Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 570:c642cbafc07a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 02 May 2022 12:43:42 +0900 |
| parents | 33b1ade17f83 |
| children | 2ade91846f57 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 48 insertions(+), 26 deletions(-) [+] |
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--- a/src/zorn.agda Mon May 02 00:44:43 2022 +0900 +++ b/src/zorn.agda Mon May 02 12:43:42 2022 +0900 @@ -54,6 +54,15 @@ -- Partial Order on HOD ( possibly limited in A ) -- +_<<_ : (x y : Ordinal ) → Set n +x << y = * x < * y + +POO : IsStrictPartialOrder _≡_ _<<_ +POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } + ; trans = IsStrictPartialOrder.trans PO + ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y + ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } + _≤_ : (x y : HOD) → Set (Level.suc n) x ≤ y = ( x ≡ y ) ∨ ( x < y ) @@ -222,14 +231,11 @@ ay : odef B y x=fy : x ≡ f y -record IsSup (A B : HOD) (T : IsTotalOrderSet B) ( B⊆A : B ⊆' A ) - {x : Ordinal } (xa : odef A x) (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) ( f : Ordinal → Ordinal ) : Set n where +record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where field chain : Ordinal chain⊆B : (* chain) ⊆' B - x=sup : x ≡ sup (* chain) ( λ lt → B⊆A (chain⊆B lt ) ) - ( ⊆-IsTotalOrderSet {B} {* chain} record { incl = chain⊆B } T ) - -- ¬prev : ¬ HasPrev A (* chain) xa f + x<sup : {y : Ordinal} → odef (* chain) y → (y ≡ x ) ∨ (y << x ) record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where @@ -242,7 +248,7 @@ 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< osuc z → (ab : odef A b) - → HasPrev A chain ab f ∨ IsSup A chain f-total chain⊆A ab sup f -- ((sup chain chain⊆A f-total) ≡ b ) + → HasPrev A chain ab f ∨ IsSup A chain ab -- ((sup chain chain⊆A f-total) ≡ b ) → * a < * b → odef chain b record Maximal ( A : HOD ) : Set (Level.suc n) where @@ -333,7 +339,7 @@ chain = ZChain.chain zc sp1 = sp0 f mf zc z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) - → HasPrev A chain ab f ∨ IsSup A chain (ZChain.f-total zc) (ZChain.chain⊆A zc) {b} ab supO f -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) + → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) → * a < * b → odef chain b z10 = ZChain.is-max zc z11 : & (SUP.sup sp1) o< & A @@ -342,11 +348,18 @@ 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 ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) - (case2 record { chain = & chain ; chain⊆B = λ z → subst (λ k → odef k _ ) *iso z ; x=sup = cong (&) (sup== (sym *iso)) } ) z13 where + (case2 z19 ) z13 where z13 : * (& s) < * (& (SUP.sup sp1)) z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc ) ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt + z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) + z19 = record { chain = & chain ; chain⊆B = λ z → subst (λ k → odef k _ ) *iso z ; x<sup = z20 } where + z20 : {y : Ordinal} → odef (* (& chain)) y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) + z20 {y} zy with SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) *iso (sym &iso) zy) + ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) + ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) + -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } 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 @@ -399,7 +412,7 @@ ; 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< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → + HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) @@ -408,7 +421,7 @@ ... | 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< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → + HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b @@ -416,12 +429,19 @@ 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 ; initial = ZChain.initial 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 (IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ax supO f) - ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain + ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) + ... | case1 is-sup = -- previous ordinal is a sup of a smaller ZChain record { chain = schain ; chain⊆A = s⊆A ; f-total = scmp ; f-next = scnext - ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup - sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) + ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax } where -- x is sup + -- sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) + -- sp = SUP.sup sup0 + -- x=sup : IsSup A (ZChain.chain zc0) {& (* x)} ax → x ≡ & sp -- sup is not minimum, so this may wrong + sup0 : SUP A (ZChain.chain zc0) + sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = {!!} } + sp : HOD sp = SUP.sup sup0 + x=sup : x ≡ & sp + x=sup = sym &iso chain = ZChain.chain zc0 sc<A : {y : Ordinal} → odef chain y ∨ FClosure A f (& sp) y → y o< & A sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc0 (subst (λ k → odef chain k) (sym &iso) zx ))) @@ -488,33 +508,35 @@ ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) → - HasPrev A schain ab f ∨ IsSup A schain scmp s⊆A ab (λ C C⊆A TC → & (SUP.sup (supP C C⊆A TC))) f + HasPrev A schain ab f ∨ IsSup A schain ab → * a < * b → odef schain b - s-ismax {a} {b} (case1 za) b<x ab (case1 p) a<b with osuc-≡< b<x - ... | case1 b=x = case2 {!!} -- (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) - ... | case2 b<x = z21 p where + s-ismax {a} {b} (case1 za) b<ox ab P a<b with osuc-≡< b<ox | ODC.p∨¬p O (HasPrev A schain ab f)-- b is some previous + ... | case1 b=x | _ = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) + ... | case2 b<x | case1 p = z21 p where z21 : HasPrev A schain ab f → odef schain b z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) - s-ismax {a} {b} (case1 za) b<x ab (case2 p) a<b with osuc-≡< b<x - ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) {!!} (init (SUP.A∋maximal sup0) )) - ... | case2 b<x = {!!} where - z22 : IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → odef schain b - z22 p = {!!} - -- case1 (ZChain.is-max zc0 za (zc0-b<x b lt) ab {!!} a<b ) + s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x | case2 ¬pr = ⊥-elim ( ¬pr p ) + s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x | case2 ¬pr = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where + -- cahin of IsSup A schain ab may larger than chain of zc0 if it has a previous but it is not + z23 : * (IsSup.chain p) ⊆' ZChain.chain zc0 + z23 = {!!} + z22 : IsSup A (ZChain.chain zc0) ab + z22 = record { chain = IsSup.chain p ; chain⊆B = z23 ; x<sup = {!!} } s-ismax {a} {b} (case2 sa) b<x ab p a<b = {!!} ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 } where -- no extention z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) (ZChain.f-total zc0) (ZChain.chain⊆A zc0) ab supO f → + HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab p a<b ... | case1 b=x with p ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (trans &iso (sym b=x) ) (HasPrev.x=fy pr ) } ) - ... | case2 b=sup = ⊥-elim ( ¬x=sup {!!} ) + ... | case2 b=sup = ⊥-elim ( ¬x=sup record { chain = IsSup.chain b=sup ; chain⊆B = IsSup.chain⊆B b=sup + ; x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) ... | no ¬ox = {!!} where --- limit ordinal case record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x field