Members/kono/Proof/ZF-in-agda: src/zorn.agda comparison

comparison src/zorn.agda @ 570:c642cbafc07a

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author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Mon, 02 May 2022 12:43:42 +0900
parents	33b1ade17f83
children	2ade91846f57

comparison

equal deleted inserted replaced

-:33b1ade17f83
+:c642cbafc07a
 --
 -- Partial Order on HOD ( possibly limited in A )
 --
+_<<_ : (x y : Ordinal ) → Set n
+x << y = * x < * y
+POO : IsStrictPartialOrder _≡_ _<<_
+POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
+; trans = IsStrictPartialOrder.trans PO
+; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y
+; <-resp-≈ =  record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } }
 _≤_ : (x y : HOD) → Set (Level.suc n)
 x ≤ y = ( x ≡ y ) ∨ ( x < y )
 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z
 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl
 field
 y : Ordinal
 ay : odef B y
 x=fy :  x ≡ f y
-record IsSup (A B : HOD) (T : IsTotalOrderSet B) ( B⊆A :  B ⊆' A )
+record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x)     : Set n where
-{x : Ordinal } (xa : odef A x)  (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) ( f : Ordinal → Ordinal )  : Set n where
 field
 chain : Ordinal
 chain⊆B : (* chain) ⊆' B
-x=sup :  x ≡ sup  (* chain) ( λ lt → B⊆A (chain⊆B lt ) )
+x<sup : {y : Ordinal} → odef (* chain) y → (y ≡ x ) ∨ (y << x )
-( ⊆-IsTotalOrderSet {B} {* chain} record { incl = chain⊆B } T )
--- ¬prev : ¬ HasPrev A (* chain) xa f
 record ZChain ( A : HOD )  {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
 (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where
 field
 chain : HOD
 initial : {y : Ordinal } → odef chain y → * x ≤ * y
 f-total : IsTotalOrderSet chain
 f-next : {a : Ordinal } → odef chain a → odef chain (f a)
 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) )
 is-max :  {a b : Ordinal } → (ca : odef chain a ) →  b o< osuc z  → (ab : odef A b)
-→ HasPrev A chain ab f ∨  IsSup A chain f-total chain⊆A ab sup f       --  ((sup  chain  chain⊆A  f-total) ≡ b )
+→ HasPrev A chain ab f ∨  IsSup A chain ab        --  ((sup  chain  chain⊆A  f-total) ≡ b )
 → * a < * b  → odef chain b
 record Maximal ( A : HOD )  : Set (Level.suc n) where
 field
 maximal : HOD
 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc ))
 z03 f mf zc = z14 where
 chain = ZChain.chain zc
 sp1 = sp0 f mf zc
 z10 :  {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b )
-→  HasPrev A chain ab f ∨  IsSup A chain (ZChain.f-total zc) (ZChain.chain⊆A zc) {b} ab supO f -- (supO  chain  (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b )
+→  HasPrev A chain ab f ∨  IsSup A chain {b} ab -- (supO  chain  (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b )
 → * a < * b  → odef chain b
 z10 = ZChain.is-max zc
 z11 : & (SUP.sup sp1) o< & A
 z11 = c<→o< ( SUP.A∋maximal sp1)
 z12 : odef chain (& (SUP.sup sp1))
 z12 with o≡? (& s) (& (SUP.sup sp1))
 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc )
 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1)
-(case2 record { chain = & chain ; chain⊆B = λ z → subst (λ  k → odef k _ ) *iso z ;  x=sup = cong (&) (sup== (sym *iso)) }  ) z13 where
+(case2 z19 ) z13 where
 z13 :  * (& s) < * (& (SUP.sup sp1))
 z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc )
 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
+z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1)
+z19 = record { chain = & chain ; chain⊆B = λ z → subst (λ  k → odef k _ ) *iso z ;  x<sup = z20 }  where
+z20 :  {y : Ordinal} → odef (* (& chain)) y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1))
+z20 {y} zy with SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) *iso (sym &iso) zy)
+... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p ))
+... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p )
+-- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ?  (SUP.x<sup sp1 ? ) }
 z14 :  f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc))
 z14 with ZChain.f-total zc  (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12
 ... | tri< a ¬b ¬c = ⊥-elim z16 where
 z16 : ⊥
 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
 ... | no noax =  -- ¬ A ∋ p, just skip
 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0
 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
 ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc11 }  where -- no extention
 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
-HasPrev A (ZChain.chain zc0) ab f ∨  IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f  →
+HasPrev A (ZChain.chain zc0) ab f ∨  IsSup A (ZChain.chain zc0) ab →
 * a < * b → odef (ZChain.chain zc0) b
 zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox
 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) )
 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt)  ab P a<b
 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) ax f )   -- we have to check adding x preserve is-max ZChain A ay f mf supO x
 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next
