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

comparison src/zorn.agda @ 543:f0b45446c7ea

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author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Wed, 27 Apr 2022 04:04:30 +0900
parents	3826221c61a6
children	27bb51b7f012

comparison

equal deleted inserted replaced

-:3826221c61a6
+:f0b45446c7ea
 field
 maximal : HOD
 A∋maximal : A ∋ maximal
 ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x       -- A is Partial, use negative
 --
 -- inductive maxmum tree from x
 -- tree structure
 --
 (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
+¬chain∋x>z : { a : Ordinal } → z o< osuc a → ¬ odef chain a
 f-total : IsTotalOrderSet chain
 f-next : {a : Ordinal } → odef chain a → a o< z  → 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) → a o< z
 → Prev< A chain ba f
 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← =  λ {y} lt → ⊥-elim ( ¬x<0 lt )}
 x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m))
 x-is-maximal nmx {x} ax nogt m am  = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) ,  ¬x<m  ⟫ where
 ¬x<m :  ¬ (* x < * m)
 ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫  (≡o∅→=od∅ nogt)
+-- Uncountable acending chain by axiom of choice
 cf : ¬ Maximal A → Ordinal → Ordinal
 cf  nmx x with ODC.∋-p O A (* x)
 ... | no _ = o∅
 ... | yes ax with is-o∅ (& ( Gtx ax ))
 ... | yes nogt = -- no larger element, so it is maximal
 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))
 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 (& 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))
 →  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 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 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 ) → a o< (& A)
 →  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) (c<→o< (subst (λ k → odef A (& k) ) *iso sa) ) (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))
-z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 {!!} ))) (me z12 )
+z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 z11 ))) (me z12 )
 ... | tri< a ¬b ¬c = ⊥-elim z16 where
 z16 : ⊥
 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) ))
 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt ))
 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b )
 ... | tri> ¬a ¬b c = ⊥-elim z17 where
 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) <  SUP.sup sp1)
-z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 {!!} ) )
+z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 z11 ) )
 z17 : ⊥
 z17 with z15
 ... | case1 eq = ¬b eq
 ... | case2 lt = ¬a lt
 z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥
 (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 y )
-→ {a b : Ordinal} (ca : odef (ZChain.chain zc0) a)
+→ {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y
-→ odef A b → a o< x → Prev< A (ZChain.chain zc0) (incl (ZChain.chain⊆A zc0) (subst (odef (ZChain.chain zc0)) (sym &iso) ca)) f
+→  Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0))
-∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b)
+(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<x P a<b = ZChain.is-max zc0 ca ab {!!} {!!} a<b
+premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab a<y P a<b
 -- Union of ZFChain
 UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal)
 → ( (y : Ordinal) → y o< B → ZChain A sa f mf supO y ) → HOD
 UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B)  ∧ ( (y<b : y o< B) → odef (ZChain.chain (prev y y<b)) y) }
 ; odmax = & A ; <odmax = z07 }
 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
 px = Oprev.oprev op
 zc0 : ZChain A sa f mf supO (Oprev.oprev op)
 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
 zc1 : ZChain A sa f mf supO x
-zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!} ; f-immediate = {!!}
+zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
-; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} }
+; f-next = zc20 (ZChain.f-next zc0) ; f-immediate =  ZChain.f-immediate zc0
+; ¬chain∋x>z =  λ {a} x<oa → ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa )
+; chain∋x  = ZChain.chain∋x zc0 ; is-max = λ za ba a<x → zc20 (λ za a<x → ZChain.is-max zc0 za ba a<x ) za a<x } where
+zc20 : {P : Ordinal →  Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a)
+→ {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) →  P a
+zc20 {P} prev {a} za a<x with trio< a px
+... | tri< a₁ ¬b ¬c = prev za a₁
+... | tri≈ ¬a b ¬c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (subst (λ k → k o< osuc a) b <-osuc ) za )
+... | tri> ¬a ¬b c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (ordtrans c <-osuc ) za )
 ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x
 px = Oprev.oprev op
 zc0 : ZChain A sa f mf supO (Oprev.oprev op)
 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
+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 sa f mf supO x
 zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ax f )
 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x  = case1 (ZChain.chain∋x zc0) ; 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 : zc5 ⊆ A
-⊆-zc5 = record { incl = λ lt → {!!} }
+⊆-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} (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 ax ) )
 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  c with mf x (subst (λ k → odef A k) &iso  ax )
-... | ⟪ case1 x=fx , afx ⟫ = tri≈ {!!} zc13 {!!} where
+... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax (Afx ax) (case1 (sym zc13))) zc13 (z01 (Afx ax) ax (case1 zc13)) where
 zc13 : * (f x) ≡ * x
 zc13 = subst (λ k → k ≡ * x ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx ))
-... | ⟪ case2 x<fx , afx ⟫ = tri> {!!} {!!} zc14 where
+... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax (Afx ax) (case2 zc14)) (λ lt → z01 (Afx ax) ax (case1 lt) zc14) zc14 where
 zc14 : * x < * (f x)
-zc14 = subst₂ (λ j k → j < k ) {!!} (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
+zc14 = subst (λ k → * x < k ) (subst (λ k → * (f x) ≡ 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)))
 ... | tri< a₁ ¬b ¬c = {!!}
-... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx=zc (subst (λ k → odef chain k) {!!} c )) where
+... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where
-zc11 : & a ≡ f x
-zc11 = fx
 zc10 : * x ≡ b
 zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax y ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁)
-zc12 : Tri (a < b) (a ≡ b) (b < a)
+zc11 : * (f x) ≡ a
-zc12 with mf x (subst (λ k → odef A k) &iso  ax )
+zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx))
-... | ⟪ case1 x=fx , afx ⟫ = tri≈ {!!} zc13 {!!} where
+zc12 : odef chain x
-zc13 : a ≡ b
+zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10)))  c
-zc13 = subst₂ (λ j k → j ≡ k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym zc11))) zc10 (sym ( x=fx ))
-... | ⟪ case2 x<fx , afx ⟫ = tri> {!!} {!!} zc14 where
-zc14 : b < a
-zc14 = subst₂ (λ j k → j < k ) zc10 (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym zc11))) x<fx
 ... | 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
 ... | 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  = ZChain.chain∋x zc0 ; is-max = {!!} }  -- no extention
 ind f mf x prev | no ¬ox with trio< (& A) x   --- limit ordinal case
-... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  {!!}
+... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
-; f-immediate = {!!} ; chain∋x  = {!!}  ; is-max = {!!} } where
+; f-next = {!!}
+; f-immediate = {!!} ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} } where
 zc0 = prev (& A) a
 ... | tri≈ ¬a b ¬c = {!!}
-... | tri> ¬a ¬b c = {!!}
+... | tri> ¬a ¬b c =  record { chain = uzc ; chain⊆A = record { incl = λ {x} lt → proj1 lt } ; f-total = {!!} ; f-next =  {!!}
+; f-immediate = {!!} ; chain∋x  = {!!}  ; is-max = {!!} } where
+uzc : HOD
+uzc = UZFChain f mf x prev
 zorn00 : Maximal A
 zorn00 with is-o∅ ( & HasMaximal )  -- we have no Level (suc n) LEM
 ... | no not = record { maximal = ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x  = zorn02 } where
 -- yes we have the maximal
 zorn03 :  odef HasMaximal ( & ( ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ) )

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

comparison src/zorn.agda @ 543:f0b45446c7ea