# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1678755023 -32400
# Node ID a7dfcbbd07fff1c1dcd39d932453cef548e0396f
# Parent  50fcf7f047d16f161dbc5b5ca9632e2b49795740
f1 f2 done

diff -r 50fcf7f047d1 -r a7dfcbbd07ff src/generic-filter.agda
--- 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}