Mercurial > hg > Members > kono > Proof > ZF-in-agda

view src/ZProduct.agda @ 1279:7e7d8d825632

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
P x Q ⇆ Q x P done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 06 Apr 2023 09:16:52 +0900
parents 2cbe0db250da
children 45cd80181a29
line wrap: on
line source

{-# OPTIONS --allow-unsolved-metas #-}

open import Level
open import Ordinals
module ZProduct {n : Level } (O : Ordinals {n})   where

open import zf
open import logic
import OD
import ODUtil
import OrdUtil

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 renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ )

open OD O
open OD.OD
open OD.HOD
open ODAxiom odAxiom

open Ordinals.Ordinals  O
open Ordinals.IsOrdinals isOrdinal
open Ordinals.IsNext isNext
open OrdUtil O
open ODUtil O

open _∧_
open _∨_
open Bool

open _==_

<_,_> : (x y : HOD) → HOD
< x , y > = (x , x ) , (x , y )

exg-pair : { x y : HOD } → (x , y ) =h= ( y , x )
exg-pair {x} {y} = record { eq→ = left ; eq← = right } where
    left : {z : Ordinal} → odef (x , y) z → odef (y , x) z
    left (case1 t) = case2 t
    left (case2 t) = case1 t
    right : {z : Ordinal} → odef (y , x) z → odef (x , y) z
    right (case1 t) = case2 t
    right (case2 t) = case1 t

ord≡→≡ : { x y : HOD } → & x ≡ & y → x ≡ y
ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong ( λ k → * k ) eq )

od≡→≡ : { x y : Ordinal } → * x ≡ * y → x ≡ y
od≡→≡ eq = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong ( λ k → & k ) eq )

eq-prod : { x x' y y' : HOD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
eq-prod refl refl = refl

xx=zy→x=y : {x y z : HOD } → ( x , x ) =h= ( z , y ) → x ≡ y
xx=zy→x=y {x} {y} eq with trio< (& x) (& y)
xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c with eq← eq {& y} (case2 refl)
xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a )
xx=zy→x=y {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a )
xx=zy→x=y {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b
xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c  with eq← eq {& y} (case2 refl)
xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c )
xx=zy→x=y {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c )

prod-eq : { x x' y y' : HOD } → < x , y > =h= < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
prod-eq {x} {x'} {y} {y'} eq = ⟪ lemmax , lemmay ⟫ where
    lemma2 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → z ≡ y
    lemma2 {x} {y} {z} eq = trans (sym (xx=zy→x=y lemma3 )) ( xx=zy→x=y eq )  where
        lemma3 : ( x , x ) =h= ( y , z )
        lemma3 = ==-trans eq exg-pair
    lemma1 : {x y : HOD } → ( x , x ) =h= ( y , y ) → x ≡ y
    lemma1 {x} {y} eq with eq← eq {& y} (case2 refl)
    lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s)
    lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s)
    lemma4 : {x y z : HOD } → ( x , y ) =h= ( x , z ) → y ≡ z
    lemma4 {x} {y} {z} eq with eq← eq {& z} (case2 refl)
    lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z
    ... | refl with lemma2 (==-sym eq )
    ... | refl = refl
    lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z
    lemmax : x ≡ x'
    lemmax with eq→ eq {& (x , x)} (case1 refl)
    lemmax | case1 s = lemma1 (ord→== s )  -- (x,x)≡(x',x')
    lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y'
    ... | refl = lemma1 (ord→== s )
    lemmay : y ≡ y'
    lemmay with lemmax
    ... | refl with lemma4 eq -- with (x,y)≡(x,y')
    ... | eq1 = lemma4 (ord→== (cong (λ  k → & k )  eq1 ))

prod-≡ : { x x' y y' : HOD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
prod-≡ eq = prod-eq (ord→== (cong (&) eq ))

--
-- unlike ordered pair, ZFPair is not a HOD

data ord-pair : (p : Ordinal) → Set n where
   pair : (x y : Ordinal ) → ord-pair ( & ( < * x , * y > ) )

