Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/generic-filter.agda @ 460:d407cc8499fc
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 19 Mar 2022 12:16:48 +0900 |
parents | 3d84389cc43f |
children | 0e018784bed3 |
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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 : Set (suc (suc n)) where field ctl-M : Ordinal 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 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) ∧ q ⊆ p(n) -- 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 (* x) y → odef (* p) y ) } ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } --- -- p(n+1) = if (f n) != ∅ then (f n) 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 → p(n) ⊆ r } -- record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where field gr : Nat pn<gr : (y : Ordinal) → odef (* (find-p L C gr (& p))) y → odef (* x) 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 open _⊆_ 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} with is-o∅ (& (PGHOD n L C (find-p L 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 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-⊆ p-monotonic L p C {Zero} {Suc m} z≤n = trans-⊆ (p-monotonic1 L p C {m} ) (p-monotonic L p C {Zero} {m} z≤n ) p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m ... | tri< a ¬b ¬c = trans-⊆ (p-monotonic1 L p C {m}) (p-monotonic L p C {Suc n} {m} a) ... | tri≈ ¬a refl ¬c = refl-⊆ ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter LP P-GenericFilter P L p0 L⊆PP Lp0 C = record { genf = record { filter = PDHOD L p0 C ; f⊆L = f⊆PL ; filter1 = λ L∋q PD∋p p⊆q → f1 L∋q PD∋p p⊆q ; filter2 = f2 } ; generic = fdense } where f⊆PL : PDHOD L p0 C ⊆ L -- Power P f⊆PL = record { incl = λ {x} lt → x∈PP lt } Lq : {q : HOD} → L ∋ q → q ⊆ P Lq {q} lt = ODC.power→⊆ O P q ( incl L⊆PP lt ) Pp0 : p0 ∈ Power P Pp0 = incl L⊆PP Lp0 PGHOD∈PL : (i : Nat) → (x : Ordinal) → PGHOD i L C x ⊆ Power P PGHOD∈PL i x = record { incl = λ {x} p → incl L⊆PP (proj1 p) } 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 = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where f04 : (y : Ordinal) → odef (* (find-p L 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 L p0 C ∋ p → PDHOD L p0 C ∋ q → PDHOD L p0 C ∋ (p ∩ q) f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋q) (gr PD∋p) ... | 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 = {!!} } where f3 : (y : Ordinal) → odef (* (find-p L 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 L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) &iso ( incl (p-monotonic L p0 C {gr PD∋q} {gr PD∋p} (<to≤ a)) (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &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 = {!!} } where f4 : (y : Ordinal) → odef (* (find-p L 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 = {!!} } where -- f3 : (y : Ordinal) → odef (* (find-p L 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 L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) &iso ( incl (p-monotonic L p0 C {gr PD∋p} {gr PD∋q} (<to≤ c)) (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) fdense : (D : Dense L⊆PP ) → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅ fdense D eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where open Dense fd : HOD fd = dense-f D Lp0 PP∋D : dense D ⊆ Power P PP∋D = {!!