Mercurial > hg > Members > kono > Proof > ZF-in-agda
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Generic Filter done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 13 Mar 2022 19:22:12 +0900 |
parents | 81691a6b352b |
children | b27d92694ed5 |
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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 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 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 import OPair open OPair O record CountableModel (P : HOD) : Set (suc (suc n)) where field ctl-M : Ordinal ctl→ : Nat → Ordinal ctl← : (x : Ordinal )→ x o< ctl-M → Nat ctl<M : (x : Nat) → ctl→ x o< ctl-M ctl-iso→ : { x : Ordinal } → (lt : x o< ctl-M) → ctl→ (ctl← x lt ) ≡ x ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x ctl-P∈M : Power P ∈ * ctl-M -- -- almmost universe -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x -- -- we expect P ∈ * ctl-M ∧ G ⊆ Power P , ¬ G ∈ * ctl-M, open CountableModel ---- -- a(n) ∈ M -- ∃ q ∈ Power P → q ∈ a(n) ∧ p(n) ⊆ q -- PGHOD : (i : Nat) (P : HOD) (C : CountableModel P) → (p : Ordinal) → HOD PGHOD i P C p = record { od = record { def = λ x → odef (Power P) x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) (proj1 lt) } --- -- p(n+1) = if (f n) != ∅ then (f n) otherwise p(n) -- find-p : (P : HOD ) (C : CountableModel P) (i : Nat) → (x : Ordinal) → Ordinal find-p P C Zero x = x find-p P C (Suc i) x with is-o∅ ( & ( PGHOD i P C (find-p P C i x)) ) ... | yes y = find-p P C i x ... | no not = & (ODC.minimal O ( PGHOD i P C (find-p P C i x)) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice --- -- G = { r ∈ Power P | ∃ n → p(n) ⊆ q } -- record PDN (P p : HOD ) (C : CountableModel P) (x : Ordinal) : Set n where field gr : Nat pn<gr : (y : Ordinal) → odef (* (find-p P C gr (& p))) y → odef (* x) y x∈PP : odef (Power P) x open PDN --- -- G as a HOD -- PDHOD : (P p : HOD ) (C : CountableModel P ) → HOD PDHOD P p C = record { od = record { def = λ x → PDN P p C x } ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) {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 open _⊆_ p-monotonic1 : (P p : HOD ) (C : CountableModel P ) → {n : Nat} → (* (find-p P C n (& p))) ⊆ (* (find-p P C (Suc n) (& p))) p-monotonic1 P p C {n} with is-o∅ (& (PGHOD n P C (find-p P C n (& p)))) ... | yes y = refl-⊆ ... | no not = record { incl = λ {x} lt → proj2 (proj2 fmin∈PGHOD) (& x) lt } where fmin : HOD fmin = ODC.minimal O (PGHOD n P C (find-p P C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) fmin∈PGHOD : PGHOD n P C (find-p P C n (& p)) ∋ fmin fmin∈PGHOD = ODC.x∋minimal O (PGHOD n P C (find-p P C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) p-monotonic : (P p : HOD ) (C : CountableModel P ) → {n m : Nat} → n ≤ m → (* (find-p P C n (& p))) ⊆ (* (find-p P C m (& p))) p-monotonic P p C {Zero} {Zero} n≤m = refl-⊆ p-monotonic P p C {Zero} {Suc m} z≤n = trans-⊆ (p-monotonic P p C {Zero} {m} z≤n ) (p-monotonic1 P p C {m} ) p-monotonic P p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m ... | tri< a ¬b ¬c = trans-⊆ (p-monotonic P p C {Suc n} {m} a) (p-monotonic1 P p C {m} ) ... | tri≈ ¬a refl ¬c = refl-⊆ ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) P-GenericFilter : (P p0 : HOD ) → Power P ∋ p0 → (C : CountableModel P) → GenericFilter P P-GenericFilter P p0 Pp0 C = record { genf = record { filter = PDHOD P p0 C ; f⊆PL = f⊆PL ; filter1 = f1 ; filter2 = f2 } ; generic = fdense } where PGHOD∈PL : (i : Nat) → (x : Ordinal) → PGHOD i P C x ⊆ Power P PGHOD∈PL i x = record { incl = λ {x} p → proj1 p } f⊆PL : PDHOD P p0 C ⊆ Power P f⊆PL = record { incl = λ {x} lt → x∈PP lt } f1 : {p q : HOD} → q ⊆ P → PDHOD P p0 C ∋ p → p ⊆ q → PDHOD P p0 C ∋ q f1 {p} {q} q⊆P PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = power← _ _ (incl q⊆P) } where f04 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (incl p⊆q (subst₂ (λ j k → odef k j ) (sym &iso) *iso ( pn<gr PD∋p y lt1 ))) -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y f2 : {p q : HOD} → PDHOD P p0 C ∋ p → PDHOD P p0 C ∋ q → PDHOD P p0 C ∋ (p ∩ q) f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋p) (gr PD∋q) ... | tri< a ¬b ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt); x∈PP = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y (f5 lt)) ⟫ where f5 : odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (find-p P C (gr PD∋q) (& p0))) y f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋p} {gr PD∋q} (<to≤ a)) (subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) -- subst (λ k → odef k y) *iso (pn<gr PD∋q y (subst (λ k → odef _ k ) &iso (incl (p-monotonic _ _ C a ) (subst (λ k → odef _ k) &iso lt) ))) ⟫ ... | tri≈ ¬a refl ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f4 y lt); x∈PP = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where f4 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y f4 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ ... | tri> ¬a ¬b c = record { gr = gr PD∋q ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt) ; x∈PP = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (p ∩ q) y f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y (f5 lt)) , subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ where f5 : odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (* (find-p P C (gr PD∋p) (& p0))) y f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋q} {gr PD∋p} (<to≤ c)) (subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) fdense : (D : Dense P ) → ¬ (filter.Dense.dense D ∩ PDHOD P p0 C) ≡ od∅ fdense D eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD P p0 C} fd01 (≡od∅→=od∅ eq0 )) where open Dense fd : HOD fd = dense-f D p0 PP∋D : dense D ⊆ Power P PP∋D = d⊆P D fd00 : PDHOD P p0 C ∋ p0 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = Pp0 } fd02 : dense D ∋ dense-f D p0 fd02 = dense-d D (ODC.power→⊆ O _ _ Pp0 ) fd04 : dense-f D p0 ⊆ P fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 ) fd03 : PDHOD P p0 C ∋ dense-f D p0 fd03 = f1 {p0} {dense-f D p0} fd04 fd00 ( dense-p D (ODC.power→⊆ O _ _ Pp0 ) ) fd01 : (dense D ∩ PDHOD P p0 C) ∋ fd fd01 = ⟪ fd02 , fd03 ⟫ open GenericFilter open Filter record Incompatible (P : HOD ) : Set (suc (suc n)) where field q : {p : HOD } → Power P ∋ p → HOD r : {p : HOD } → Power P ∋ p → HOD incompatible : { p : HOD } → (P∋p : Power P ∋ p) → Power P ∋ q P∋p → Power P ∋ r P∋p → ( p ⊆ q P∋p) ∧ ( p ⊆ r P∋p) → ∀ ( s : HOD ) → Power P ∋ s → ¬ (( q P∋p ⊆ s ) ∧ ( r P∋p ⊆ s )) lemma725 : (P p : HOD ) (C : CountableModel P) → * (ctl-M C) ∋ Power P → Incompatible P → ¬ ( * (ctl-M C) ∋ filter ( genf ( P-GenericFilter P p {!!} C ))) lemma725 = {!!} open import PFOD O -- HODω2 : HOD -- -- ω→2 : HOD -- ω→2 = Power infinite lemma725-1 : Incompatible HODω2 lemma725-1 = {!!} lemma726 : (C : CountableModel HODω2) → Union ( Replace HODω2 (λ p → filter ( genf ( P-GenericFilter HODω2 p {!!} C )))) =h= ω→2 lemma726 = {!!} -- -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } -- record valR (x : HOD) {P : HOD} (G : GenericFilter P) : 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 : HOD } → (G : GenericFilter P) → 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 = {!!} } -- -- W (ω , H ( ω , 2 )) = { p ∈ ( Nat → H (ω , 2) ) | { i ∈ Nat → p i ≠ i1 } is finite } --