ZFPair : OD
ZFPair = record { def = λ x → ord-pair x }

_⊗_ : (A B : HOD) → HOD
A ⊗ B  = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) ))

product→ : {A B a b : HOD} → A ∋ a → B ∋ b  → ( A ⊗ B ) ∋ < a , b >
product→ {A} {B} {a} {b} A∋a B∋b = record { owner = _ ; ao = lemma1 ; ox = subst (λ k → odef k _) (sym *iso) lemma2  } where
    lemma1 :  odef (Replace B (λ b₁ → Replace A (λ a₁ → < a₁ , b₁ >))) (& (Replace A (λ a₁ → < a₁ , b >)))
    lemma1 = replacement← B b B∋b
    lemma2 : odef (Replace A (λ a₁ → < a₁ , b >)) (& < a , b >)
    lemma2 = replacement← A a A∋a

data ZFProduct  (A B : HOD) : (p : Ordinal) → Set n where
    ab-pair : {a b : Ordinal } → odef A a → odef B b → ZFProduct A B ( & ( < * a , * b > ) )

ZFP  : (A B : HOD) → HOD
ZFP  A B = record { od = record { def = λ x → ZFProduct A B x  }
        ; odmax = odmax ( A ⊗ B ) ; <odmax = λ {y} px → <odmax ( A ⊗ B ) (lemma0 px) }
   where
        lemma0 :  {A B : HOD} {x : Ordinal} → ZFProduct A B x → odef (A ⊗ B) x
        lemma0 {A} {B} {px} ( ab-pair {a} {b} ax by ) = product→ (d→∋ A ax) (d→∋ B by)

ZFP→ : {A B a b : HOD} → A ∋ a → B ∋ b  → ZFP A B ∋ < a , b >
ZFP→ {A} {B} {a} {b} aa bb = subst (λ k → ZFProduct A B k ) (cong₂ (λ j k → & < j , k >) *iso *iso ) ( ab-pair aa bb )

zπ1 : {A B : HOD} → {x : Ordinal } → odef (ZFP A B) x → Ordinal
zπ1 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb) = a

zp1 : {A B : HOD} → {x : Ordinal } → (zx : odef (ZFP A B) x) → odef A (zπ1 zx)
zp1 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb ) = aa

zπ2 : {A B : HOD} → {x : Ordinal } → odef (ZFP A B) x → Ordinal
zπ2 (ab-pair {a} {b} aa bb) = b

zp2 : {A B : HOD} → {x : Ordinal } → (zx : odef (ZFP A B) x) → odef B (zπ2 zx)
zp2 {A} {B} {.(& < * _ , * _ >)} (ab-pair {a} {b} aa bb ) = bb

zp-iso :  { A B : HOD } → {x : Ordinal } → (p : odef (ZFP A B) x ) → & < * (zπ1 p) , * (zπ2 p) > ≡ x
zp-iso {A} {B} {_} (ab-pair {a} {b} aa bb)  = refl

zp-iso1 :  { A B : HOD } → {a b : Ordinal } → (p : odef (ZFP A B) (& < * a , * b > )) → (* (zπ1 p) ≡ (* a)) ∧ (* (zπ2 p) ≡ (* b))
zp-iso1 {A} {B} {a} {b} pab = prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) zz11) ) where
      zz11 : & < * (zπ1 pab) , * (zπ2 pab) > ≡ & < * a , * b >
      zz11 = zp-iso pab

zp-iso0 :  { A B : HOD } → {a b : Ordinal } → (p : odef (ZFP A B) (& < * a , * b > )) → (zπ1 p ≡ a) ∧ (zπ2 p ≡ b)
zp-iso0 {A} {B} {a} {b} pab = ⟪ subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj1 (zp-iso1 pab) ))
                              , subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj2 (zp-iso1 pab) ) )  ⟫

ZFP⊆⊗ :  {A B : HOD} {x : Ordinal} → odef (ZFP A B) x → odef (A ⊗ B) x
ZFP⊆⊗ {A} {B} {px} ( ab-pair {a} {b} ax by ) = product→ (d→∋ A ax) (d→∋ B by)

