# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1678960907 -32400
# Node ID 0b7e4eb68afcbe0a7c99a921a4f96b487a4fc76b
# Parent  afecaee48825007cb581a979e279952ecc1ff99a
change to Ideal

diff -r afecaee48825 -r 0b7e4eb68afc src/filter.agda
--- a/src/filter.agda	Thu Mar 16 17:46:36 2023 +0900
+++ b/src/filter.agda	Thu Mar 16 19:01:47 2023 +0900
@@ -156,7 +156,7 @@
        ideal : HOD   
        i⊆L :  ideal ⊆ L 
        ideal1 : { p q : HOD } →  L ∋ q  → ideal ∋ p →  q ⊆ p  → ideal ∋ q
-       ideal2 : { p q : HOD } → ideal ∋ p →  ideal ∋ q  → L ∋ (p ∩ q) → ideal ∋ (p ∪ q)
+       ideal2 : { p q : HOD } → ideal ∋ p →  ideal ∋ q  → L ∋ (p ∪ q) → ideal ∋ (p ∪ q)
 
 open Ideal
 
diff -r afecaee48825 -r 0b7e4eb68afc src/generic-filter.agda
--- a/src/generic-filter.agda	Thu Mar 16 17:46:36 2023 +0900
+++ b/src/generic-filter.agda	Thu Mar 16 19:01:47 2023 +0900
@@ -179,265 +179,106 @@
        d⊆P :  dense ⊆ L
        has-expansion : {p : HOD} → (Lp : L ∋ p) → Expansion L dense Lp
 
-record GenericFilter1 {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where
+record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where
     field
        genf : Ideal {L} {P} LP
        generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Ideal.ideal genf ) ≡ od∅ )
 
