Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/zorn.agda @ 537:e12add1519ec
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sun, 24 Apr 2022 08:04:42 +0900 |
parents | c43375ade2c5 |
children | 854908eb70f2 |
line wrap: on
line source
{-# 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 : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where field y : Ordinal ay : odef A 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 f-total : IsTotalOrderSet chain f-next : {a : Ordinal } → odef chain a → odef chain (f a) is-max : {a b : Ordinal } → (ca : odef chain a ) → odef A b → a o< z → ( Prev< A (incl chain⊆A (subst (λ k → odef chain k ) (sym &iso) ca)) 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) 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 = ⊥-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 larger element, so it is maximal ... | 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 ) -- zsup zc f mf = & ( SUP.sup (supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc f mf ) ) ) 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 ) ) z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) → f (& ( SUP.sup (zsup f mf zc ))) ≡ & (SUP.sup (zsup f mf zc )) z03 f mf zc = {!!} 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 (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ))))) (A∋zsup nmx zc ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) (proj1 (cf-is-<-monotonic nmx c ((subst λ k → odef A k ) &iso (A∋zsup nmx zc )))) where c = & (SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc )) -- 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 = ZChain.f-next zc0 ; is-max = {!!} } ... | 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) -- 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 = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; is-max = {!!} } 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 = ZChain.f-next zc0 ; is-max = {!!} } where zc0 = prev (& A) a ... | tri≈ ¬a b ¬c = {!!} ... | tri> ¬a ¬b c = {!!} 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