 chain = ZChain.chain zc0
 zc17 :  {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
-HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f →
+HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab →
 * a < * b → odef (ZChain.chain zc0) b
 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox
 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b
 ... | case1 b=x = subst (λ k → odef chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr))
 zc9 :  ZChain A ay f mf supO x
 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
 ; initial = ZChain.initial zc0 ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc17 }  -- no extention
-... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ax supO f)
+... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax )
-... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain
+... | case1 is-sup = -- previous ordinal is a sup of a smaller ZChain
 record { chain = schain ; chain⊆A = s⊆A  ; f-total = scmp ; f-next = scnext
-; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup
+; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax } where -- x is sup
-sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0)
+-- sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0)
+-- sp = SUP.sup sup0
+-- x=sup : IsSup A (ZChain.chain zc0) {& (* x)} ax → x ≡ & sp -- sup is not minimum, so this may wrong
+sup0 : SUP A (ZChain.chain zc0)
+sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = {!!}  }
+sp : HOD
 sp = SUP.sup sup0
+x=sup : x ≡ & sp
+x=sup = sym &iso
 chain = ZChain.chain zc0
 sc<A : {y : Ordinal} → odef chain y ∨ FClosure A f (& sp) y → y o< & A
 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc0 (subst (λ k → odef chain k) (sym &iso) zx )))
 sc<A {y} (case2 fx) = subst (λ k → k o< (& A)) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fc (& sp) f mf fx )) )
 schain : HOD
 ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a)  b<sp ) ) (proj1 p )
 ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ  k → k < * a ) (trans *iso (sym b=sp)) sp<a  )) (proj1 p )
 ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p )
 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p
 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) →
-HasPrev A schain ab f ∨ IsSup A schain scmp s⊆A ab (λ C C⊆A TC → & (SUP.sup (supP C C⊆A TC))) f
+HasPrev A schain ab f ∨ IsSup A schain ab
 → * a < * b → odef schain b
-s-ismax {a} {b} (case1 za) b<x ab (case1 p) a<b with osuc-≡< b<x
+s-ismax {a} {b} (case1 za) b<ox ab P a<b with osuc-≡< b<ox | ODC.p∨¬p O (HasPrev A schain ab f)-- b is some previous
-... | case1 b=x = case2 {!!} -- (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) ))
+... | case1 b=x | _ = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) ))
-... | case2 b<x = z21 p where
+... | case2 b<x | case1 p = z21 p where
 z21 : HasPrev A schain ab f → odef schain b
 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } =
 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b )
 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) )
-s-ismax {a} {b} (case1 za) b<x ab (case2 p) a<b with osuc-≡< b<x
+s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x | case2 ¬pr = ⊥-elim ( ¬pr p )
-... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) {!!} (init (SUP.A∋maximal sup0) ))
+s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x | case2 ¬pr = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where
-... | case2 b<x = {!!} where
+-- cahin of IsSup A schain ab may larger than chain of zc0 if it has a previous but it is not
-z22 : IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → odef schain b
+z23 : * (IsSup.chain p) ⊆' ZChain.chain zc0
-z22 p = {!!}
+z23 = {!!}
--- case1 (ZChain.is-max zc0 za (zc0-b<x b lt) ab {!!} a<b )
+z22 : IsSup A (ZChain.chain zc0)   ab
+z22 = record { chain = IsSup.chain p ; chain⊆B = z23 ; x<sup = {!!} }
 s-ismax {a} {b} (case2 sa) b<x ab p a<b = {!!}
 ... | case2 ¬x=sup =  -- x is not f y' nor sup of former ZChain from y
 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0
 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x  =  ZChain.chain∋x zc0 ; is-max = z18 }  where -- no extention
 z18 :  {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
-HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f →
+HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0)   ab →
 * a < * b → odef (ZChain.chain zc0) b
 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x
 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab p a<b
 ... | case1 b=x with p
 ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (trans &iso (sym b=x) ) (HasPrev.x=fy pr ) } )
-... | case2 b=sup = ⊥-elim ( ¬x=sup {!!} )
+... | case2 b=sup = ⊥-elim ( ¬x=sup record { chain = IsSup.chain b=sup ; chain⊆B =  IsSup.chain⊆B b=sup
+;  x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy)  } )
 ... | no ¬ox =  {!!}  where --- limit ordinal case
 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x
 field
 u : Ordinal
 u<x : u o< x

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

comparison src/zorn.agda @ 570:c642cbafc07a