--- a/src/generic-filter.agda	Thu Dec 22 15:10:31 2022 +0900
+++ b/src/generic-filter.agda	Fri Dec 23 12:54:05 2022 +0900
@@ -125,36 +125,35 @@
 
 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
+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 : 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 {Zero} {Zero} n≤m = refl-⊆ {* (find-p L C Zero (& p))}
+p-monotonic L p C {Zero} {Suc m} z≤n lt = (p-monotonic L p C {Zero} {m} z≤n ) (p-monotonic1 L p C {m} lt ) 
 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 = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt ) 
+... | tri≈ ¬a refl ¬c = λ x → x
 ... | 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 ( ctl-M C )
+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 = 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 
-        f⊆PL = record { incl = λ {x} lt → x∈PP lt  }
+        f⊆PL lt = x∈PP lt  
         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 )))
+           f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (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 → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q)
         f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p)
@@ -162,7 +161,7 @@
             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))
+               f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) &iso ( (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 = L∋pq } where
             f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y
@@ -171,7 +170,7 @@
             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))
+               f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) &iso ( (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 ) → (ctl-M C ) ∋ Dense.dense D  → ¬ (filter.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
@@ -204,11 +203,11 @@
               fd11 : Ordinal
               fd11 = & ( dense-f D fd12 )
               fd13 : L ∋ ( dense-f D fd12 )
-              fd13 = incl (d⊆P D) (  dense-d D fd12 )
+              fd13 = (d⊆P D) (  dense-d D fd12 )
               fd14 : (* (ctl→ C an)) ∋ ( dense-f D fd12 )
               fd14 = subst (λ k → odef k (& ( dense-f D fd12 ) )) d=an (  dense-d D fd12 ) 
               fd15 :  (y : Ordinal) → odef (* (& (dense-f D fd12))) y → odef (* (find-p L C an (& p0))) y
-              fd15 y lt = subst (λ k → odef  (* (find-p L C an (& p0)))  k ) &iso ( incl (dense-p D  fd12 ) fd16  ) where
+              fd15 y lt = subst (λ k → odef  (* (find-p L C an (& p0)))  k ) &iso ( (dense-p D  fd12 ) fd16  ) where
                   fd16 : odef (dense-f D fd12) (& ( * y))
                   fd16 = subst₂ (λ j k → odef j k ) (*iso) (sym &iso) lt
               fd10 :  PGHOD an L C (find-p L C an (& p0)) =h= od∅
@@ -243,7 +242,7 @@
 --   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 LP (ctl-M C) ) : Set (suc n) where
+record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (suc n) where
    field
      valx : HOD
 
@@ -254,7 +253,7 @@
      is-val : odef (* ox) ( & < * oy , * op >  )
 
 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P}
-    →  (G : GenericFilter LP {!!} )
+    →  (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