⊗⊆ZFP : {A B x : HOD} → ( A ⊗ B ) ∋ x → odef (ZFP A B) (& x)
⊗⊆ZFP {A} {B} {x} record { owner = owner ; ao = record { z = a ; az = ba ; x=ψz = x=ψa } ; ox = ox } = zfp01 where
       zfp02 : Replace A (λ z → < z , * a >) ≡ * owner
       zfp02 = subst₂ ( λ j k → j ≡ k ) *iso refl (sym (cong (*) x=ψa ))
       zfp01 : odef (ZFP A B) (& x)
       zfp01 with subst (λ k → odef k (& x) ) (sym zfp02) ox
       ... | record { z = b ; az = ab ; x=ψz = x=ψb } = subst (λ k → ZFProduct A B k ) (sym x=ψb) (ab-pair ab ba)

ZPI1 : (A B : HOD) → HOD
ZPI1 A B = Replace' (ZFP A B) ( λ x px → * (zπ1 px ))

ZPI2 : (A B  : HOD) → HOD
ZPI2 A B = Replace' (ZFP A B) ( λ x px → * (zπ2 px ))

ZFProj1-iso : {P Q : HOD} {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ1 p ≡ a
ZFProj1-iso {P} {Q} {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) a=c)

ZFProj2-iso : {P Q : HOD} {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ2 p ≡ b
ZFProj2-iso {P} {Q} {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) b=d)

ZPI1-iso : (A B : HOD) → {b : Ordinal } → odef B b → ZPI1 A B ≡ A
ZPI1-iso P Q {q} qq = ==→o≡ record { eq→ = ty20 ; eq← = ty22 } where
     ty21 : {a b : Ordinal } → (pz : odef P a) → (qz : odef Q b) → ZFProduct P Q (& (* (& < * a , * b >)))
     ty21 pz qz = subst (odef (ZFP P Q)) (sym &iso) (ab-pair pz qz )
     ty32 : {a b : Ordinal } → (pz : odef P a) → (qz : odef Q b) → zπ1 (ty21 pz qz) ≡ a
     ty32 {a} {b} pz qz  = ty33 (ty21 pz qz) (cong (&) *iso) where
         ty33 : {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ1 p ≡ a
         ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
         ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) a=c)
     ty20 : {x : Ordinal} → odef (ZPI1 P Q) x → odef P x
     ty20 {x} record { z = _ ; az = ab-pair {a} {b} pz qz ; x=ψz = x=ψz } = subst (λ k → odef P k) a=x pz  where
         ty24 : * x  ≡ * a
         ty24 = begin
            * x ≡⟨ cong (*) x=ψz ⟩
            _ ≡⟨ *iso  ⟩
            * (zπ1 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair pz qz))) ≡⟨ cong (*) (ty32 pz qz) ⟩
            * a ∎ where open ≡-Reasoning
         a=x : a  ≡ x
         a=x = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (sym ty24))
     ty22 : {x : Ordinal} → odef P x → odef (ZPI1 P Q) x
     ty22 {x} px = record { z = _ ; az = ab-pair px qq ; x=ψz = subst₂ (λ j k → j ≡ k) &iso refl (cong (&) ty12 ) }  where
         ty12 : * x ≡ * (zπ1 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair px qq )))
         ty12 = begin
            * x ≡⟨ sym (cong (*) (ty32 px qq )) ⟩
            * (zπ1 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair px qq ))) ∎ where open ≡-Reasoning

ZPI2-iso : (A B : HOD) → {b : Ordinal } → odef A b → ZPI2 A B ≡ B
ZPI2-iso P Q {p} pp = ==→o≡ record { eq→ = ty20 ; eq← = ty22 } where
     ty21 : {a b : Ordinal } → (pz : odef P a) → (qz : odef Q b) → ZFProduct P Q (& (* (& < * a , * b >)))
     ty21 pz qz = subst (odef (ZFP P Q)) (sym &iso) (ab-pair pz qz )
     ty32 : {a b : Ordinal } → (pz : odef P a) → (qz : odef Q b) → zπ2 (ty21 pz qz) ≡ b
     ty32 {a} {b} pz qz  = ty33 (ty21 pz qz) (cong (&) *iso) where
         ty33 : {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ2 p ≡ b
         ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq))
         ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) b=d)
     ty20 : {x : Ordinal} → odef (ZPI2 P Q) x → odef Q x
     ty20 {x} record { z = _ ; az = ab-pair {a} {b} pz qz ; x=ψz = x=ψz } = subst (λ k → odef Q k) a=x qz  where
         ty24 : * x  ≡ * b
         ty24 = begin
            * x ≡⟨ cong (*) x=ψz ⟩
            _ ≡⟨ *iso  ⟩
            * (zπ2 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair pz qz))) ≡⟨ cong (*) (ty32 pz qz) ⟩
            * b ∎ where open ≡-Reasoning
         a=x : b  ≡ x
         a=x = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (sym ty24))
     ty22 : {x : Ordinal} → odef Q x → odef (ZPI2 P Q) x
     ty22 {x} qx = record { z = _ ; az = ab-pair pp qx  ; x=ψz = subst₂ (λ j k → j ≡ k) &iso refl (cong (&) ty12 ) }  where
         ty12 : * x ≡ * (zπ2 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair pp qx)))
         ty12 = begin
            * x ≡⟨ sym (cong (*) (ty32 pp qx  )) ⟩
            * (zπ2 (subst (odef (ZFP P Q)) (sym &iso) (ab-pair pp qx  ))) ∎ where open ≡-Reasoning