-P-GenericFilter1 : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0
-      → (C : CountableModel ) → GenericFilter1 {L} {P} LP ( ctl-M C )
-P-GenericFilter1 P L p0 L⊆PP Lp0 C = record {
-      genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; ideal2 =  ? }
-    ; generic = ?
-   } where
-       ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q
-       ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = 
-                 record { gr = gr ; pn<gr = λ y qy → pn<gr y ? ; x∈PP = ? }  where
-            gf00 : {y : Ordinal } →  odef (* (& q)) y → odef (* (& q)) y  
-            gf00 {y} qy = subst (λ k → odef k y ) ? (q⊆p (subst (λ k → odef k y) ? qy ))
-
-record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where
-    field
-       genf : Filter {L} {P} LP
-    rgen : HOD
-    rgen = Replace (Filter.filter genf) (λ x → P ＼ x )
-    field
-       generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ rgen ) ≡ od∅ )
-       gideal1 : {p q : HOD} → rgen ∋ p → q ⊆ p  → L ∋ ( P ＼ q) → rgen ∋ q
-       gideal2 : {p q : HOD} → (rgen ∋ p ) ∧ (rgen ∋ q) → rgen ∋ (p ∪ q)
-
 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 =  gf01 ; filter1 = f1 ; filter2 = f2 }
-    ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd )
-    ; gideal1 = gideal1
-    ; gideal2 = gideal2
+      genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; ideal2 =  ? }
+    ; generic = fdense
    } where
-    GP =  Replace (PDHOD L p0 C) (λ x → P ＼ x)
-    GPR = Replace GP (_＼_ P) 
-    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 = ? } ; 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)) ? } 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 = ? } ; 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 = ?
-          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)) ?  } 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
-       open Expansion
-       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 )
-          exp = has-expansion D L∋pn
-          L∋df : L ∋ ( expansion exp )
-          L∋df = (d⊆P D) (dense∋exp exp)
-          pn∋df : (* (ctl→ C an)) ∋ ( expansion exp)
-          pn∋df = subst (λ k → odef k (& ( expansion exp))) d=an (dense∋exp exp )
-          pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (expansion exp))) y
-          pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp exp py)
-          fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (expansion exp))
-          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 ⟫  
-    gpx→⊆P : {p : Ordinal } → odef GP p → (* p) ⊆ P
-    gpx→⊆P {p} record { z = z ; az = az ; x=ψz = x=ψz } {x} px with subst (λ k → odef k x ) 
-       (trans (cong (*) x=ψz) *iso) px
-    ... | ⟪ Px , npz ⟫ = Px
-    L∋gpr : {p : HOD } → GPR ∋ p → (L ∋ p) ∧ ( L ∋ (P ＼ p))
-    L∋gpr {p} record { z = zp ; az = record { z = z ; az = az ; x=ψz = x=ψzp } ; x=ψz = x=ψz } 
-      = ⟪ subst (λ k → odef L k) fd40 (PDN.x∈PP az) , ? ⟫ where
-        fd41 : * z ⊆ P
-        fd41 {x} lt = L⊆PP ( PDN.x∈PP az ) _ lt
-        fd40 : z ≡ & p
-        fd40 = begin
-           z ≡⟨ sym &iso ⟩ 
-           & (* z) ≡⟨ cong (&) (sym (L＼Lx=x fd41 )) ⟩ 
-           & (P ＼ ( P ＼ * z ) )  ≡⟨ cong (λ k →  & (P ＼ k)) (sym *iso) ⟩ 
-           & (P ＼ * (& ( P ＼ * z )))  ≡⟨ cong (λ k → & (P ＼  * k )) (sym x=ψzp)  ⟩ 
-           & (P ＼ * zp)  ≡⟨ sym x=ψz ⟩ 
-           & p  ∎ where open ≡-Reasoning
-    gpr→gp : {p : HOD} → GPR ∋ p → GP ∋ (P ＼ p ) 
-    gpr→gp {p} record { z = zp ; az = azp ; x=ψz = x=ψzp } = gfp where
-        open ≡-Reasoning
-        gfp : GP ∋ (P ＼ p ) 
-        gfp = subst (λ k → odef GP k) (begin
-           zp ≡⟨ sym &iso ⟩
-           & (* zp) ≡⟨ cong (&) (sym (L＼Lx=x (gpx→⊆P azp) )) ⟩
-           & (P ＼ (P ＼ (* zp) )) ≡⟨ cong (λ k → & ( P ＼ k)) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym x=ψzp)))  ⟩
-           & (P ＼ p) ∎ ) azp
-    gideal1 : {p q : HOD} → GPR ∋ p → q ⊆ p → L ∋  ( P ＼ q) → GPR ∋ q
-    gideal1 {p} {q} record { z = np ; az = record { z = z ; az = pdz ; x=ψz = x=ψznp } ; x=ψz = x=ψz } q⊆p Lpq 
-      = record { z = _ ; az = gf30 ; x=ψz = cong (&) fd42 } where
-        gp =  record { z = np ; az = record { z = z ; az = pdz ; x=ψz = x=ψznp } ; x=ψz = x=ψz } 
-        open ≡-Reasoning
-        fd41 : * z ⊆ P
-        fd41 {x} lt = L⊆PP ( PDN.x∈PP pdz ) _ lt
-        p=*z : p ≡ * z
-        p=*z = trans (sym *iso) ( cong (*) (sym ( begin
-           z ≡⟨ sym &iso ⟩ 
-           & (* z) ≡⟨ cong (&) (sym (L＼Lx=x fd41 )) ⟩ 
-           & (P ＼ ( P ＼ * z ) )  ≡⟨ cong (λ k →  & (P ＼ k)) (sym *iso) ⟩ 
-           & (P ＼ * (& ( P ＼ * z )))  ≡⟨ cong (λ k → & (P ＼  * k )) (sym x=ψznp)  ⟩ 
-           & (P ＼ * np)  ≡⟨ sym x=ψz ⟩ 
-           & p ∎ ))) 
-        q⊆P : q ⊆ P
-        q⊆P {x} lt = L⊆PP ( PDN.x∈PP pdz ) _ (subst (λ k → odef k x) p=*z (q⊆p lt))
-        fd42 : q ≡  P ＼ * (& (P ＼ q))
-        fd42 = trans (sym (L＼Lx=x  q⊆P )) (cong (λ k → P ＼ k) (sym *iso) )
-        gf32 : (P ＼ p) ⊆ (P ＼ q)
-        gf32 = proj1 (＼-⊆ {P} {q} {p} q⊆P ) q⊆p 
-        gf30 : GP ∋ (P ＼ q )
-        gf30 = f1 Lpq (gpr→gp gp) gf32
-    gideal2 : {p q : HOD} → (GPR ∋ p) ∧ (GPR ∋ q) → Replace GP (_＼_ P) ∋ (p ∪ q)
-    gideal2 {p} {q} ⟪ gp , gq ⟫ 
-       = record { z = _ ; az = gf31 ; x=ψz = cong (&) gf32  } where
-        open ≡-Reasoning
-        gf31 : GP ∋ ( (P ＼ p ) ∩ (P ＼ q ) )
