Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 568:666377324edd
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 01 May 2022 09:36:44 +0900 |
| parents | 4d8a54e2861e |
| children | 33b1ade17f83 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 82 insertions(+), 79 deletions(-) [+] |
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--- a/src/zorn.agda Sun May 01 05:35:36 2022 +0900 +++ b/src/zorn.agda Sun May 01 09:36:44 2022 +0900 @@ -110,10 +110,10 @@ fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y fc00 zero zero refl (init _) (init x₁) i=x i=y = refl - fc00 zero zero refl (init sa) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) - ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init sa) cy i=x i=y ) - fc00 zero zero refl (fsuc x cx) (init sa) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) - ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init sa) i=x i=y ) + fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) + ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) + fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) + ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) @@ -184,7 +184,7 @@ cxx : FClosure A f s (f x) cxx = fsuc x cx fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) - fc16 x (init sa) with proj1 (mf s sa ) + fc16 x (init as) with proj1 (mf s as ) ... | case1 _ = case1 refl ... | case2 _ = case2 refl fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) @@ -206,13 +206,10 @@ IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B -⊆-IsTotalOrderSet = {!!} +⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) -record Maximal ( A : HOD ) : Set (Level.suc n) where - field - maximal : HOD - A∋maximal : A ∋ maximal - ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative +_⊆'_ : ( A B : HOD ) → Set n +_⊆'_ A B = {x : Ordinal } → odef A x → odef B x -- -- inductive maxmum tree from x @@ -225,16 +222,34 @@ ay : odef B y x=fy : x ≡ f y -_⊆'_ : ( A B : HOD ) → Set n -_⊆'_ A B = (x : Ordinal ) → odef A x → odef B x - -record IsSup (A : HOD) (T : IsTotalOrderSet A) {x : Ordinal } - (xa : odef A x) (sup : (C : HOD ) → ( C ⊆ A) → IsTotalOrderSet C → Ordinal) ( f : Ordinal → Ordinal ) : Set n where +record IsSup (A B : HOD) (T : IsTotalOrderSet B) {x : Ordinal } ( B⊆A : B ⊆' A ) + (xa : odef A x) (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) ( f : Ordinal → Ordinal ) : Set n where field chain : Ordinal - chain⊆A : (* chain) ⊆' A + chain⊆B : (* chain) ⊆' B + x=sup : x ≡ sup (* chain) ( λ lt → B⊆A (chain⊆B lt ) ) + ( ⊆-IsTotalOrderSet {B} {* chain} record { incl = chain⊆B } T ) -- ¬prev : ¬ HasPrev A (* chain) xa f - x=sup : x ≡ sup (* chain) record { incl = λ {x} → chain⊆A (& x) } ( ⊆-IsTotalOrderSet {A} {* chain} record { incl = λ {x} → chain⊆A (& x) } T ) + +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 + chain⊆A : chain ⊆' A + chain∋x : odef chain x + 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 ) + → * a < * b → odef chain b + +record Maximal ( A : HOD ) : Set (Level.suc n) where + field + maximal : HOD + A∋maximal : A ∋ maximal + ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative record SUP ( A B : HOD ) : Set (Level.suc n) where field @@ -245,26 +260,12 @@ 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 : HOD ) → ( C ⊆ A) → IsTotalOrderSet C → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where - field - chain : HOD - chain⊆A : chain ⊆ A - chain∋x : odef chain x - 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 → (ba : odef A b) - → HasPrev A chain ba f ∨ ((sup chain chain⊆A f-total) ≡ b ) - → * a < * b → odef chain b - Zorn-lemma : { A : HOD } → o∅ o< & A - → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition + → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition → Maximal A Zorn-lemma {A} 0<A supP = zorn00 where - supO : (C : HOD ) → C ⊆ A → IsTotalOrderSet C → Ordinal + supO : (C : HOD ) → C ⊆' A → IsTotalOrderSet C → Ordinal supO C C⊆A TC = & ( SUP.sup ( supP C C⊆A TC )) z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ z01 {a} {b} A∋a A∋b = <-irr @@ -272,10 +273,10 @@ 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 )) ) + as : A ∋ * ( & s ) + as = 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 ) + s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) 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) )) → ⊥ @@ -309,12 +310,12 @@ 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 : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A as 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∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A as (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 (& A) ) → SUP A (ZChain.chain zc) + sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso as ) f mf supO (& A) ) → SUP A (ZChain.chain zc) sp0 f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (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) @@ -322,13 +323,13 @@ --- --- the maximum chain has fix point of any ≤-monotonic function --- - z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) + z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso as ) 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 ) → b o< osuc (& A) → (ab : odef A b ) - → HasPrev A chain ab f ∨ (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) + → HasPrev A chain ab f ∨ IsSup A chain (ZChain.f-total zc) (ZChain.chain⊆A zc) ab supO f -- (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 @@ -336,11 +337,13 @@ 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 refl ) z13 where + ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) (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 (ZChain.f-total zc) (ZChain.chain⊆A zc) (SUP.A∋maximal sp1) supO f + z19 = {!!