p ... | yes y = case1 y ... | no np with ∋-p (filter mf) (P \ p) ... | yes y = case2 y ... | no n-p = ⊥-elim (MaximumFilter.is-maximum mx FisFilter FisProper ⟪ F< , FisGreater ⟫ ) where F⊆L : {x : Ordinal } → Fp {L} {P} LP mx (& p) x ∨ odef (filter mf) x → odef L x F⊆L (case2 mfx) = f⊆L mf mfx F⊆L {x} (case1 pmf) = mu04 pmf where mu05 : (fp : Fp LP mx (& p) x ) → odef L (Fp.y fp ) mu05 fp = f⊆L mf ( Fp.mfy fp ) mu04 : Fp LP mx (& p) x → odef L x mu04 fp = subst (λ k → odef L k ) (sym (trans (Fp.x=y∪p fp ) mu06 )) ( CUP (subst (λ k → odef L k ) (sym &iso) (mu05 fp)) Lp ) where mu06 : & (* (Fp.y fp) ∪ * (& p)) ≡ & (* (Fp.y fp) ∪ p) mu06 = cong (λ k → & (* (Fp.y fp) ∪ k)) *iso F : HOD -- Replace (filter mf) (λ y → y ∪ p ) ∪ filter mf F = record { od = record { def = λ x → Fp {L} {P} LP mx (& p) x ∨ odef (filter mf) x } ; odmax = & L ; <odmax = λ lt → odef< (F⊆L lt) } mu01 : {r : HOD} {q : HOD} → L ∋ q → F ∋ r → r ⊆ q → F ∋ q mu01 {r} {q} Lq (case1 record { y = y ; mfy = mfy ; x=y∪p = x=y∪p }) r⊆q = mu03 where y+q-r : HOD y+q-r = * y ∪ ( q \ r ) Ly : L ∋ * y Ly = subst (λ k → odef L k) (sym &iso) (f⊆L mf mfy) mu08 : L ∋ y+q-r mu08 = CUP Ly (NEG Lq (subst (λ k → odef L k) (trans (cong (λ k → & (* y ∪ k)) (sym *iso)) (sym x=y∪p) ) (CUP Ly Lp )) ) mu09 : * y ⊆ y+q-r mu09 {x} yx = case1 yx mu07 : filter mf ∋ y+q-r mu07 = filter1 mf {_} {y+q-r} mu08 (subst (λ k → odef (filter mf) k) (sym &iso) mfy) mu09 mu03 : odef F (& q) -- y+q-r + p ≡ y+q-(y+p)+ p = q so it is in F mu03 = case1 record { y = _ ; mfy = mu07 ; x=y∪p = cong (&) (==→o≡ record { eq→ = mu10 ; eq← = mu11 } ) } where mu12 : r ≡ (* y ∪ p) mu12 = subst₂ (λ j k → j ≡ k ) *iso (trans *iso (cong (λ k → (* y ∪ k)) *iso)) (cong (*) x=y∪p ) mu10 : {x : Ordinal} → odef q x → odef (* (& y+q-r) ∪ * (& p)) x mu10 {x} qx with ODC.∋-p O r (* x) ... | no nrx = case1 (subst (λ k → odef k x) (sym *iso) mu13) where mu13 : odef (* y ∪ (q \ r)) x mu13 = case2 ⟪ qx , (λ rx → nrx (subst (λ k → odef r k ) (sym &iso) rx)) ⟫ ... | yes rx with subst₂ (λ j k → odef j k ) mu12 (sym &iso) rx ... | case1 yx = case1 (subst (λ k → odef k x) (sym *iso) (case1 (subst (λ k → odef (* y) k) (trans &iso &iso) yx) ) ) ... | case2 px = case2 (subst₂ (λ j k → odef j k ) (sym *iso) (trans &iso &iso) px ) mu11 : {x : Ordinal} → odef (* (& y+q-r) ∪ * (& p)) x → odef q x mu11 {x} (case2 px) = r⊆q (subst (λ k → odef k x) (sym mu12) (case2 (subst (λ k → odef k x) *iso px) )) mu11 {x} (case1 m06x) with subst (λ k → odef k x) *iso m06x ... | case1 yx = r⊆q (subst (λ k → odef k x) (sym mu12) (case1 yx)) ... | case2 q-rx = proj1 q-rx mu01 {r} {q} Lq (case2 mfr) r⊆q = case2 ( filter1 mf Lq mfr r⊆q) mu02 : {r : HOD} {q : HOD} → F ∋ r → F ∋ q → L ∋ (r ∩ q) → F ∋ (r ∩ q) mu02 {r} {q} (case1 record { y = y₁ ; mfy = mfy₁ ; x=y∪p = x=y∪p₁ }) (case1 record { y = y ; mfy = mfy ; x=y∪p = x=y∪p }) Lrq = ? mu02 {r} {q} (case1 record { y = y ; mfy = mfy ; x=y∪p = x=y∪p }) (case2 mfq) Lrq = ? mu02 {r} {q} (case2 mfr) (case1 record { y = y ; mfy = mfp ; x=y∪p = x=y∪p }) Lrq = ? mu02 {r} {q} (case2 mfr) (case2 mfq ) Lrq = ? FisFilter : Filter {L} {P} LP FisFilter = record { filter = F ; f⊆L = F⊆L ; filter1 = mu01 ; filter2 = mu02 } FisGreater : {x : Ordinal } → odef (filter (MaximumFilter.mf mx)) x → odef (filter FisFilter ) x FisGreater {x} mfx = case2 mfx F< : & (filter (MaximumFilter.mf mx)) o< & F F< = ? FisProper : ¬ (filter FisFilter ∋ od∅) FisProper = {!!