record Func (A B : HOD) : Set n where
    field
       func : {x : Ordinal } → odef A x → Ordinal
       is-func : {x : Ordinal } → (ax : odef A x) → odef B (func ax )

data FuncHOD (A B : HOD) : (x : Ordinal) →  Set n where
     felm :  (F : Func A B) → FuncHOD A B (& ( Replace' A ( λ x ax → < x , (* (Func.func F {& x} ax )) > )))

FuncHOD→F : {A B : HOD} {x : Ordinal} → FuncHOD A B x → Func A B
FuncHOD→F {A} {B} (felm F) = F

FuncHOD=R : {A B : HOD} {x : Ordinal} → (fc : FuncHOD A B x) → (* x) ≡  Replace' A ( λ x ax → < x , (* (Func.func (FuncHOD→F fc) ax)) > )
FuncHOD=R {A} {B}  (felm F) = *iso

--
--  Set of All function from A to B
--

open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )

Funcs : (A B : HOD) → HOD
Funcs A B = record { od = record { def = λ x → FuncHOD A B x } ; odmax = osuc (& (ZFP A B))
       ; <odmax = λ {y} px → subst ( λ k → k o≤ (& (ZFP A B)) ) &iso (⊆→o≤ (lemma1 px)) } where
    lemma1 : {y : Ordinal } → FuncHOD A B y → {x : Ordinal} → odef (* y) x → odef (ZFP A B) x
    lemma1 {y} (felm F) {x} yx with subst (λ k → odef k x) *iso yx
    ... | record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → ZFProduct A B k)
          (sym x=ψz) lemma4 where
       lemma4 : ZFProduct A B (& < * z , * (Func.func F (subst (λ k → odef A k) (sym &iso) az)) > )
       lemma4 = ab-pair az (Func.is-func F (subst (λ k → odef A k) (sym &iso) az))

record Injection (A B : Ordinal ) : Set n where
   field
       i→  : (x : Ordinal ) → odef (* A)  x → Ordinal
       iB  : (x : Ordinal ) → ( lt : odef (* A)  x ) → odef (* B) ( i→ x lt )
       iiso : (x y : Ordinal ) → ( ltx : odef (* A)  x ) ( lty : odef (* A)  y ) → i→ x ltx ≡ i→ y lty → x ≡ y

record HODBijection (A B : HOD ) : Set n where
   field
       fun←  : (x : Ordinal ) → odef A  x → Ordinal
       fun→  : (x : Ordinal ) → odef B  x → Ordinal
       funB  : (x : Ordinal ) → ( lt : odef A  x ) → odef B ( fun← x lt )
       funA  : (x : Ordinal ) → ( lt : odef B  x ) → odef A ( fun→ x lt )
       fiso← : (x : Ordinal ) → ( lt : odef B  x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x
       fiso→ : (x : Ordinal ) → ( lt : odef A  x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x

hodbij-refl : { a b : HOD } → a ≡ b → HODBijection a b
hodbij-refl {a} refl = record {
       fun←  = λ x _ → x
     ; fun→  = λ x _ → x
     ; funB  = λ x lt → lt
     ; funA  = λ x lt → lt
     ; fiso← = λ x lt → refl
     ; fiso→ = λ x lt → refl
    }

