Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 28 Apr 2022 19:00:40 +0900 |
| parents | fb210e812eba |
| children | 0687736285ce |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level hiding ( suc ; zero ) open import Ordinals open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality import OD module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where open import zf open import logic -- open import partfunc {n} O open import Relation.Nullary open import Data.Empty import BAlgbra open inOrdinal O open OD O open OD.OD open ODAxiom odAxiom import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal open Ordinals.IsNext isNext open OrdUtil O open ODUtil O import ODC open _∧_ open _∨_ open Bool open HOD _≤_ : (x y : HOD) → Set (Level.suc n) x ≤ y = ( x ≡ y ) ∨ ( x < y ) open _==_ open _⊆_ -- open import Relation.Binary.Properties.Poset as Poset IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) 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 -- -- inductive maxmum tree from x -- tree structure -- ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where init : {x : Ordinal} → odef A s → FClosure A f s s fsuc : {x : Ordinal} ( p : FClosure A f s x ) → FClosure A f s (f x) record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where field y : Ordinal ay : odef B y x=fy : x ≡ f y record SUP ( A B : HOD ) : Set (Level.suc n) where field sup : HOD A∋maximal : A ∋ sup x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive 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) ( 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 ) → b 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 } → o∅ o< & A → ( ( 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 : Ordinal ) → (* 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 (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl (IsStrictPartialOrder.trans PO b<a a<b) z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A 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 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } z08 : ¬ Maximal A → HasMaximal =h= od∅ z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) ; ¬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 ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) is-cf nmx {x} ax with ODC.∋-p O A (* x) ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) ... | yes ax with is-o∅ (& ( Gtx ax )) ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) ... | 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 (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 (& 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< & 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 ) z11 (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)) 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 )) ... | 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 )) z17 : ⊥ z17 with z15 ... | case1 eq = ¬b eq ... | case2 lt = ¬a lt -- 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) -- 3cases : {x y : Ordinal} → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) -- → (ax : odef A x )→ (ay : odef A y ) -- → (zc0 : ZChain A ay f mf supO x ) -- → Prev< A (ZChain.chain zc0) ax f -- ∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ x) -- ∨ ( ( z : Ordinal) → odef (ZChain.chain zc0) z → ¬ ( * z < * x )) -- 3cases {x} {y} f mf ax ay zc0 = {!!} -- create all ZChains under o< x ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((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 {y} ay with Oprev-p x ... | yes op = zc4 where px = Oprev.oprev op zc0 : ZChain A ay f mf supO (Oprev.oprev op) 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 (* px) ... | no noapx = 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 = zc11 } where -- no extention zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (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 zc11 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b ... | yes apx with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO px ... | 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< x → (ba : odef A b) → Prev< A (ZChain.chain zc0) ba 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 zc17 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b ... | case1 b=px = subst (λ k → odef chain k ) (trans (sym (Prev<.x=fy pr )) (trans &iso (sym b=px))) ( ZChain.f-next zc0 (Prev<.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 ; 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) ) )) ... | case1 x=sup = record { chain = schain ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup sp = SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) chain = ZChain.chain zc0 schain : HOD schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = {!!} } ... | case2 ¬x=sup = 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 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (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 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} z18 {a} {b} za b<x ba (case2 p) a<b = {!!} ... | no ¬ox = {!!} where --- limit ordinal case -- Union of ZFChain record UZFChain (y : Ordinal) : Set n where field u : Ordinal u<x : u o< x zuy : odef (ZChain.chain (prev u u<x ay )) y uzc : HOD uzc = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } u-total : IsTotalOrderSet uzc u-total {x} {y} ux uy = {!!} 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)) ) ) zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) zorn01 = proj1 zorn03 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) 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 (& 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 ) zorn99 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) → (x y : Ordinal) (ay : odef A y) → (zc0 : ZChain A {!!} f mf supO x) → Prev< A (ZChain.chain zc0) {!!} f → {!!} zorn99 f mf x y ay zc0 pr = {!!} where 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)))) chain = ZChain.chain zc0 zc7 : ZChain A ay f mf supO x zc7 with trio< (Prev<.y pr) x ... | tri< a ¬b ¬c = {!!} -- already x ∈ chain because of is-max ... | tri≈ ¬a b ¬c = {!!} -- x ≡ z ∈ chain ... | tri> ¬a ¬b x<z = record { chain = zc5 ; chain⊆A = ⊆-zc5 --- ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 {!!} ; is-max = {!!} } where -- extend with x ≡ f z where cahin ∋ z 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 {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 {!!} ) ) 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) y → Tri (* (f y) < * y ) (* (f y) ≡ * y) (* y < * (f y) ) fx=zc c with mf y (subst (λ k → odef A k) &iso ay0 ) ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ay0 (Afx ay0) (case1 (sym zc13))) zc13 (z01 (Afx ay0) ay0 (case1 zc13)) where zc13 : * (f y) ≡ * y 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 ay0 (Afx ay0) (case2 zc14)) (λ lt → z01 (Afx ay0) ay0 (case1 lt) zc14) zc14 where zc14 : * y < * (f y) zc14 = subst (λ k → * y < k ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx zc6 : IsTotalOrderSet zc5 zc6 {a} {b} ( case1 x ) ( case1 x₁ ) = ZChain.f-total zc0 x x₁ zc6 {a} {b} ( case2 fx ) ( case2 fx₁ ) = tri≈ {!!} (subst₂ (λ j k → j ≡ k ) *iso *iso (trans (cong (*) fx) (sym (cong (*) fx₁ ) ))) {!!} zc6 {a} {b} ( case1 c ) ( case2 fx ) = {!!} -- subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx>zc (subst (λ k → odef chain k) {!!} c )) zc6 {a} {b} ( case2 fx ) ( case1 c ) with ODC.p∨¬p O ( Prev< A chain (incl (ZChain.chain⊆A zc0) c) f ) ... | case2 n = {!!} ... | case1 fb with ZChain.f-total zc0 (subst (λ k → odef chain k) (sym &iso) (Prev<.ay pr)) (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 : * y ≡ b 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 y) ≡ a zc11 = subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym {!!} )) 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 = {!!} -- usage (see filter.agda ) -- -- _⊆'_ : ( A B : HOD ) → Set n -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x -- MaximumSubset : {L P : HOD} -- → o∅ o< & L → o∅ o< & P → P ⊆ L -- → IsPartialOrderSet P _⊆'_ -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) -- → Maximal P (_⊆'_) -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP