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, 14 Mar 2023 14:41:39 +0900 |
| parents | dd3debafba2d |
| children | b1d024385208 |
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{-# OPTIONS --allow-unsolved-metas #-} import Level open import Ordinals module generic-filter {n : Level.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 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 (Level.suc (Level.suc n)) where field ctl-M : HOD ctl→ : ℕ → Ordinal ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) ctl← : (x : Ordinal )→ odef (ctl-M ) x → ℕ ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x -- we have no otherway round -- ctl-iso← : { x : ℕ } → 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 abs-minus : {p q : HOD} → (C : CountableModel) → ctl-M C ∋ (p ＼ q) abs-minus {p} {q} C = ? where p-q : {x : Ordinal } → odef (p ＼ q) x → ℕ p-q pqx = ctl← C _ ? ---- -- a(n) ∈ M -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q -- PGHOD : (i : ℕ) (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 : ℕ) → (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 : ℕ 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. ℕ → 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 gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥ gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx gf02 : {P a b : HOD } → (P ＼ a) ∩ (P ＼ b) ≡ ( P ＼ (a ∪ b) ) gf02 {P} {a} {b} = ==→o≡ record { eq→ = gf03 ; eq← = gf04 }where gf03 : {x : Ordinal} → odef ((P ＼ a) ∩ (P ＼ b)) x → odef (P ＼ (a ∪ b)) x gf03 {x} ⟪ ⟪ Px , ¬ax ⟫ , ⟪ _ , ¬bx ⟫ ⟫ = ⟪ Px , (λ pab → gf05 {a} {b} {x} pab ¬ax ¬bx ) ⟫ gf04 : {x : Ordinal} → odef (P ＼ (a ∪ b)) x → odef ((P ＼ a) ∩ (P ＼ b)) x gf04 {x} ⟪ Px , abx ⟫ = ⟪ ⟪ Px , (λ ax → abx (case1 ax) ) ⟫ , ⟪ Px , (λ bx → abx (case2 bx) ) ⟫ ⟫ open import Data.Nat.Properties open import nat p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc 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 = λ lt → proj2 (proj2 fmin∈PGHOD) _ 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 : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& 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-monotonic1 L p C {m} (p-monotonic L p C {zero} {m} z≤n 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-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a 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 (Level.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) → p ⊆ dense-f lt record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.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 → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) -- L is a Boolean Algebra → (UNI : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q )) → (NEG : ({p : HOD} → L ∋ p → L ∋ ( P ＼ p))) → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) P-GenericFilter P L p0 L⊆PP Lp0 CAP UNI NEG C = record { genf = record { filter = Replace (PDHOD L p0 C) (λ x → P ＼ x) ; f⊆L = gf01 ; filter1 = f1 ; filter2 = f2 } ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd ) } where GP = Replace (PDHOD L p0 C) (λ x → P ＼ x) f⊆PL : PDHOD L p0 C ⊆ L f⊆PL lt = x∈PP lt gf01 : Replace (PDHOD L p0 C) (λ x → P ＼ x) ⊆ L gf01 {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef L k) (sym x=ψz) ( NEG (subst (λ k → odef L k) (sym &iso) (f⊆PL az)) ) gf141 : {xp xq : Ordinal } → (Pp : PDN L p0 C xp) (Pq : PDN L p0 C xq) → (* xp ∪ * xq) ⊆ P gf141 Pp Pq {x} (case1 xpx) = L⊆PP (PDN.x∈PP Pp) _ xpx gf141 Pp Pq {x} (case2 xqx) = L⊆PP (PDN.x∈PP Pq) _ xqx gf121 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → p ∩ q ≡ P ＼ * (& (* (Replaced.z gp) ∪ * (Replaced.z gq))) gf121 {p} {q} gp gq = begin p ∩ q ≡⟨ cong₂ (λ j k → j ∩ k ) (sym *iso) (sym *iso) ⟩ (* (& p)) ∩ (* (& q)) ≡⟨ cong₂ (λ j k → ( * j ) ∩ ( * k)) (Replaced.x=ψz gp) (Replaced.x=ψz gq) ⟩ * (& (P ＼ (* xp ))) ∩ (* (& (P ＼ (* xq )))) ≡⟨ cong₂ (λ j k → j ∩ k ) *iso *iso ⟩ (P ＼ (* xp )) ∩ (P ＼ (* xq )) ≡⟨ gf02 {P} {* xp} {* xq} ⟩ P ＼ ((* xp) ∪ (* xq)) ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ P ＼ * (& (* xp ∪ * xq)) ∎ where