pj12 : (A B : HOD) {x : Ordinal} → (ab : odef (ZFP A B) x ) →
   (zπ1  (subst (odef (ZFP A B)) (sym &iso) ab) ≡ & (* (zπ1 ab ))) ∧
   (zπ2  (subst (odef (ZFP A B)) (sym &iso) ab) ≡ & (* (zπ2 ab )))
pj12 A B (ab-pair {x} {y} ax by) = ⟪ subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj1 (prod-≡ pj24 )))
      , subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj2 (prod-≡ pj24)))  ⟫ where
   pj22 : odef (ZFP A B) (& (* (& < * x , * y >)))
   pj22 = subst (odef (ZFP A B)) (sym &iso) (ab-pair ax by)
   pj23 : & < * (zπ1 pj22 ) , * (zπ2 pj22) > ≡ & (* (& < * x , * y >) )
   pj23 = zp-iso pj22
   pj24 : < * (zπ1 (subst (odef (ZFP A B)) (sym &iso) (ab-pair ax by))) , * (zπ2 (subst (odef (ZFP A B)) (sym &iso) (ab-pair ax by))) >
    ≡ < * (& (* x)) ,  * (& (* y)) >
   pj24 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ( trans pj23 (trans &iso
       (sym (cong (&) (cong₂ (λ j k → < j , k >) *iso *iso)) ))))
pj02 : (A B : HOD) (x : Ordinal) → (ab : odef (ZFP A B) x ) → odef (ZPI2 A B) (zπ2 ab)
pj02 A B x ab = record { z = _ ; az = ab ; x=ψz = trans (sym &iso) (trans ( sym (proj2 (pj12 A B ab))) (sym &iso))  }
pj01 : (A B : HOD) (x : Ordinal) → (ab : odef (ZFP A B) x ) → odef (ZPI1 A B) (zπ1 ab)
pj01 A B x ab = record { z = _ ; az = ab ; x=ψz = trans (sym &iso) (trans ( sym (proj1 (pj12 A B ab))) (sym &iso))  }

pj2 :  (A B : HOD) (x : Ordinal) (lt : odef (ZFP A B) x) → odef (ZFP (ZPI2 A B) (ZPI1 A B)) (& < * (zπ2 lt) , * (zπ1 lt) >)
pj2 A B x ab = ab-pair (pj02 A B x ab)  (pj01 A B x ab)

aZPI1 : (A B : HOD) {y : Ordinal} → odef (ZPI1 A B) y → odef A y
aZPI1 A B {y} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef A k) (trans (
    trans (sym &iso) (trans (sym (proj1 (pj12 A B az))) (sym &iso))) (sym x=ψz)  ) ( zp1 az )
aZPI2 : (A B : HOD) {y : Ordinal} → odef (ZPI2 A B) y → odef B y
aZPI2 A B {y} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef B k) (trans (
    trans (sym &iso) (trans (sym (proj2 (pj12 A B az))) (sym &iso))) (sym x=ψz)  ) ( zp2 az )

pj1 :  (A B : HOD) (x : Ordinal) (lt : odef (ZFP (ZPI2 A B) (ZPI1 A B)) x) → odef (ZFP A B) (& < * (zπ2 lt) , * (zπ1 lt) >)
pj1 A B _ (ab-pair ax by) = ab-pair (aZPI1 A B by) (aZPI2 A B ax)

ZFPsym1 : (A B  : HOD) → HODBijection (ZFP A B) (ZFP (ZPI2 A B) (ZPI1 A B))
ZFPsym1 A B = record {
       fun←  = λ xy ab → & < * ( zπ2 ab) , * ( zπ1 ab) >
     ; fun→  = λ xy ab → & < * ( zπ2 ab) , * ( zπ1 ab) >
     ; funB  = pj2 A B
     ; funA  = pj1 A B
     ; fiso← = λ xy ab → pj00 A B ab
     ; fiso→ = λ xy ab → zp-iso ab
    } where
       pj10 : (A B : HOD) → {xy : Ordinal} → (ab : odef (ZFP (ZPI2 A B) (ZPI1 A B)) xy )
           → & < * (zπ1 ab) , * (zπ2 ab) > ≡ & < *  (zπ2 (pj1 A B xy ab)) ,  * (zπ1 (pj1 A B xy ab)) >
       pj10 A B {.(& < * _ , * _ >)} (ab-pair ax by ) = refl
       pj00 : (A B : HOD) → {xy : Ordinal} → (ab : odef (ZFP (ZPI2 A B) (ZPI1 A B)) xy )
           → & < * (zπ2 (pj1 A B xy ab)) , * (zπ1 (pj1 A B xy ab)) > ≡ xy
       pj00 A B {xy} ab = trans (sym (pj10 A B ab)) (zp-iso {ZPI2 A B} {ZPI1 A B} {xy} ab)

