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 | Tue, 12 Apr 2022 14:10:44 +0900 |
parents | 8ec0b88b022f |
children | a97a1f1e27fa |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level 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 record Element (A : HOD) : Set (suc n) where field elm : HOD is-elm : A ∋ elm open Element IsPartialOrderSet : ( A : HOD ) → Set (suc n) IsPartialOrderSet A = IsStrictPartialOrder _≡A_ _<A_ where _<A_ : (x y : Element A ) → Set n x <A y = elm x < elm y _≡A_ : (x y : Element A ) → Set (suc n) x ≡A y = elm x ≡ elm y 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 (suc n) IsTotalOrderSet A = IsStrictTotalOrder _≡A_ _<A_ where _<A_ : (x y : Element A ) → Set n x <A y = elm x < elm y _≡A_ : (x y : Element A ) → Set (suc n) x ≡A y = elm x ≡ elm y me : { A a : HOD } → A ∋ a → Element A me {A} {a} lt = record { elm = a ; is-elm = lt } record SUP ( A B : HOD ) : Set (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 record Maximal ( A : HOD ) : Set (suc n) where field maximal : HOD A∋maximal : A ∋ maximal ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative record ZChain ( A : HOD ) (y : Ordinal) : Set (suc n) where field max : HOD A∋max : A ∋ max y<max : y o< & max chain : HOD chain⊆A : chain ⊆ A total : IsTotalOrderSet chain chain-max : (x : HOD) → chain ∋ x → (x ≡ max ) ∨ ( x < max ) data IChain (A : HOD) : Ordinal → Set n where ifirst : {ox : Ordinal} → odef A ox → IChain A ox inext : {ox oy : Ordinal} → odef A oy → * ox < * oy → IChain A ox → IChain A oy ic-connet : {A : HOD} {x : Ordinal} → (x : IChain A x) → Ordinal → Set n ic-connet {A} (ifirst {ox} ax) oy = ox ≡ oy ic-connet {A} (inext {ox} {oy} ay x<y iy) oz = (ox ≡ oz) ∨ ic-connet iy oz IChainSet : {A : HOD} → Element A → HOD IChainSet {A} ax = record { od = record { def = λ y → odef A y ∧ ( (iy : IChain A y ) → ic-connet iy (& (elm ax))) } ; odmax = & A ; <odmax = λ {y} lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k) (sym &iso) (proj1 lt))) } -- there is a y, & y > & x record Sup> {A : HOD} (x : Element A) : Set (suc n) where field y : Element A y>x : IChainSet x ∋ elm y → & (elm x) o< & (elm y) -- finite IChain record InFiniteIChain {A : HOD} (x : Element A) (z : Ordinal) : Set (suc n) where field ny : (y : Element A ) → & (elm y) o< z → Element A y>x : {y : Element A} → (lt : & (elm y) o< z )→ IChainSet x ∋ elm y → elm y < elm (ny y lt ) Zorn-lemma-case : { A : HOD } → o∅ o< & A → IsPartialOrderSet A → (x : Element A) → Sup> x ∨ Dec ( InFiniteIChain x (& A)) Zorn-lemma-case {A} 0<A PO x = zc2 where Gtx : HOD Gtx = record { od = record { def = λ y → odef ( IChainSet x ) y ∧ ( & (elm x) o< y ) } ; odmax = & A ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k) (sym &iso) (proj1 (proj1 lt)))) } zc2 : Sup> x ∨ Dec (InFiniteIChain x (& A)) zc2 with is-o∅ (& Gtx) ... | no not = case1 record { y = record { elm = ODC.minimal O Gtx (λ eq → not (=od∅→≡o∅ eq) ); is-elm = {!!} } ; y>x = {!!} } ... | yes nogt = case2 {!!} where zc3 : {y : Element A} → IChainSet x ∋ elm y → ¬ (& (elm x) o< & (elm y)) zc3 = {!!} zcind : (z : Ordinal ) → ((y : Ordinal) → y o< z → Dec (InFiniteIChain x y ) ) → Dec (InFiniteIChain x z) zcind z prev = {!!} zc4 : Dec (InFiniteIChain x (& A)) zc4 = TransFinite zcind (& A) 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 someA : HOD someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) isSomeA : A ∋ someA isSomeA = 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 y → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))} ; odmax = & A ; <odmax = z08 } where z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P))) no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥ no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ ) Gtx : { x : HOD} → A ∋ x → HOD Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P))) 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) -- ZChain is not compatible with the SUP condition record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where field bx : Ordinal bx<y : bx o< y is-fb : x ≡ & (fb bx ) bx<A : (z : ZChain A (& A) ) → {x : Ordinal } → (bx : BX x (& A) {!!}) → BX.bx bx o< & A bx<A z {x} bx = BX.bx<y bx z12 : (z : ZChain A (& A) ) → {y : Ordinal} → BX y (& A) {!!} → y o< & A z12 z {y} bx = subst (λ k → k o< & A) (sym (BX.is-fb bx)) (c<→o< {!!}) B : (z : ZChain A (& A) ) → HOD B z = {!!} z11 : (z : ZChain A (& A) ) → (x : Element (B z)) → elm x ≡ {!!