} fd00 : PDHOD P p0 C ∋ p0 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = {!!} } fd02 : dense D ∋ dense-f D Lp0 fd02 = dense-d D Lp0 fd04 : dense-f D Lp0 ⊆ P fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 ) fd03 : PDHOD L p0 C ∋ dense-f D Lp0 fd03 = {!!} fd01 : (dense D ∩ PDHOD L p0 C) ∋ fd fd01 = ⟪ fd02 , fd03 ⟫ open GenericFilter open Filter record Incompatible (P p : HOD ) (PP∋p : p ⊆ P ) : Set (suc (suc n)) where field q r : HOD PP∋q : q ⊆ P PP∋r : r ⊆ P p⊆q : p ⊆ q p⊆r : p ⊆ r incompatible : ∀ ( s : HOD ) → s ⊆ P → (¬ ( q ⊆ s )) ∨ (¬ ( r ⊆ s )) lemma725 : (P p : HOD ) (C : CountableModel ) → (PP∋p : Power P ∋ p ) → * (ctl-M C) ∋ (Power P ∩ * (ctl-M C)) -- M is a Model of ZF → * (ctl-M C) ∋ {!!} -- ( (Power P ∩ * (ctl-M C)) \ filter ( genf ( P-GenericFilter P ? p ? C ? )) ) -- M ∋ G and M is a Model of ZF → ((p : HOD) → (PP∋p : p ⊆ P ) → Incompatible P p PP∋p ) → ¬ ( * (ctl-M C) ∋ filter {!!} ) -- ( genf ( P-GenericFilter P ? ? p PP∋p C ))) lemma725 P p C PP∋p M∋PM M∋D I M∋G = D∩G≠∅ D∩G=∅ where G = filter ( genf ( P-GenericFilter P {!!} p {!!} {!!} C )) M = * (ctl-M C) D : HOD D = Power P \ G p⊆P : p ⊆ P p⊆P = ODC.power→⊆ O _ _ PP∋p df : {x : HOD} → x ⊆ P → HOD df {x} PP∋x with ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) ... | yes y = Incompatible.q (I x PP∋x) ... | no n = Incompatible.r (I x PP∋x) df¬⊆P : {x : HOD} → (lt : x ⊆ P) → df lt ⊆ P df¬⊆P {x} PP∋x with ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) ... | yes _ = Incompatible.PP∋q (I x PP∋x) ... | no _ = Incompatible.PP∋r (I x PP∋x) ¬G∋df : {x : HOD} → (lt : x ⊆ P) → ¬ G ∋ (df lt ) ¬G∋df {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt ) ) ... | no n = n ... | yes y with Incompatible.incompatible (I x lt ) (Incompatible.q (I x lt )) (Incompatible.PP∋q (I x lt )) ... | case1 ¬q⊆pn = λ _ → ¬q⊆pn refl-⊆ ... | case2 ¬r⊆pn = {!!} df-d : {x : HOD} → (lt : x ⊆ P) → D ∋ df lt df-d {x} lt = ⟪ power← P _ (incl (df¬⊆P lt)) , ¬G∋df lt ⟫ df-p : {x : HOD} → (lt : x ⊆ P) → x ⊆ df lt df-p {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt) ) ... | yes _ = Incompatible.p⊆q (I x lt) ... | no _ = Incompatible.p⊆r (I x lt) D-Dense : Dense {!!} D-Dense = record { dense = D ; d⊆P = record { incl = λ {x} lt → {!!} } ; dense-f = {!!} ; dense-d = {!!} ; dense-p = {!!} } D∩G=∅ : ( D ∩ G ) =h= od∅ D∩G=∅ = ≡od∅→=od∅ ([a-b]∩b=0 {Power P} {G}) D∩G≠∅ : ¬ (( D ∩ G ) =h= od∅ ) D∩G≠∅ eq = generic (P-GenericFilter P {!!} {!!} {!!} {!!} C) D-Dense ( ==→o≡ eq ) open import PFOD O -- HODω2 : HOD -- -- ω→2 : HOD -- ω→2 = Power infinite lemma725-1 : (p : HOD) → (PP∋p : p ⊆ HODω2 ) → Incompatible HODω2 p PP∋p lemma725-1 = {!!} lemma726 : (C : CountableModel ) → Union ( Replace' (Power (ω→2 \ HODω2)) (λ p lt → filter ( genf ( P-GenericFilter {!!} (ω→2 \ HODω2) p {!!} {!!} C )))) =h= ω→2 -- HODω2 ∋ p lemma726 = {!!} -- -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } -- record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (G : GenericFilter LP) : 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 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 = {!!} } -- -- W (ω , H ( ω , 2 )) = { p ∈ ( Nat → H (ω , 2) ) | { i ∈ Nat → p i ≠ i1 } is finite } --