-        gf31 = f2 (gpr→gp gp) (gpr→gp gq) ? -- (CAP (proj2 (L∋gpr gp)) (proj2 (L∋gpr gq))  ) 
-        gf33 : (p ∪ q) ⊆ P
-        gf33 {x} (case1 px) = L⊆PP (proj1 (L∋gpr gp)) _ (subst (λ k → odef k x) (sym *iso) px )
-        gf33 {x} (case2 qx) = L⊆PP (proj1 (L∋gpr gq)) _ (subst (λ k → odef k x) (sym *iso) qx )
-        gf32 :  (p ∪ q) ≡ (P ＼ * (& ((P ＼ p) ∩ (P ＼ q))))
-        gf32 = begin
-          p ∪ q ≡⟨ sym ( L＼Lx=x gf33 ) ⟩ 
-          P ＼ (P ＼ (p ∪ q)) ≡⟨ cong (λ k → P ＼ k) (sym (gf02 {P} {p}{q} ) ) ⟩ 
-          P ＼ ((P ＼ p) ∩ (P ＼ q)) ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩ 
-          P ＼ * (& ((P ＼ p) ∩ (P ＼ q))) ∎
+       ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q
+       ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p =  
+                 record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq }  where
+            gf00 : {y : Ordinal } →  odef (* (& q)) y → odef (* (& p)) y  
+            gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy ))
+       ideal2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∪ q) → PDHOD L p0 C ∋ (p ∪ q)
+       ideal2 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } 
+                      record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } Lpq = gf01 where
+            Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } 
+            Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } 
+            gf01 : PDHOD L p0 C ∋ (p ∪ q)
+            gf01 with <-cmp pgr qgr
+            ... | tri< a ¬b ¬c = record { gr = qgr ; pn<gr = gf03 ; x∈PP = Lpq } where
+               gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C qgr (& p0))) y
+               gf03 y pqy = gf15 y pqy where
+                 gf16 : gr Pp ≤ gr Pq
+                 gf16 = <to≤ a
+                 gf15 :  (y : Ordinal) → odef (* (& (p ∪ q))) 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 (subst (λ k → odef k y) (sym *iso) xpy) )
+                 ... | case2 xqy = PDN.pn<gr Pq _ (subst (λ k → odef k y) (sym *iso) xqy)
+            ... | tri≈ ¬a refl ¬c = record { gr = qgr ; pn<gr = gf03 ; x∈PP = Lpq } where
+               gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C qgr (& p0))) y
+               gf03 y pqy = gf15 y pqy where
+                 gf16 : gr Pp ≤ gr Pq
+                 gf16 = ≤-refl
+                 gf15 :  (y : Ordinal) → odef (* (& (p ∪ q))) 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 (subst (λ k → odef k y) (sym *iso) xpy) )
+                 ... | case2 xqy = PDN.pn<gr Pq _ (subst (λ k → odef k y) (sym *iso) xqy)
+            ... | tri> ¬a ¬b c = record { gr = pgr ; pn<gr = gf03 ; x∈PP = Lpq } where
+               gf03 : (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C pgr (& p0))) y
+               gf03 y ppy = gf15 y ppy where
+                 gf16 : gr Pq ≤ gr Pp
+                 gf16 = <to≤ c
+                 gf15 :  (y : Ordinal) → odef (* (& (p ∪ q))) y → odef (* (find-p L C (gr Pp) (& p0))) y
+                 gf15 y gppy with subst (λ k → odef k y ) *iso gppy
+                 ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y (subst (λ k → odef k y) (sym *iso) xqy) )
+                 ... | case1 xpy = PDN.pn<gr Pp _ (subst (λ k → odef k y) (sym *iso) xpy)
+       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
+           open Expansion
+           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 )
+              exp = has-expansion D L∋pn
+              L∋df : L ∋ ( expansion exp )
+              L∋df = (d⊆P D) (dense∋exp exp)
+              pn∋df : (* (ctl→ C an)) ∋ ( expansion exp)
+              pn∋df = subst (λ k → odef k (& ( expansion exp))) d=an (dense∋exp exp )
+              pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (expansion exp))) y
+              pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp exp py)
+              fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (expansion exp))
+              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
@@ -455,12 +296,12 @@
       → (C : CountableModel ) 
       → ctl-M C ∋ L
       → ( {p : HOD} → (Lp : L ∋ p ) → NotCompatible L p Lp )
-      →  ¬ ( ctl-M C ∋  rgen ( P-GenericFilter P L p0 LPP Lp0 C ))
+      →  ¬ ( ctl-M C ∋  Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C )))
 lemma232 P L p0 LPP Lp0 C ML NC MF = ¬rgf∩D=0 record { eq→ = λ {x} rgf∩D → ⊥-elim( proj2 (proj1 rgf∩D) (proj2 rgf∩D)) 
         ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where
     PG = P-GenericFilter P L p0 LPP Lp0 C 
     GF =  genf PG
-    rgf =  rgen PG
+    rgf =  Ideal.ideal (genf PG)
     M = ctl-M C
     D : HOD  
     D = L ＼ rgf
@@ -493,9 +334,9 @@
                 ll05 : ¬ ( (q ⊆ (q ∪ r) ∧ (r ⊆ (q ∪ r)) ))
                 ll05 = NotCompatible.¬compat (NC Lp )  (q ∪ r) ?
                 ll03 : rgf ∋ p → rgf ∋ q → rgf ∋ (p ∪ q)
-                ll03 rp rq = gideal2 PG ⟪ rp , rq ⟫ 
+                ll03 rp rq = ? -- Ideal.ideal2 GF ⟪ rp , rq ⟫ 
                 ll04 : rgf ∋ p → q ⊆ p → rgf ∋ q
-                ll04 rp q⊆p = gideal1 PG rp q⊆p ?
+                ll04 rp q⊆p = ? -- Ideal.ideal1 GF rp q⊆p ?
     ¬rgf∩D=0 : ¬ ( (D ∩ rgf ) =h= od∅ )
     ¬rgf∩D=0 eq =  generic PG DD M∋D (==→o≡ eq)
 
@@ -526,5 +367,5 @@
     →  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 = {!!} }
+  ind x valy = record { od = record { def = λ y → valS x y (& (Ideal.ideal (genf G))) } ; odmax = {!!} ; <odmax = {!!} }