Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 626:35d8aca1a2b7
failed again
monotonicity only happens on Minimum ZChain
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 20 Jun 2022 13:47:06 +0900 |
| parents | 886e1f82cca0 |
| children | bc970dabf75e 0b5ff1c7032c |
| files | src/zorn.agda |
| diffstat | 1 files changed, 127 insertions(+), 124 deletions(-) [+] |
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--- a/src/zorn.agda Mon Jun 20 08:43:23 2022 +0900 +++ b/src/zorn.agda Mon Jun 20 13:47:06 2022 +0900 @@ -233,7 +233,11 @@ field x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) -record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) (supf : Ordinal → HOD) ( z : Ordinal ) : Set (Level.suc n) where +record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where + field + supf : Ordinal → HOD + chain-mono : {x y : Ordinal} → x o≤ y → y o≤ z → supf x ⊆' supf y + f-total : {x y : Ordinal} → x o≤ z → IsTotalOrderSet (supf x) chain : HOD chain = supf z field @@ -246,12 +250,9 @@ → HasPrev A chain ab f ∨ IsSup A chain ab → * a < * b → odef chain b -record ZChain1 ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where - field - supf : Ordinal → HOD - zc : ZChain A x f supf z - chain-mono : {x y : Ordinal} → x o≤ y → y o≤ z → supf x ⊆' supf y - f-total : {x y : Ordinal} → x o≤ z → IsTotalOrderSet (supf x) +ZChainSupUnique : ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) ( a b : Ordinal ) + → ( za : ZChain A x f a ) → (zb : ZChain A x f b ) → a o< b → i o≤ a → ZChain.supf za i ≡ ZChain.supf zb i +ZChainSupUnique = ? record Maximal ( A : HOD ) : Set (Level.suc n) where field @@ -324,27 +325,24 @@ 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) → (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) - zsup f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain1.f-total zc1 {& A} {& A} o≤-refl ) where - zc = ZChain1.zc zc1 - A∋zsup : (nmx : ¬ Maximal A ) (zc1 : ZChain1 A (& s) (cf nmx) (& A) ) - → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ))) - A∋zsup nmx zc1 = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ) ) - sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) - sp0 f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain1.f-total zc1 {& A} {& A} o≤-refl ) where - zc = ZChain1.zc zc1 + zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A (& s) f (& A) ) → SUP A (ZChain.chain zc) + zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc {& A} {& A} o≤-refl ) + A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A (& s) (cf nmx) (& 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 (& s) f (& A) ) → SUP A (ZChain.chain zc) + sp0 f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc {& A} {& A} o≤-refl ) 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) --- --- the maximum chain has fix point of any ≤-monotonic function --- - fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) - → f (& (SUP.sup (sp0 f mf zc1 ))) ≡ & (SUP.sup (sp0 f mf zc1 )) - fixpoint f mf zc1 = z14 where - zc = ZChain1.zc zc1 + fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (& s) f (& A) ) + → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) + fixpoint f mf zc = z14 where chain = ZChain.chain zc - sp1 = sp0 f mf zc1 + 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 {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) → * a < * b → odef chain b @@ -367,8 +365,8 @@ ... | 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 zc1))) ≡ & (SUP.sup (sp0 f mf zc1)) - z14 with ZChain1.f-total zc1 {& A} {& A} o≤-refl (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 + z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) + z14 with ZChain.f-total zc {& A} {& A} o≤-refl (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 ... | tri< a ¬b ¬c = ⊥-elim z16 where z16 : ⊥ z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) @@ -388,13 +386,12 @@ -- ZChain forces fix point on any ≤-monotonic function (fixpoint) -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain -- - z04 : (nmx : ¬ Maximal A ) → (zc : ZChain1 A (& s) (cf nmx) (& A)) → ⊥ - z04 nmx zc1 = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) + z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (& s) (cf nmx) (& A)) → ⊥ + z04 nmx zc = <-irr0 {* (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 (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc1 ))) -- x ≡ f x ̄ + (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) -- x ≡ f x ̄ (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where -- x < f x - zc = ZChain1.zc zc1 - sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc1 + sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc c = & (SUP.sup sp1) -- @@ -402,45 +399,41 @@ -- ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → { y₁ : Ordinal} (ay : odef A y₁) - → ZChain1 A y₁ f y) → {y : Ordinal} (ay : odef A y) → ZChain1 A y f x + → ZChain A y₁ f y) → {y : Ordinal} (ay : odef A y) → ZChain A y f x ind f mf x prev {y} ay with Oprev-p x ... | yes op = zc4 where -- -- we have previous ordinal to use induction -- - open ZChain - px = Oprev.oprev op supf : Ordinal → HOD - supf = ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) - zc1 : ZChain1 A y f (Oprev.oprev op) - zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay - zc0 : ZChain A y f (ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay)) (Oprev.oprev op) - zc0 = ZChain1.zc (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) - zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px - zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt + supf = ZChain.