# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1678681942 -32400
# Node ID e843ace90b313767fa8c41b20b51ddd02cf7238b
# Parent  5f1572d1f19afed735bc95ab8a1274aca86f91ef
...

diff -r 5f1572d1f19a -r e843ace90b31 src/generic-filter.agda
--- a/src/generic-filter.agda	Mon Mar 13 01:30:55 2023 +0900
+++ b/src/generic-filter.agda	Mon Mar 13 13:32:22 2023 +0900
@@ -1,7 +1,7 @@
 {-# OPTIONS --allow-unsolved-metas #-}
-open import Level
+import Level 
 open import Ordinals
-module generic-filter {n : Level } (O : Ordinals {n})   where
+module generic-filter {n : Level.Level } (O : Ordinals {n})   where
 
 import filter 
 open import zf
@@ -15,7 +15,7 @@
 open import Relation.Binary
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality
-open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import Data.Nat 
 import BAlgebra 
 
 open BAlgebra O
@@ -54,15 +54,15 @@
 
 open import ZProduct O
 
-record CountableModel : Set (suc (suc n)) where
+record CountableModel : Set (Level.suc (Level.suc n)) where
    field
        ctl-M : HOD
-       ctl→ : Nat → Ordinal
-       ctl<M : (x : Nat) → odef (ctl-M) (ctl→ x) 
-       ctl← : (x : Ordinal )→  odef (ctl-M ) x → Nat
+       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 : Nat }  →  ctl← (ctl→ x ) (ctl<M x)  ≡ x
+       -- 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
@@ -76,7 +76,7 @@
 --   a(n) ∈ M
 --   ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q
 --
-PGHOD :  (i : Nat) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD
+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) }  
@@ -84,9 +84,9 @@
 ---
 --   p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n)
 --  
-find-p :  (L : HOD ) (C : CountableModel )  (i : Nat) → (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)) )
+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
 
@@ -95,7 +95,7 @@
 --
 record PDN  (L p : HOD ) (C : CountableModel )  (x : Ordinal) : Set n where
    field
-       gr : Nat
+       gr : ℕ
        pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y 
        x∈PP  : odef L x
 
@@ -111,7 +111,7 @@
 open PDN
 
 ----
---  Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → P → Set )
+--  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 ) )
@@ -124,27 +124,38 @@
 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 : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p)))
+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 = ? where -- λ  lt →   proj2 (proj2 fmin∈PGHOD) _ ?   where
+... | 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-⊆ {* (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 = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt ) 
+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 (suc n) where
+record Dense  {L P : HOD } (LP : L ⊆ Power P)  : Set (Level.suc n) where
    field
        dense : HOD
        d⊆P :  dense ⊆ L
@@ -152,7 +163,7 @@
        dense-d :  { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt
        dense-p :  { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p  
 
-record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where
+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∅ )
@@ -172,16 +183,77 @@
         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 → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q
         f1 {p} {q} L∋q PD∋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 = ?
-        ... | tri≈ ¬a eq ¬c = ?
-        ... | tri> ¬a ¬b c = ? 
+        ... | 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 refl ¬c =  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 , ? ⟫   }
+        ... | 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 = ?
+        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  = ? where
            open Dense
@@ -189,7 +261,7 @@
 open GenericFilter
 open Filter
 
-record NonAtomic  (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where
+record NonAtomic  (L a : HOD ) (L∋a : L ∋ a ) : Set (Level.suc (Level.suc n)) where
    field
       b : HOD
       0<b : ¬ o∅ ≡ & b
@@ -215,7 +287,7 @@
 --   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 {L} {P} 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 (Level.suc n) where
    field
      valx : HOD