} z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) ) obx : (z : ZChain A (& A) ) → {x : HOD} → B z ∋ x → Ordinal obx z {x} bx = BX.bx bx obx=fb : (z : ZChain A (& A) ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ {!!} obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx)) B⊆A : (z : ZChain A (& A) ) → B z ⊆ A B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) {!!} } -- PO-B : (z : ZChain A (& A) ) → IsPartialOrderSet (B z) _<_ -- PO-B z = ⊆-IsPartialOrderSet (B⊆A z) PO bx-monotonic : (z : ZChain A (& A) ) → {x y : Element (B z)} → obx z (is-elm x) o< obx z (is-elm y) → elm x < elm y bx-monotonic z {x} {y} a = subst₂ (λ j k → j < k ) (sym (z11 z x)) (sym (z11 z y)) {!!} bcmp : (z : ZChain A (& A) ) → Trichotomous (λ (x : Element (B z)) y → elm x ≡ elm y ) (λ x y → elm x < elm y ) bcmp z x y with trio< (obx z (is-elm x)) (obx z (is-elm y)) ... | tri< a ¬b ¬c = tri< z15 (λ eq → z01 (incl (B⊆A z) (is-elm y)) (incl (B⊆A z) (is-elm x))(case1 (sym eq)) z15 ) z17 where z15 : elm x < elm y z15 = bx-monotonic z {x} {y} a z17 : elm y < elm x → ⊥ z17 lt = z01 (incl (B⊆A z) (is-elm y)) (incl (B⊆A z) (is-elm x))(case2 lt) z15 ... | tri≈ ¬a b ¬c = tri≈ (IsStrictPartialOrder.irrefl PO {isA (B⊆A z) x} {isA (B⊆A z) y} z14) z14 z16 where z14 : elm x ≡ elm y z14 = {!!} z16 = IsStrictPartialOrder.irrefl PO {isA (B⊆A z) y} {isA (B⊆A z) x} (sym z14) ... | tri> ¬a ¬b c = tri> z17 (λ eq → z01 (incl (B⊆A z) (is-elm x)) (incl (B⊆A z) (is-elm y))(case1 eq) z15 ) z15 where z15 : elm y < elm x z15 = bx-monotonic z {y} {x} c z17 : elm x < elm y → ⊥ z17 lt = z01 (incl (B⊆A z) (is-elm x)) (incl (B⊆A z) (is-elm y))(case2 lt) z15 B-is-total : (z : ZChain A (& A) ) → IsTotalOrderSet (B z) B-is-total zc = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {x} {y} {z} x<y y<z → IsStrictPartialOrder.trans PO {isA (B⊆A zc) x} {isA (B⊆A zc) y} {isA (B⊆A zc) z} x<y y<z ; compare = bcmp zc } ind : (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y ∨ Maximal A ) → ZChain A x ∨ Maximal A -- has previous ordinal -- has maximal use this -- else has chain -- & A < y A is a counter example of assumption -- chack y is maximal -- y < max use previous chain -- y = max ( y > max cannot happen ) -- ¬ A ∋ y use previous chain -- A ∋ y is use oridinaly min of y or previous -- y is limit ordinal -- has maximal in some lower use this -- no maximal in all lower -- & A < y A is a counter example of assumption -- A ∋ y is maximal use this -- ¬ A ∋ y use previous chain -- check some y ≤ max -- if none A < y is the counter example -- else use the ordinaly smallest max as next chain ind 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 zc1 : ZChain A x ∨ Maximal A zc1 with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) ... | case2 x = case2 x -- we have the Maximal ... | case1 z with trio< x (& (ZChain.max z)) ... | tri< a ¬b ¬c = case1 record { max = ZChain.max z ; y<max = a } ... | tri≈ ¬a b ¬c = {!!} -- x = max so ¬ A ∋ max ... | tri> ¬a ¬b c = {!!} -- can't happen ... | yes ax = zc1 where -- we have previous ordinal and A ∋ x px = Oprev.oprev op zc1 : ZChain A x ∨ Maximal A zc1 with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) ... | case2 x = case2 x ... | case1 x with is-o∅ ( & (Gtx ax )) ... | yes no-sup = case2 {!!} ... | no sup = case1 {!!} ind x prev | no ¬ox with trio< (& A) x --- limit ordinal case ... | tri< a ¬b ¬c = {!!} where zc1 : ZChain A (& A) zc1 with prev (& A) a ... | t = {!!} ... | tri≈ ¬a b ¬c = {!!} where ... | tri> ¬a ¬b c with ODC.∋-p O A (* x) ... | no ¬Ax = {!!} where ... | yes ax with is-o∅ (& (Gtx ax)) ... | yes nogt = ⊥-elim {!!} where -- no larger element, so it is maximal x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m) x-is-maximal m am = ¬x<m where ¬x<m : ¬ (* x < * m) ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) ... | no not = {!!} where zorn03 : (x : Ordinal) → ZChain A x ∨ Maximal A zorn03 x with TransFinite ind x ... | t = {!!} zorn04 : Maximal A zorn04 with zorn03 (& A) ... | case1 chain = {!!} ... | case2 m = m zorn00 : Maximal A zorn00 with is-o∅ ( & HasMaximal ) ... | 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 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) hasm = 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 hasm zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = {!!} where -- if we have no maximal, make ZChain, which contradict SUP condition z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x ∨ Maximal A z x nomx with TransFinite {!!} x ... | t = {!!} -- _⊆'_ : ( 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