} open _==_ -- open import Relation.Binary.Definitions ultra→max : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P \ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (U : Filter {L} {P} LP) → ultra-filter U → MaximumFilter LP ultra→max {L} {P} LP NG CAP U u = record { mf = U ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where is-maximum : (F : Filter {L} {P} LP) → (¬ (filter F ∋ od∅)) → (U⊂F : filter U ⊂ filter F ) → ⊥ is-maximum F Prop ⟪ U<F , U⊆F ⟫ = Prop f0 where GT : HOD GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = & L ; <odmax = um02 } where um02 : {y : Ordinal } → odef (filter F) y ∧ (¬ odef (filter U) y) → y o< & L um02 {y} Fy = odef< ( f⊆L F (proj1 Fy ) ) GT≠∅ : ¬ (GT =h= od∅) GT≠∅ eq = ⊥-elim (U≠F ( ==→o≡ ((⊆→= {filter U} {filter F}) U⊆F (U-F=∅→F⊆U {filter F} {filter U} gt01)))) where U≠F : ¬ ( filter U ≡ filter F ) U≠F eq = o<¬≡ (cong (&) eq) U<F gt01 : (x : Ordinal) → ¬ ( odef (filter F) x ∧ (¬ odef (filter U) x)) gt01 x not = ¬x<0 ( eq→ eq not ) p : HOD p = ODC.minimal O GT GT≠∅ ¬U∋p : ¬ ( filter U ∋ p ) ¬U∋p = proj2 (ODC.x∋minimal O GT GT≠∅) L∋p : L ∋ p L∋p = f⊆L F ( proj1 (ODC.x∋minimal O GT GT≠∅)) um00 : ¬ odef (filter U) (& p) um00 = proj2 (ODC.x∋minimal O GT GT≠∅) L∋-p : L ∋ ( P \ p ) L∋-p = NG L∋p U∋-p : filter U ∋ ( P \ p ) U∋-p with ultra-filter.ultra u {p} L∋p L∋-p ... | case1 ux = ⊥-elim ( ¬U∋p ux ) ... | case2 u-x = u-x F∋p : filter F ∋ p F∋p = proj1 (ODC.x∋minimal O GT GT≠∅) F∋-p : filter F ∋ ( P \ p ) F∋-p = U⊆F U∋-p f0 : filter F ∋ od∅ f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP L∋p L∋-p) ) import zorn open import Relation.Binary PO : IsStrictPartialOrder _≡_ _⊂_ PO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {a} {b} {c} → trans-⊂ {a} {b} {c} ; irrefl = λ x=y x<y → ? ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → ? ; snd = λ {x} {x1} {y} x=x1 x1y → ? } } open zorn O _⊂_ PO open import Relation.Binary.Structures SUP⊆ : (P B : HOD) → B ⊆ P → IsTotalOrderSet B → SUP P B SUP⊆ P B B⊆P OB = record { sup = Union B ; isSUP = record { ax = ? ; x≤sup = ? } } MaximumSubset : {L P : HOD} → o∅ o< & L → o∅ o< & P → P ⊆ L → Maximal P MaximumSubset {L} {P} 0<L 0<P P⊆L = Zorn-lemma 0<P (SUP⊆ P) MaximumFilterExist : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P \ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F : Filter {L} {P} LP) → o∅ o< & L → o∅ o< & (filter F) → (¬ (filter F ∋ od∅)) → MaximumFilter {L} {P} LP MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = record { mf = {!!} ; proper = {!!} ; is-maximum = {!!} } where mf01 : Maximal {!!} mf01 = MaximumSubset 0<L {!!} {!!}