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

comparison src/zorn.agda @ 547:379bd9b4610c

...

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Wed, 27 Apr 2022 23:21:21 +0900
parents	3234a5f6bfcf
children	5ad7a31df4f4

comparison

equal deleted inserted replaced

-:3234a5f6bfcf
+:379bd9b4610c
 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n)
 SupCond A B _ _ = SUP A B
 record ZChain ( A : HOD )  {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
-(sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) : Set (Level.suc n) where
+(sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where
 field
 chain : HOD
 chain⊆A : chain ⊆ A
 chain∋x : odef chain x
 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 ) → (ba : odef A b)
+is-max :  {a b : Ordinal } → (ca : odef chain a ) →  a o< z  → (ba : odef A b)
 → Prev< A chain ba f
 ∨  (sup (& chain) (subst (λ k → k  ⊆ A) (sym *iso) chain⊆A)  (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b )
 → * a < * b  → odef chain b
 Zorn-lemma : { A : HOD }
 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
 s : HOD
 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
 sa : A ∋ * ( & s  )
 sa =  subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))  )
+s<A : & s o< & A
+s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa )
 HasMaximal : HOD
 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) →  odef A m → ¬ (* x < * m)) }  ; odmax = & A ; <odmax = z07 }
 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) →  odef A m →  odef A x ∧ (¬ (* x < * m) )) →  ⊥
 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ )
 Gtx : { x : HOD} → A ∋ x → HOD
 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) →  odef A x → ( * x < * (cf nmx x) ) ∧  odef A (cf nmx x )
 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫
 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) →  ≤-monotonic-f A ( cf nmx )
 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫
-zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) →  (zc : ZChain A sa f mf supO ) → SUP A  (ZChain.chain zc)
+zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) →  (zc : ZChain A sa f mf supO (& A) ) → SUP A  (ZChain.chain zc)
 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc  )
-A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO )
+A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) )
 →  A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ))
 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )
-sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO ) → SUP A (* (& (ZChain.chain zc)))
+sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) → SUP A (* (& (ZChain.chain zc)))
 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc))
 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) )
 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y)
 ---
 --- sup is fix point in maximum chain
 ---
-z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO )
+z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) )
 → 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 ) → (ab : odef A b )
+z10 :  {a b : Ordinal } → (ca : odef chain a ) → a o< & A → (ab : odef A b )
 →  Prev< A chain ab f
 ∨  (supO (& chain) (subst (λ k → k  ⊆ A) (sym *iso) (ZChain.chain⊆A zc))  (subst (λ k → IsTotalOrderSet k) (sym *iso) (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 ) (SUP.A∋maximal sp1)  (case2 refl ) z13 where
+... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) s<A (SUP.A∋maximal sp1)  (case2 refl ) z13 where
 z13 :  * (& s) < * (& (SUP.sup sp1))
 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc ))
 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
 z14 :  f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc))
 z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 ))
 z17 : ⊥
 z17 with z15
 ... | case1 eq = ¬b eq
 ... | case2 lt = ¬a lt
-z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO ) → ⊥
+-- ZChain requires the Maximal
+z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥
 z04 nmx zc = z01  {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso)
 (proj1 (is-cf nmx (SUP.A∋maximal  sp1))))
 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc )))
 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where
 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc
 c = & (SUP.sup sp1)
-premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 :  ZChain A sa f mf supO  )
+premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 :  ZChain A sa f mf supO y )
 → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y
 →  Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0))
 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0))
 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b)
 → * a < * b → odef (ZChain.chain zc0) b
-premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab P a<b -- ca ab y P a<b
+premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca a<y ab P a<b -- ca ab y P a<b
 -- Union of ZFChain
 UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal)
-→ ( (y : Ordinal) → y o< B → (ya : odef A y) → ZChain A ya f mf supO ) → HOD
+→ ( (z : Ordinal) → z o< B → {y : Ordinal} →  (ya : odef A y) → ZChain A ya f mf supO z ) → HOD
 UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B)  ∧ ( (y<b : y o< B) → (ya : odef A y) → odef (ZChain.chain (prev y y<b ya )) y) }
 ; odmax = & A ; <odmax = z07 }
--- ZChain is not compatible with the SUP condition
+-- create all ZChains under o< x
 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) →
-((y : Ordinal) → y o< x → (ya : odef A y) → ZChain A ya f mf supO) → (ya : odef A x) → ZChain A ya f mf supO
+((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → { y : Ordinal } → (ya : odef A y) → ZChain A ya f mf supO x
-ind f mf x prev ax with Oprev-p x
+ind f mf x prev {y} ay with Oprev-p x
 ... | yes op with ODC.∋-p O A (* (Oprev.oprev op))
 ... | yes apx = zc4 where -- we have previous ordinal and A ∋ op
 px = Oprev.oprev op
 apx0 = subst (λ k → odef A k ) &iso apx
-zc0 : ZChain A apx0 f mf supO
+zc0 : ZChain A ay f mf supO (Oprev.oprev op)
-zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) apx0
+zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay
-ax0 : odef A (& (* x))
+zc1 : {y : Ordinal } → (ay : odef A y ) → ZChain A ay f mf supO (Oprev.oprev op)
-ax0 = {!!}
+zc1 {y} ay = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay
+ay0 : odef A (& (* y))
+ay0 = (subst (λ k → odef A k ) (sym &iso) ay )
 Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x)
 Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax))))
 --   x is in the previous chain, use the same
 --   x has some y which y < x ∧ f y ≡ x
 --   x has no y which y < x
-zc4 : ZChain A ax f mf supO
+zc4 : ZChain A ay f mf supO x
-zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ax f )
+zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ay f )
-... | case1 y = zc7 where -- we have previous <
+... | case1 pr = zc7 where -- we have previous <
 chain = ZChain.chain zc0
-zc7 :  ZChain A ax f mf supO
+zc7 :  ZChain A ay f mf supO x
 zc7 with ODC.∋-p O  (ZChain.chain zc0) (* ( f x ) )
-... | yes y = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
+... | yes pr = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A 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 = {!!} }  where -- no extention
 z22 : {a : Ordinal} → x o< osuc a → ¬ odef (ZChain.chain zc0) a
 z22 {a} x<oa = {!!}
 zc20 : {P : Ordinal →  Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a)
 ... | tri> ¬a ¬b c = ⊥-elim ( o<> c a<x )
 ... | no not = record { chain = zc5 ; chain⊆A =  ⊆-zc5
 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x  = case1 {!!} ; ¬chain∋x>z = {!!} ; is-max = {!!} } where
 --   extend with f x -- cahin ∋ y ∧  chain ∋ f y ∧ x ≡ f ( f y )
 zc5 : HOD
-zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f x) } ; odmax = & A ; <odmax = {!!} }
+zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f y) } ; odmax = & A ; <odmax = {!!} }
 ⊆-zc5 : zc5 ⊆ A
 ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where
-zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f x) ) → odef A z
+zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f y) ) → odef A z
 zc15 {z} (case1 zz) = subst (λ k → odef A k ) &iso ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain  k ) (sym &iso) zz ) )
-zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso {!!} ) )
+zc15 (case2 refl) = proj2 ( mf y (subst (λ k → odef A k ) &iso {!!} ) )
 IPO = ⊆-IsPartialOrderSet  ⊆-zc5 PO
 zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x
 zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P)))
-fx=zc :  odef (ZChain.chain zc0) x → Tri  (* (f x) < * x ) (* (f x) ≡ * x) (* x < * (f x) )
+fx=zc :  odef (ZChain.chain zc0) y → Tri  (* (f y) < * y ) (* (f y) ≡ * y) (* y < * (f y) )
-fx=zc  c with mf x (subst (λ k → odef A k) &iso  ax0 )
+fx=zc  c with mf y (subst (λ k → odef A k) &iso  ay0 )
-... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax0 (Afx ax0) (case1 (sym zc13))) zc13 (z01 (Afx ax0) ax0 (case1 zc13)) where
+... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ay0 (Afx ay0) (case1 (sym zc13))) zc13 (z01 (Afx ay0) ay0 (case1 zc13)) where
-zc13 : * (f x) ≡ * x
+zc13 : * (f y) ≡ * y
-zc13 = subst (λ k → k ≡ * x ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx ))
+zc13 = subst (λ k → k ≡ * y ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx ))
-... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax0 (Afx ax0) (case2 zc14)) (λ lt → z01 (Afx ax0) ax0 (case1 lt) zc14) zc14 where
+... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ay0 (Afx ay0) (case2 zc14)) (λ lt → z01 (Afx ay0) ay0 (case1 lt) zc14) zc14 where
-zc14 : * x < * (f x)
+zc14 : * y < * (f y)
-zc14 = subst (λ k → * x < k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
+zc14 = subst (λ k → * y < k ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
 cmp : Trichotomous _ _
 cmp record { elm = a ; is-elm = aa } record { elm = b ; is-elm = ab } with aa | ab
 ... | case1 x | case1 x₁ = IsStrictTotalOrder.compare (ZChain.f-total zc0) (me x) (me x₁)
 ... | case2 fx | case2 fx₁ = tri≈ {!!} (subst₂ (λ j k → j ≡ k ) *iso *iso (trans (cong (*) fx) (sym (cong (*) fx₁ ) ))) {!!}
 ... | case1 c | case2 fx = {!!} -- subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx>zc (subst (λ k → odef chain k) {!!} c ))
 ... | case2 fx | case1 c with ODC.p∨¬p O ( Prev< A chain (incl (ZChain.chain⊆A zc0) c) f )
 ... | case2 n = {!!}
-... | case1 fb with IsStrictTotalOrder.compare (ZChain.f-total zc0) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay y))) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb)))
+... | case1 fb with IsStrictTotalOrder.compare (ZChain.f-total zc0) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay pr))) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb)))
 ... | tri< a₁ ¬b ¬c = {!!}
 ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where
-zc10 : * x ≡ b
+zc10 : * y ≡ b
-zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax {!!} ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁)
+zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ay {!!} ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁)
-zc11 : * (f x) ≡ a
+zc11 : * (f y) ≡ a
-zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx))
+zc11 = subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym fx))
-zc12 : odef chain x
+zc12 : odef chain y
 zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10)))  c
 ... | tri> ¬a ¬b c₁ = {!!}
 zc6 : IsTotalOrderSet zc5
 zc6 =  record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z}
 ; compare = cmp }
 ... | case2 not with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) {!!} {!!} ) ))
-... | case1 y = {!!} -- x is sup
+... | case1 pr = {!!} -- x is sup
 ... | case2 not = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!}
 ; f-immediate = {!!} ; chain∋x  = {!!} ; is-max = {!!} }  -- no extention
 ... | no noapx = {!!} -- we have previous ordinal but ¬ A ∋ op
 ind f mf x prev ya | no ¬ox with trio< (& A) x   --- limit ordinal case
 ... | tri< a ¬b ¬c = {!!} where
 -- if we have no maximal, make ZChain, which contradict SUP condition
 nmx : ¬ Maximal A
 nmx mx =  ∅< {HasMaximal} zc5 ( ≡o∅→=od∅  ¬Maximal ) where
 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) →  odef A y → ¬ (* (& (Maximal.maximal mx)) < * y))
 zc5 = ⟪  Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫
-zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO
+zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A)
-zorn03 f mf = TransFinite {λ y →  (ya : odef A y ) → ZChain A ya f mf supO  } (ind f mf)  (& s )
+zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A)
-zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO
+zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)
 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa )
 -- usage (see filter.agda )
 --
 -- _⊆'_ : ( A B : HOD ) → Set n

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

comparison src/zorn.agda @ 547:379bd9b4610c