Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/generic-filter.agda @ 1239:5223f0b40d91
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 12 Mar 2023 11:58:55 +0900 |
parents | 362e43a1477c |
children | fbe072447243 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module generic-filter {n : Level } (O : Ordinals {n}) where import filter open import zf open import logic -- open import partfunc {n} O import OD open import Relation.Nullary open import Relation.Binary open import Data.Empty open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) 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 filter O open _∧_ open _∨_ open Bool open HOD ------- -- the set of finite partial functions from ω to 2 -- -- open import Data.List hiding (filter) open import Data.Maybe open import ZProduct O record CountableModel : Set (suc (suc n)) where field ctl-M : HOD ctl→ : Nat → Ordinal ctl<M : (x : Nat) → odef (ctl-M) (ctl→ x) ctl← : (x : Ordinal )→ odef (ctl-M ) x → Nat ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x -- we have no otherway round -- ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x -- -- almmost universe -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x -- -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, open CountableModel ---- -- a(n) ∈ M -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q -- PGHOD : (i : Nat) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD PGHOD i L C p = record { od = record { def = λ x → odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } --- -- p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n) -- find-p : (L : HOD ) (C : CountableModel ) (i : Nat) → (x : Ordinal) → Ordinal find-p L C Zero x = x find-p L C (Suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) ... | yes y = find-p L C i x ... | no not = & (ODC.minimal O ( PGHOD i L C (find-p L C i x)) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice --- -- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) } -- record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where field gr : Nat pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y x∈PP : odef L x open PDN --- -- G as a HOD -- PDHOD : (L p : HOD ) (C : CountableModel ) → HOD PDHOD L p C = record { od = record { def = λ x → PDN L p C x } ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } open PDN ---- -- Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → P → Set ) -- -- p 0 ≡ ∅ -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) --- else p n P∅ : {P : HOD} → odef (Power P) o∅ P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt open import Data.Nat.Properties open import nat p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p))) p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) ... | yes y = refl-⊆ {* (find-p L C n (& p))} ... | no not = ? where -- λ lt → proj2 (proj2 fmin∈PGHOD) _ ? where fmin : HOD fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : Nat} → n ≤ m → (* (find-p L C m (& p))) ⊆ (* (find-p L C n (& p))) p-monotonic L p C {Zero} {Zero} n≤m = refl-⊆ {* (find-p L C Zero (& p))} p-monotonic L p C {Zero} {Suc m} z≤n lt = (p-monotonic L p C {Zero} {m} z≤n ) (p-monotonic1 L p C {m} lt ) p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m ... | tri< a ¬b ¬c = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt ) ... | tri≈ ¬a refl ¬c = λ x → x ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) 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 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 ∩ Replace (Filter.filter genf) (λ x → P \ x )) ≡ od∅ ) rgen : HOD rgen = Replace (Filter.filter genf) (λ x → P \ x ) P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) P-GenericFilter P L p0 L⊆PP Lp0 C = record { genf = record { filter = Replace (PDHOD L p0 C) (λ x → P \ x) ; f⊆L = ? ; filter1 = ? ; filter2 = ? } ; generic = ? } where f⊆PL : PDHOD L p0 C ⊆ L f⊆PL lt = x∈PP lt f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q f1 {p} {q} L∋q PD∋p p⊆q = ? f2 : {p q : HOD} → ? ∋ p → ? ∋ q → L ∋ (p ∩ q) → ? ∋ (p ∩ q) f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p) ... | tri< a ¬b ¬c = ? ... | tri≈ ¬a eq ¬c = ? ... | tri> ¬a ¬b c = ? gf : HOD gf = Replace (Replace (PDHOD L p0 C) (λ x → P \ x)) (_\_ P) fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ gf ) ≡ od∅ fdense D MD eq0 = ? where open Dense open GenericFilter open Filter record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where field b : HOD 0<b : ¬ o∅ ≡ & b b<a : b ⊆ a lemma232 : (P L p : HOD ) (C : CountableModel ) → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) → ¬ ( (ctl-M C) ∋ rgen ( P-GenericFilter P L p LP Lp0 C )) lemma232 P L p C LP Lp0 NA MG = {!!} where D : HOD -- P - G D = ? -- -- P-Generic Filter defines a countable model D ⊂ C from P -- -- -- in D, we have V ≠ L -- -- -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } -- record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (suc n) where field valx : HOD record valS (ox oy oG : Ordinal) : Set n where field op : Ordinal p∈G : odef (* oG) op is-val : odef (* ox) ( & < * oy , * op > ) val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} → (G : GenericFilter {L} {P} LP {!!} ) → HOD val x G = TransFinite {λ x → HOD } ind (& x) where ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} }