open ≡-Reasoning xp = Replaced.z gp xq = Replaced.z gq gf131 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q) → P ＼ (p ∩ q) ≡ * (Replaced.z gp) ∪ * (Replaced.z gq) gf131 {p} {q} gp gq = trans (cong (λ k → P ＼ k) (gf121 gp gq)) (trans ( L＼Lx=x (subst (λ k → k ⊆ P) (sym *iso) (gf141 (Replaced.az gp) (Replaced.az gq))) ) *iso ) f1 : {p q : HOD} → L ∋ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ p → p ⊆ q → Replace (PDHOD L p0 C) (λ x → P ＼ x) ∋ q f1 {p} {q} L∋q record { z = z ; az = az ; x=ψz = x=ψz } p⊆q = record { z = _ ; az = record { gr = gr az ; pn<gr = f04 ; x∈PP = NEG L∋q } ; x=ψz = f05 } where open ≡-Reasoning f04 : (y : Ordinal) → odef (* (& (P ＼ q))) y → odef (* (find-p L C (gr az ) (& p0))) y f04 y qy = PDN.pn<gr az _ (subst (λ k → odef k y ) f06 (f03 qy )) where f06 : * (& (P ＼ p)) ≡ * z f06 = begin * (& (P ＼ p)) ≡⟨ *iso ⟩ P ＼ p ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ P ＼ (* (& p)) ≡⟨ cong (λ k → P ＼ k) (cong (*) x=ψz) ⟩ P ＼ (* (& (P ＼ * z))) ≡⟨ cong ( λ k → P ＼ k) *iso ⟩ P ＼ (P ＼ * z) ≡⟨ L＼Lx=x (λ {x} lt → L⊆PP (x∈PP az) _ lt ) ⟩ * z ∎ f03 : odef (* (& (P ＼ q))) y → odef (* (& (P ＼ p))) y f03 pqy with subst (λ k → odef k y ) *iso pqy ... | ⟪ Py , nqy ⟫ = subst (λ k → odef k y ) (sym *iso) ⟪ Py , (λ py → nqy (p⊆q py) ) ⟫ f05 : & q ≡ & (P ＼ * (& (P ＼ q))) f05 = cong (&) ( begin q ≡⟨ sym (L＼Lx=x (λ {x} lt → L⊆PP L∋q _ (subst (λ k → odef k x) (sym *iso) lt) )) ⟩ P ＼ (P ＼ q ) ≡⟨ cong ( λ k → P ＼ k) (sym *iso) ⟩ P ＼ * (& (P ＼ q)) ∎ ) f2 : {p q : HOD} → GP ∋ p → GP ∋ q → L ∋ (p ∩ q) → GP ∋ (p ∩ q) f2 {p} {q} record { z = xp ; az = Pp ; x=ψz = peq } record { z = xq ; az = Pq ; x=ψz = qeq } L∋pq with <-cmp (gr Pp) (gr Pq) ... | tri< a ¬b ¬c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq) } where gp = record { z = xp ; az = Pp ; x=ψz = peq } gq = record { z = xq ; az = Pq ; x=ψz = qeq } gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) gf10 = record { gr = PDN.gr Pq ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where gf16 : gr Pp ≤ gr Pq gf16 = <to≤ a gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pq) (& p0))) y gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy ... | case1 xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y xpy ) ... | case2 xqy = PDN.pn<gr Pq _ xqy ... | tri≈ ¬a eq ¬c = record { z = & (* xp ∪ * xq) ; az = record { gr = gr Pp ; pn<gr = gf21 ; x∈PP = gf22 } ; x=ψz = gf23 } where gp = record { z = xp ; az = Pp ; x=ψz = peq } gq = record { z = xq ; az = Pq ; x=ψz = qeq } gf22 : odef L (& (* xp ∪ * xq)) gf22 = UNI (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pp)) (subst (λ k → odef L k ) (sym &iso) (PDN.x∈PP Pq)) gf21 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y gf21 y xpqy with subst (λ k → odef k y) *iso xpqy ... | case1 xpy = PDN.pn<gr Pp _ xpy ... | case2 xqy = subst (λ k → odef (* (find-p L C k (& p0))) y ) (sym eq) ( PDN.pn<gr Pq _ xqy ) gf25 : odef L (& p) gf25 = subst (λ k → odef L k ) (sym peq) ( NEG (subst (λ k → odef L k) (sym &iso) (PDN.x∈PP Pp) )) gf27 : {x : Ordinal} → odef p x → odef (P ＼ * xp) x gf27 {x} px = subst (λ k → odef k x) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) peq)) px -- gf02 : {P a b : HOD } → (P ＼ a) ∩ (P ＼ b) ≡ ( P ＼ (a ∪ b) ) gf23 : & (p ∩ q) ≡ & (P ＼ * (& (* xp ∪ * xq))) gf23 = cong (&) (gf121 gp gq ) ... | tri> ¬a ¬b c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10 ; x=ψz = cong (&) (gf121 gp gq ) } where gp = record { z = xp ; az = Pp ; x=ψz = peq } gq = record { z = xq ; az = Pq ; x=ψz = qeq } gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq)) gf10 = record { gr = PDN.gr Pp ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where gf16 : gr Pq ≤ gr Pp gf16 = <to≤ c gf15 : (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy ... | case1 xpy = PDN.pn<gr Pp _ xpy ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y xqy ) gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P ＼ x)) (_＼_ P) ≡ PDHOD L p0 C gf00 = ==→o≡ record { eq→ = gf20 ; eq← = gf22 } where gf20 : {x : Ordinal} → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x → PDN L p0 C x gf20 {x} record { z = z₁ ; az = record { z = z ; az = az ; x=ψz = x=ψz₁ } ; x=ψz = x=ψz } = subst (λ k → PDN L p0 C k ) (begin z ≡⟨ sym &iso ⟩ & (* z) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ & (P ＼ ( P ＼ (* z) )) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ & (P ＼ (* ( & (P ＼ (* z ))))) ≡⟨ cong (λ k → & (P ＼ (* k))) (sym x=ψz₁) ⟩ & (P ＼ (* z₁)) ≡⟨ sym x=ψz ⟩ x ∎ ) az where open ≡-Reasoning gf21 : {x : Ordinal } → odef (* z) x → odef P x gf21 {x} lt = L⊆PP ( PDN.x∈PP az) _ lt gf22 : {x : Ordinal} → PDN L p0 C x → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x gf22 {x} pdx = record { z = _ ; az = record { z = _ ; az = pdx ; x=ψz = refl } ; x=ψz = ( begin x ≡⟨ sym &iso ⟩ & (* x) ≡⟨ cong (&) (sym (L＼Lx=x gf21 )) ⟩ & (P ＼ (P ＼ * x)) ≡⟨ cong (λ k → & ( P ＼ k)) (sym *iso) ⟩ & (P ＼ * (& (P ＼ * x))) ∎ ) } where open ≡-Reasoning gf21 : {z : Ordinal } → odef (* x) z → odef P z gf21 {z} lt = L⊆PP ( PDN.x∈PP pdx ) z lt fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where open Dense fd09 : (i : ℕ ) → odef L (find-p L C i (& p0)) fd09 zero = Lp0 fd09 (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) ... | yes _ = fd09 i ... | no not = fd17 where fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) fd17 = proj1 fd18 an : ℕ an = ctl← C (& (dense D)) MD pn : Ordinal pn = find-p L C an (& p0) pn+1 : Ordinal pn+1 = find-p L C (suc an) (& p0) d=an : dense D ≡ * (ctl→ C an) d=an = begin dense D ≡⟨ sym *iso ⟩ * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ * (ctl→ C an) ∎ where open ≡-Reasoning fd07 : odef (dense D) pn+1 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where L∋pn : L ∋ * (find-p L C an (& p0)) L∋pn = subst (λ k → odef L k) (sym &iso) (fd09 an ) L∋df : L ∋ ( dense-f D L∋pn ) L∋df = (d⊆P D) ( dense-d D L∋pn ) pn∋df : (* (ctl→ C an)) ∋ ( dense-f D L∋pn ) pn∋df = subst (λ k → odef k (& ( dense-f D L∋pn ) )) d=an ( dense-d D L∋pn ) pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (dense-f D L∋pn))) y pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (dense-p D L∋pn py) fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (dense-f D L∋pn)) fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ fd10 = ≡o∅→=od∅ y ... | no not = fd27 where fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) fd27 : odef (dense D) (& fd29) fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) fd03 : odef (PDHOD L p0 C) pn+1 fd03 = record { gr = suc an ; pn<gr = λ y lt → lt ; x∈PP = fd09 (suc an)} fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ open GenericFilter open Filter record NotCompatible (L p : HOD ) (L∋a : L ∋ p ) : Set (Level.suc (Level.suc n)) where field q r : HOD Lq : L ∋ q Lr : L ∋ r p⊆q : p ⊆ q p⊆r : p ⊆ r ¬compat : (s : HOD) → ¬ ( (q ⊆ s) ∧ (r ⊆ s) ) lemma232 : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) -- L is a Boolean Algebra → (UNI : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q )) → (NEG : ({p : HOD} → L ∋ p → L ∋ ( P ＼ p))) → (C : CountableModel ) → ( MP : ctl-M C ∋ P ) → ( {p : HOD} → (Lp : L ∋ p ) → NotCompatible L p Lp ) → ¬ ( ctl-M C ∋ rgen ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C )) lemma232 P L p0 LPP Lp0 CAP UNI NEG C MP NC M∋gf = ¬gf∩D=0 record { eq→ = λ {x} gf∩D → ⊥-elim( proj2 (proj2 gf∩D) (proj1 gf∩D)) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where gf = rgen ( P-GenericFilter P L p0 LPP Lp0 CAP UNI NEG C ) M = ctl-M C D : HOD D = L ＼ gf M∋D : M ∋ D M∋D = subst (λ k → odef M k) ? (ctl<M C ?) D⊆PP : D ⊆ Power P D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx DD : Dense {L} {P} LPP Dense.dense DD = D Dense.d⊆P DD ⟪ Lx , _ ⟫ = Lx Dense.dense-f DD Lp = ? where ll00 : HOD ll00 with NotCompatible.¬compat (NC Lp) ... | nc = ? where ll01 : {q r : HOD } → (s : HOD) → ¬ ( (q ⊆ s) ∧ (r ⊆ s)) → (¬ (gf ∋ q)) ∨ (¬ (gf ∋ q)) ll01 = ? Dense.dense-d DD = ? Dense.dense-p DD = ? ¬gf∩D=0 : ¬ ( (gf ∩ D) =h= od∅ ) ¬gf∩D=0 = ? -- -- 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 (Level.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 = {!!} }