} 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 @@ -362,7 +365,7 @@ -- ZChain forces fix point on any ≤-monotonic function (z03) -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain -- - 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 : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso as ) (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 ))) @@ -385,42 +388,42 @@ zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay zc4 : ZChain A ay f mf supO x zc4 with ODC.∋-p O A (* x) - ... | no noapx = -- ¬ A ∋ p, just skip + ... | 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 → (ba : odef A b) → - HasPrev A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0))) ≡ b) → + 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 → * a < * b → odef (ZChain.chain zc0) b - zc11 {a} {b} za b<ox ba P a<b with osuc-≡< b<ox - ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso)) ba ) ) - ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt ) ba P a<b - ... | yes apx with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO x + 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 (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) 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 → (ba : odef A b) → - HasPrev A (ZChain.chain zc0) ba f ∨ (supO (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0) ≡ b) → + 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 → * a < * b → odef (ZChain.chain zc0) b - zc17 {a} {b} za b<ox ba P a<b with osuc-≡< b<ox - ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt) ba P a<b + zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox + ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) 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 ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) + ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ax supO f) ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain - record { chain = schain ; chain⊆A = record { incl = A∋schain } ; f-total = scmp ; f-next = scnext - ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax } where -- x is sup + record { chain = schain ; chain⊆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 sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) sp = SUP.sup sup0 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< ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx ))) + sc<A {y} (case1 zx) = {!!} -- subst (λ k → k o< (& A)) &iso ( c<→o< ( incl (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 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } A∋schain : {x : HOD } → schain ∋ x → A ∋ x - A∋schain (case1 zx ) = subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx )) + A∋schain (case1 zx ) = {!!} -- subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx )) A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx cmp : {a b : HOD} (za : odef chain (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb @@ -455,7 +458,7 @@ ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) A∋za : {a : Ordinal } → odef chain a → odef A a - A∋za za = (subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) za) ) ) + A∋za za = {!!} -- (subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) za) ) ) za<sup : {a : Ordinal } → odef chain a → ( * a ≡ sp ) ∨ ( * a < sp ) za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) za ) simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) @@ -475,34 +478,34 @@ ... | 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 → (ba : odef A b) → - HasPrev A schain ba f ∨ (& (SUP.sup (supP schain record { incl = A∋schain } scmp)) ≡ b) + s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) → + HasPrev A schain ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → * a < * b → odef schain b - s-ismax {a} {b} (case1 za) b<x ba (case1 p) a<b with osuc-≡< b<x - ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) + s-ismax {a} {b} (case1 za) b<x ab (case1 p) a<b with osuc-≡< b<x + ... | 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 - z21 : HasPrev A schain ba f → odef schain b + 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 (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) b<x ) ba (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) + case1 (ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) 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 ba (case2 p) a<b with osuc-≡< b<x - ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) + s-ismax {a} {b} (case1 za) b<x ab (case2 p) a<b with osuc-≡< b<x + ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) {!!} (init (SUP.A∋maximal sup0) )) ... | case2 b<x = z22 p where - z22 : & (SUP.sup (supP schain record { incl = A∋schain } scmp)) ≡ b → odef schain b + z22 : IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → odef schain b z22 p = {!!} - -- case1 (ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) b<x ) ba {!!} a<b ) - s-ismax {a} {b} (case2 sa) b<x ba p a<b = {!!} + -- case1 (ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) b<x ) ab {!!} a<b ) + 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 → (ba : odef A b) → - HasPrev A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0))) ≡ b) → + 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 → * a < * b → odef (ZChain.chain zc0) b - z18 {a} {b} za b<x ba p a<b with osuc-≡< b<x - ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) lt ) ba p a<b + z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x + ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) 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 (sym (trans b=sup b=x )) ) + ... | case2 b=sup = ⊥-elim ( ¬x=sup {!!} ) ... | no ¬ox = {!!} where --- limit ordinal case record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x field @@ -533,8 +536,8 @@ 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 (& A) 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 (& A) - zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) + zorn04 : ZChain A (subst (λ k → odef A k ) &iso as ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) + zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) -- usage (see filter.agda ) --