Mercurial > hg > Members > kono > Proof > ZF-in-agda
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total of B
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 09 Apr 2022 13:56:49 +0900 |
parents | 4203ba14fd53 |
children | 2a8629b5cff9 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module zorn {n : Level } (O : Ordinals {n}) where open import zf open import logic -- open import partfunc {n} O import OD 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 ) → (_<_ : (x y : HOD) → Set n ) → 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 → {_<_ : (x y : HOD) → Set n } → IsPartialOrderSet A _<_ → IsPartialOrderSet B _<_ ⊆-IsPartialOrderSet {A} {B} B⊆A {_<_} PA = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = λ eq → case1 eq ; trans = λ {x} {y} {z} → trans1 {x} {y} {z} ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; trans = trans1 ; <-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 ) → (_<_ : (x y : HOD) → Set n ) → 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 ) (_<_ : (x y : HOD) → Set n ) : 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 ) (_<_ : (x y : HOD) → Set n ) : 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) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where field fb : (x : Ordinal ) → HOD A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox total : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ( ox ≡ oz ) ∨ ( fb ox < fb oz ) ∨ ( fb oz < fb ox ) monotonic : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ox o< oz → fb ox < fb oz Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n } → 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) ( ZChain.fb z )) → BX.bx bx o< & A bx<A z {x} bx = BX.bx<y bx z12 : (z : ZChain A (& A) _<_ ) → {y : Ordinal} → BX y (& A) (ZChain.fb z) → y o< & A z12 z {y} bx = subst (λ k → k o< & A) (sym (BX.is-fb bx)) (c<→o< (ZChain.A∋fb z (BX.bx bx) (BX.bx<y bx))) B : (z : ZChain A (& A) _<_ ) → HOD B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; <odmax = z12 z } z11 : (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-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 ≡ ZChain.fb z (BX.bx bx) 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)) (ZChain.A∋fb z (BX.bx bx) (BX.bx<y 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)) (ZChain.monotonic z (bx<A z (is-elm x)) (bx<A z (is-elm y)) a ) 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 = begin elm x ≡⟨ obx=fb z (is-elm x) ⟩ ZChain.fb z (BX.bx (is-elm x)) ≡⟨ cong (ZChain.fb z) b ⟩ ZChain.fb z (BX.bx (is-elm y)) ≡⟨ sym ( obx=fb z (is-elm y)) ⟩ elm y ∎ where open ≡-Reasoning 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 } ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ ) ZChain→¬SUP z sp = ⊥-elim {!!} where z03 : & (SUP.sup sp) o< osuc (& A) z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥ z02 x xe s<x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x<sup sp xe) s<x ind : HasMaximal =h= od∅ → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ ) → ZChain A x _<_ ind nomx x prev with Oprev-p x ... | yes op with ODC.∋-p O A (* x) ... | no ¬Ax = {!!} where -- we have previous ordinal and ¬ A ∋ x, use previous Zchain px = Oprev.oprev op zc1 : ZChain A px _<_ zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as z04 sup as s<x with trio< (& sup) x ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) ) ... | tri< a ¬b ¬c = {!!} -- ZChain.¬x<sup zc1 _ as ( subst (λ k → & sup o< k ) (sym (Oprev.oprev=x op)) a ) ... | tri> ¬a ¬b c with osuc-≡< s<x ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) ) ... | case2 lt = ⊥-elim (¬a lt ) ... | yes ax = z06 where -- we have previous ordinal and A ∋ x px = Oprev.oprev op zc1 : ZChain A px _<_ zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) z06 : ZChain A x _<_ z06 with is-o∅ (& (Gtx ax)) ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) 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 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case ... | tri< a ¬b ¬c = {!!} where zc1 : ZChain A (& A) _<_ zc1 = prev (& A) a ... | 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 (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) 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 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 = ⊥-elim {!!} where -- if we have no maximal, make ZChain, which contradict SUP condition z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_ z x nomx = TransFinite (ind nomx) x _⊆'_ : ( 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