Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Wed, 27 Apr 2022 05:19:31 +0900 |
| parents | f0b45446c7ea |
| children | f8eb56442f2c |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level hiding ( suc ; zero ) open import Ordinals import OD module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) where open import zf open import logic -- open import partfunc {n} O open import Relation.Nullary open import Relation.Binary open import Data.Empty open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality 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 ) record Element (A : HOD) : Set (Level.suc n) where field elm : HOD is-elm : A ∋ elm open Element _<A_ : {A : HOD} → (x y : Element A ) → Set n x <A y = elm x < elm y _≡A_ : {A : HOD} → (x y : Element A ) → Set (Level.suc n) x ≡A y = elm x ≡ elm y IsPartialOrderSet : ( A : HOD ) → Set (Level.suc n) IsPartialOrderSet A = IsStrictPartialOrder (_≡A_ {A}) _<A_ open _==_ open _⊆_ isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) } ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → IsPartialOrderSet A → IsPartialOrderSet B ⊆-IsPartialOrderSet {A} {B} B⊆A PA = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {x} {y} {z} → trans1 {x} {y} {z} ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; <-resp-≈ = resp0 } where _<B_ : (x y : Element B ) → Set n x <B y = elm x < elm y trans1 : {x y z : Element B} → x <B y → y <B z → x <B z trans1 {x} {y} {z} x<y y<z = IsStrictPartialOrder.trans PA {isA B⊆A x} {isA B⊆A y} {isA B⊆A z} x<y y<z irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y ) irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y open import Data.Product resp0 : ({x y z : Element B} → elm y ≡ elm z → x <B y → x <B z) × ({x y z : Element B} → elm y ≡ elm z → y <B x → z <B x) resp0 = Data.Product._,_ (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) -- open import Relation.Binary.Properties.Poset as Poset IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) IsTotalOrderSet A = IsStrictTotalOrder (_≡A_ {A}) _<A_ me : { A a : HOD } → A ∋ a → Element A me {A} {a} lt = record { elm = a ; is-elm = lt } A∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (A ∋ x) ≡ (A ∋ y ) A∋x-irr A {x} {y} refl = refl me-elm-refl : (A : HOD) → (x : Element A) → elm (me {A} (d→∋ A (is-elm x))) ≡ elm x me-elm-refl A record { elm = ex ; is-elm = ax } = *iso -- <-induction : (A : HOD) { ψ : (x : HOD) → A ∋ x → Set (Level.suc n)} -- → IsPartialOrderSet A -- → ( {x : HOD } → A ∋ x → ({ y : HOD } → A ∋ y → y < x → ψ y ) → ψ x ) -- → {x0 x : HOD } → A ∋ x0 → A ∋ x → x0 < x → ψ x -- <-induction A {ψ} PO ind ax0 ax x0<a = subst (λ k → ψ k ) *iso (<-induction-ord (osuc (& x)) <-osuc ) where -- -- y < * ox → & y o< ox -- induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox) -- induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) {!!})) -- <-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy) -- <-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- Don't use Element other than Order, you'll be in a trouble -- postulate -- may be proved by transfinite induction and functional extentionality -- ∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (ax : A ∋ x) (ay : A ∋ y ) → ax ≅ ay -- odef-irr : (A : OD) {x y : Ordinal} → x ≡ y → (ax : def A x) (ay : def A y ) → ax ≅ ay -- is-elm-irr : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y -- is-elm-irr A {x} {y} eq = ∋x-irr A eq (is-elm x) (is-elm y ) El-irr2 : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y → x ≡ y El-irr2 A {x} {y} refl HE.refl = refl -- El-irr : {A : HOD} → {x y : Element A } → elm x ≡ elm y → x ≡ y -- El-irr {A} {x} {y} eq = El-irr2 A eq ( is-elm-irr A eq ) record _Set≈_ (A B : Ordinal ) : Set n where field fun← : {x : Ordinal } → odef (* A) x → Ordinal fun→ : {x : Ordinal } → odef (* B) x → Ordinal funB : {x : Ordinal } → ( lt : odef (* A) x ) → odef (* B) ( fun← lt ) funA : {x : Ordinal } → ( lt : odef (* B) x ) → odef (* A) ( fun→ lt ) fiso← : {x : Ordinal } → ( lt : odef (* B) x ) → fun← ( funA lt ) ≡ x fiso→ : {x : Ordinal } → ( lt : odef (* A) x ) → fun→ ( funB lt ) ≡ x open _Set≈_ record _OS≈_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where field iso : (& A ) Set≈ (& B) fmap : {x y : Ordinal} → (ax : odef A x) → (ay : odef A y) → * x < * y → * (fun← iso (subst (λ k → odef k x) (sym *iso) ax)) < * (fun← iso (subst (λ k → odef k y) (sym *iso) ay)) Cut< : ( A x : HOD ) → HOD Cut< A x = record { od = record { def = λ y → ( odef A y ) ∧ ( x < * y ) } ; odmax = & A ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (proj1 lt))) } Cut<T : {A : HOD} → (TA : IsTotalOrderSet A ) ( x : HOD )→ IsTotalOrderSet ( Cut< A x ) Cut<T {A} TA x = record { isEquivalence = record { refl = refl ; trans = trans ; sym = sym } ; trans = λ {x} {y} {z} → IsStrictTotalOrder.trans TA {me (proj1 (is-elm x))} {me (proj1 (is-elm y))} {me (proj1 (is-elm z))} ; compare = λ x y → IsStrictTotalOrder.compare TA (me (proj1 (is-elm x))) (me (proj1 (is-elm y))) } record _OS<_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where field x : HOD iso : TA OS≈ (Cut<T TA x) -- OS<-cmp : {x : HOD} → Trichotomous {_} {IsTotalOrderSet x} _OS≈_ _OS<_ -- OS<-cmp A 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 -- -- 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 ) record Indirect< (A : HOD) {x y : Ordinal } (xa : odef A x) (ya : odef A y) (z : Ordinal) : Set n where field az : odef A z x<z : * x < * z z<y : * z < * y 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 : 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 ¬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 ∨ (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 → IsPartialOrderSet A → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition → Maximal A Zorn-lemma {A} 0<A PO 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 {me A∋b} {me A∋a} (sym a=b) b<a z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b} 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 )) ) 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 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 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 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)) → ⊥ 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 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 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 } -- ZChain is not compatible with the SUP condition ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A sa f mf supO y ) → ZChain A sa f mf supO x ind f mf x prev with Oprev-p x ... | yes op with ODC.∋-p O A (* x) ... | no ¬Ax = zc1 where -- 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 = 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 ) ... | case1 y = zc7 where -- we have previous < chain = ZChain.chain zc0 zc7 : ZChain A sa 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 = zc20 (ZChain.f-next zc0) ; f-immediate = ZChain.f-immediate zc0 ; ¬chain∋x>z = z22 ; 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 -- no extention z22 : {a : Ordinal} → x o< osuc a → ¬ odef (ZChain.chain zc0) a z22 {a} x<oa = ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa ) 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 ) ... | no not = record { chain = zc5 ; chain⊆A = ⊆-zc5 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; ¬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 : 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 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≈ ( 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> (z01 ax (Afx ax) (case2 zc14)) (λ lt → z01 (Afx ax) ax (case1 lt) zc14) zc14 where zc14 : * x < * (f x) 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 ) ) zc11 zc10 ( fx=zc zc12 ) where 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) zc11 : * (f x) ≡ a zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx)) zc12 : odef chain x 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 ... | 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 = {!!} ; f-immediate = {!!} ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where zc0 = prev (& A) a ... | 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)) ) ) 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 (zorn03 (cf nmx) (cf-is-≤-monotonic nmx))) 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 ) → ZChain A sa f mf supO (& A) zorn03 f mf = TransFinite (ind f mf) (& A) -- 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