--
-- Bijection of (A x B) and (B x A) requires one element or axiom of choice
--
ZFPsym : (A B  : HOD) → {a b : Ordinal } → odef A a → odef B b  → HODBijection (ZFP A B) (ZFP B A)
ZFPsym A B aa bb = subst₂ ( λ j k → HODBijection (ZFP A B) (ZFP j k)) (ZPI2-iso A B aa) (ZPI1-iso A B bb) ( ZFPsym1 A B )

ZFP∩  : {A B C : HOD} → ( ZFP (A ∩ B) C ≡ ZFP A C ∩ ZFP B C ) ∧ ( ZFP C (A ∩ B) ≡ ZFP C A  ∩ ZFP C B )
proj1 (ZFP∩ {A} {B} {C} ) = ==→o≡ record { eq→ = zfp00  ; eq← = zfp01 } where
   zfp00 : {x : Ordinal} → ZFProduct (A ∩ B) C x → odef (ZFP A C ∩ ZFP B C) x
   zfp00  (ab-pair ⟪ pa , pb ⟫ qx) = ⟪ ab-pair pa qx  , ab-pair pb qx  ⟫
   zfp01 : {x : Ordinal} → odef (ZFP A C ∩ ZFP B C) x → ZFProduct (A ∩ B) C x
   zfp01 {x} ⟪ p , q ⟫  = subst (λ k → ZFProduct (A ∩ B) C k) zfp07 ( ab-pair (zfp02 ⟪ p , q ⟫ ) (zfp04 q) ) where
       zfp05 : & < * (zπ1 p) , * (zπ2 p) > ≡ x
       zfp05 = zp-iso p
       zfp06 : & < * (zπ1 q) , * (zπ2 q) > ≡ x
       zfp06 = zp-iso q
       zfp07 : & < * (zπ1 p) , * (zπ2 q) > ≡ x
       zfp07 = trans (cong (λ k → & < k , * (zπ2 q)  >  )
           (proj1 (prod-≡ (subst₂ _≡_  *iso *iso (cong (*) (trans  zfp05 (sym (zfp06)))))))) zfp06
       zfp02 : {x  : Ordinal  } → (acx : odef (ZFP A C ∩ ZFP B C) x)   → odef (A ∩ B) (zπ1 (proj1 acx))
       zfp02 {.(& < * _ , * _ >)} ⟪ ab-pair {a} {b} ax bx , bcx ⟫ = ⟪ ax , zfp03 bcx refl ⟫ where
           zfp03 : {x : Ordinal } →  (bc : odef (ZFP B C) x) → x ≡ (& < * a , * b >)  → odef B (zπ1 (ab-pair {A} {C} ax bx))
           zfp03 (ab-pair {a1} {b1} x x₁) eq = subst (λ k → odef B k ) zfp08 x  where
              zfp08 : a1 ≡ a
              zfp08 = subst₂ _≡_ &iso &iso (cong (&) (proj1 (prod-≡ (subst₂  _≡_  *iso *iso (cong (*) eq)))))
       zfp04 : {x : Ordinal } (acx : odef (ZFP B C) x )→ odef C (zπ2 acx)
       zfp04 (ab-pair x x₁) = x₁
proj2 (ZFP∩ {A} {B} {C} ) = ==→o≡ record { eq→ = zfp00 ; eq← = zfp01  } where
   zfp00 : {x : Ordinal} → ZFProduct C (A ∩ B) x → odef (ZFP C A ∩ ZFP C B) x
   zfp00  (ab-pair qx ⟪ pa , pb ⟫ ) = ⟪ ab-pair qx pa  , ab-pair qx pb   ⟫
   zfp01 : {x : Ordinal} → odef (ZFP C A ∩ ZFP C B ) x → ZFProduct C (A ∩ B)  x
   zfp01 {x} ⟪ p , q ⟫  = subst (λ k → ZFProduct C (A ∩ B)  k) zfp07 ( ab-pair (zfp04 p) (zfp02 ⟪ p , q ⟫ )  ) where
       zfp05 : & < * (zπ1 p) , * (zπ2 p) > ≡ x
       zfp05 = zp-iso p
       zfp06 : & < * (zπ1 q) , * (zπ2 q) > ≡ x
       zfp06 = zp-iso q
       zfp07 : & < * (zπ1 p) , * (zπ2 q) > ≡ x
       zfp07 = trans (cong (λ k → & < * (zπ1 p) , k  >  )
           (sym (proj2 (prod-≡ (subst₂ _≡_  *iso *iso (cong (*) (trans  zfp05 (sym (zfp06))))))))) zfp05
       zfp02 : {x  : Ordinal  } → (acx : odef (ZFP C A ∩ ZFP C B ) x)   → odef (A ∩ B) (zπ2 (proj2 acx))
       zfp02 {.(& < * _ , * _ >)} ⟪ bcx , ab-pair {b} {a} ax bx  ⟫ = ⟪ zfp03 bcx refl , bx ⟫ where
           zfp03 : {x : Ordinal } →  (bc : odef (ZFP C A ) x) → x ≡ (& < * b , * a >)  → odef A (zπ2 (ab-pair {C} {B} ax bx ))
           zfp03 (ab-pair {b1} {a1} x x₁) eq = subst (λ k → odef A k ) zfp08 x₁ where
              zfp08 : a1 ≡ a
              zfp08 = subst₂ _≡_ &iso &iso (cong (&) (proj2 (prod-≡ (subst₂  _≡_  *iso *iso (cong (*) eq)))))
       zfp04 : {x : Ordinal } (acx : odef (ZFP C A ) x )→ odef C (zπ1 acx)
       zfp04 (ab-pair x x₁) = x

open import BAlgebra O

ZFP\Q : {P Q p : HOD} → (( ZFP P Q ＼ ZFP p Q ) ≡ ZFP (P ＼ p) Q ) ∧ (( ZFP P Q ＼ ZFP P p ) ≡ ZFP P (Q ＼ p) )
ZFP\Q {P} {Q} {p} = ⟪ ==→o≡ record { eq→ = ty70 ; eq← = ty71 } , ==→o≡ record { eq→ = ty73 ; eq← = ty75 } ⟫ where
    ty70 : {x : Ordinal } → odef ( ZFP P Q ＼ ZFP p Q ) x →  odef (ZFP (P ＼ p) Q) x
    ty70 ⟪ ab-pair {a} {b} Pa pb  , npq ⟫ = ab-pair ty72 pb  where
       ty72 : odef (P ＼ p ) a
       ty72 = ⟪ Pa , (λ pa → npq (ab-pair pa pb ) ) ⟫
    ty71 : {x : Ordinal } → odef (ZFP (P ＼ p) Q) x → odef ( ZFP P Q ＼ ZFP p Q ) x
    ty71 (ab-pair {a} {b} ⟪ Pa , npa ⟫ Qb) = ⟪ ab-pair Pa Qb
        , (λ pab → npa (subst (λ k → odef p k) (proj1 (zp-iso0 pab)) (zp1 pab)) ) ⟫
    ty73 : {x : Ordinal } → odef ( ZFP P Q ＼ ZFP P p ) x →  odef (ZFP P (Q ＼ p) ) x
    ty73 ⟪ ab-pair {a} {b} pa Qb  , npq ⟫ = ab-pair pa ty72  where
       ty72 : odef (Q ＼ p ) b
       ty72 = ⟪ Qb , (λ qb → npq (ab-pair pa qb ) ) ⟫
    ty75 : {x : Ordinal } → odef (ZFP P (Q ＼ p) ) x → odef ( ZFP P Q ＼ ZFP P p ) x
    ty75 (ab-pair {a} {b} Pa ⟪ Qb , nqb ⟫ ) = ⟪ ab-pair Pa Qb
        , (λ pab → nqb (subst (λ k → odef p k) (proj2 (zp-iso0 pab)) (zp2 pab)) ) ⟫