Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1244:a7dfcbbd07ff
f1 f2 done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 14 Mar 2023 09:50:23 +0900 |
| parents | 50fcf7f047d1 |
| children | 11049e3168ad |
| files | src/generic-filter.agda |
| diffstat | 1 files changed, 89 insertions(+), 88 deletions(-) [+] |
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--- a/src/generic-filter.agda Tue Mar 14 06:19:42 2023 +0900 +++ b/src/generic-filter.agda Tue Mar 14 09:50:23 2023 +0900 @@ -1,22 +1,22 @@ {-# OPTIONS --allow-unsolved-metas #-} -import Level +import Level open import Ordinals module generic-filter {n : Level.Level } (O : Ordinals {n}) where -import filter +import filter open import zf open import logic -- open import partfunc {n} O -import OD +import OD -open import Relation.Nullary -open import Relation.Binary -open import Data.Empty +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 import Data.Nat +import BAlgebra open BAlgebra O @@ -50,7 +50,7 @@ -- open import Data.List hiding (filter) -open import Data.Maybe +open import Data.Maybe open import ZProduct O @@ -58,19 +58,19 @@ field ctl-M : HOD ctl→ : ℕ → Ordinal - ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) + 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 + 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, +-- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, -open CountableModel +open CountableModel ---- -- a(n) ∈ M @@ -79,11 +79,11 @@ 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) } + ; 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)) ) @@ -96,7 +96,7 @@ 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 + pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y x∈PP : odef L x open PDN @@ -106,7 +106,7 @@ -- 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) } + ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } open PDN @@ -125,7 +125,7 @@ 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} (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) ) @@ -133,7 +133,7 @@ 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) ) ⟫ ⟫ + gf04 {x} ⟪ Px , abx ⟫ = ⟪ ⟪ Px , (λ ax → abx (case1 ax) ) ⟫ , ⟪ Px , (λ bx → abx (case2 bx) ) ⟫ ⟫ open import Data.Nat.Properties open import nat @@ -151,7 +151,7 @@ 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 ¬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 ) @@ -161,7 +161,7 @@ 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 + dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where field @@ -172,17 +172,18 @@ -- ＼-⊆ -P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 - → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) +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 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 NEG C = record { - genf = record { filter = Replace (PDHOD L p0 C) (λ x → P ＼ x) ; f⊆L = gf01 ; filter1 = ? ; filter2 = ? } +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 + 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 @@ -195,7 +196,7 @@ * (& (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 + P ＼ * (& (* xp ∪ * xq)) ∎ where open ≡-Reasoning xp = Replaced.z gp xq = Replaced.z gq @@ -203,89 +204,89 @@ 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 → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q - f1 {p} {q} L∋q PD∋p p⊆q = ? + 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) + 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 } + 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 + 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 - gf22 : odef L (& (* xp ∩ * xq)) - gf22 = CAP (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 + ... | 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 - ... | ⟪ xpy , xqy ⟫ = PDN.pn<gr Pp _ xpy + ... | 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 - gf23 : & (p ∩ q) ≡ & (P ＼ * (& (* xp ∩ * xq))) -- != P ＼ (xp ∪ xq) - gf23 = cong (&) ( ==→o≡ record { eq→ = gf24 ; eq← = gf30 } ) where - gf24 : {x : Ordinal} → odef (p ∩ q) x → odef (P ＼ * (& (* xp ∩ * xq))) x - gf24 {x} ⟪ px , qx ⟫ = subst (λ k → odef (P ＼ k) x) (sym *iso) ⟪ L⊆PP gf25 _ (subst (λ k → odef k x) (sym *iso) px) , gf26 ⟫ where - gf26 : ¬ odef (* xp ∩ * xq) x - gf26 npx = proj2 (gf27 px) (proj1 npx) - gf30 : {x : Ordinal} → odef (P ＼ * (& (* xp ∩ * xq))) x → odef (p ∩ q) x - gf30 {x} pxp with subst (λ k → odef (P ＼ k) x) *iso pxp - ... | ⟪ Px , ¬xpqx ⟫ = ⟪ ? , gf28 ⟪ Px , (λ xqx → ¬xpqx ⟪ ? , xqx ⟫ ) ⟫ ⟫ where - gf28 : {x : Ordinal} → odef (P ＼ * xq) x → odef q x - gf28 {x} qx = subst (λ k → odef k x) (sym (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) qeq))) qx - pn : HOD - pn = * (find-p L C (gr Pp) (& p0)) - qn : HOD - qn = * (find-p L C (gr Pq) (& p0)) - gf20 : pn ≡ qn - gf20 = cong ( λ k → * (find-p L C k (& p0))) eq - gf19 : * xp ⊆ pn - gf19 = PDN.pn<gr Pp _ - gf18 : PDN L p0 C xp → PDN L p0 C xq → Replaced (PDHOD L p0 C) (λ z → & (P ＼ * z)) (& (p ∩ q)) - gf18 record { gr = gr₁ ; pn<gr = pn<gr₁ ; x∈PP = x∈PP₁ } record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } = ? - -- record { z = xp ; az = Pp ; x=ψz = trans (cong (&) gf17) peq } where - gf17 : p ∩ q ≡ p - gf17 = ==→o≡ record { eq→ = proj1 ; eq← = λ {y} px → ⟪ px , ? ⟫ } - f4 : (y : Ordinal) → odef (* (find-p L C (gr Pp ) (& p0))) y → odef (p ∩ q) y - f4 y lt = ⟪ subst (λ k → odef k y) *iso ? , subst (λ k → odef k y) *iso (pn<gr ? ? lt) ⟫ - ... | 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 } + -- 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 + 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 } = + 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 + 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 + 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 + 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 @@ -302,10 +303,10 @@ 0<b : ¬ o∅ ≡ & b b<a : b ⊆ a -lemma232 : (P L p : HOD ) (C : CountableModel ) +lemma232 : (P L p : HOD ) (C : CountableModel ) → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) ( NEG : {p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) - → ¬ ( (ctl-M C) ∋ rgen ( P-GenericFilter P L p LP Lp0 ? NEG C )) + → ¬ ( (ctl-M C) ∋ rgen ( P-GenericFilter P L p LP Lp0 ? ? NEG C )) lemma232 P L p C LP Lp0 NEG NA MG = {!!} where D : HOD -- P - G D = ? @@ -329,7 +330,7 @@ record valS (ox oy oG : Ordinal) : Set n where field op : Ordinal - p∈G : odef (* oG) op + p∈G : odef (* oG) op is-val : odef (* ox) ( & < * oy , * op > ) val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P}