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) + zc : ZChain A y f (Oprev.oprev op) + zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay + zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px + zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt px<x : px o< x px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc -- if previous chain satisfies maximality, we caan reuse it -- - no-extenion : ( {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) ab → - * a < * b → odef (ZChain.chain zc0) b ) → ZChain1 A y f x - no-extenion is-max = record { supf = supf0 ; zc = record { chain⊆A = subst (λ k → k ⊆' A ) seq (ZChain.chain⊆A zc0) - ; initial = subst (λ k → {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) seq (ZChain.initial zc0) - ; f-next = subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) seq (ZChain.f-next zc0) + no-extenion : ( {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → + HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → + * a < * b → odef (ZChain.chain zc) b ) → ZChain A y f x + no-extenion is-max = record { supf = supf0 ; chain⊆A = subst (λ k → k ⊆' A ) seq (ZChain.chain⊆A zc) + ; initial = subst (λ k → {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) seq (ZChain.initial zc) + ; f-next = subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) seq (ZChain.f-next zc) ; f-immediate = subst (λ k → {x₁ : Ordinal} {y₁ : Ordinal} → odef k x₁ → odef k y₁ → - ¬ (* x₁ < * y₁) ∧ (* y₁ < * (f x₁)) ) seq (ZChain.f-immediate zc0) ; chain∋x = subst (λ k → odef k y ) seq (ZChain.chain∋x zc0) + ¬ (* x₁ < * y₁) ∧ (* y₁ < * (f x₁)) ) seq (ZChain.f-immediate zc) ; chain∋x = subst (λ k → odef k y ) seq (ZChain.chain∋x zc) ; is-max = subst (λ k → {a b : Ordinal} → odef k a → b o< osuc x → (ab : odef A b) → - HasPrev A k ab f ∨ IsSup A k ab → * a < * b → odef k b ) seq is-max } + HasPrev A k ab f ∨ IsSup A k ab → * a < * b → odef k b ) seq is-max ; chain-mono = mono ; f-total = {!!} } where supf0 : Ordinal → HOD supf0 z with trio< z x ... | tri< a ¬b ¬c = supf z - ... | tri≈ ¬a b ¬c = ZChain.chain zc0 - ... | tri> ¬a ¬b c = ZChain.chain zc0 - seq : ZChain.chain zc0 ≡ supf0 x + ... | tri≈ ¬a b ¬c = ZChain.chain zc + ... | tri> ¬a ¬b c = ZChain.chain zc + seq : ZChain.chain zc ≡ supf0 x seq with trio< x x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = refl @@ -455,7 +448,7 @@ mono {a} {b} a≤b b<ox with osuc-≡< a≤b ... | case1 refl = λ x → x ... | case2 a<b with osuc-≡< b<ox - ... | case1 b=x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x) nc00 ( ZChain1.chain-mono zc1 a≤px <-osuc ) where + ... | case1 b=x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x) nc00 ( ZChain.chain-mono zc a≤px <-osuc ) where a<x : a o< x a<x with osuc-≡< b<ox ... | case1 b=x = subst (λ k → a o< k ) b=x a<b @@ -468,38 +461,38 @@ ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b=x ) ... | case2 b<x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x ) (seq<x b<x ) - ( ZChain1.chain-mono zc1 a≤b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x ) ) + ( ZChain.chain-mono zc a≤b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x ) ) where a<x : a o< x a<x with osuc-≡< b<ox ... | case1 b=x = subst (λ k → a o< k ) b=x a<b ... | case2 b<x = ordtrans a<b b<x - zc4 : ZChain1 A y f x + zc4 : ZChain A y f x zc4 with ODC.∋-p O A (* x) - ... | no noax = no-extenion zc11 where -- ¬ A ∋ p, just skip - 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) ab → - * a < * b → odef (ZChain.chain zc0) b - zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox + ... | no noax = no-extenion zc1 where -- ¬ A ∋ p, just skip + zc1 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → + HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → + * a < * b → odef (ZChain.chain zc) b + zc1 {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 ) ) - ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b - ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf supO x - ... | case1 pr = no-extenion zc17 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next - chain0 = 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) 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 - ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) - ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) - ... | case1 is-sup = -- x is a sup of zc0 - record { zc = record { chain⊆A = {!!} ; f-next = {!!} - ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } ; supf = supf0 ; chain-mono = mono ; f-total = {!!} } where - sup0 : SUP A (ZChain.chain zc0) + ... | case2 lt = ZChain.is-max zc za (zc-b<x b lt) ab P a<b + ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf supO x + ... | case1 pr = no-extenion zc7 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next + chain0 = ZChain.chain zc + zc7 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → + HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → + * a < * b → odef (ZChain.chain zc) b + zc7 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox + ... | case2 lt = ZChain.is-max zc za (zc-b<x b lt) ab P a<b + ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc (HasPrev.ay pr)) + ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc) ax ) + ... | case1 is-sup = -- x is a sup of zc + record { chain⊆A = {!!} ; f-next = {!!} + ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; supf = supf0 ; chain-mono = mono ; f-total = {!!} } where + sup0 : SUP A (ZChain.chain zc) sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where - x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) + x21 : {y : HOD} → ZChain.chain zc ∋ y → (y ≡ * x) ∨ (y < * x) x21 {y} zy with IsSup.x<sup is-sup zy ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) @@ -507,17 +500,17 @@ sp = SUP.sup sup0 x=sup : x ≡ & sp x=sup = sym &iso - chain0 = ZChain.chain zc0 + chain0 = ZChain.chain zc sc<A : {y : Ordinal} → odef chain0 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 chain0 k) (sym &iso) zx ))) + sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc (subst (λ k → odef chain0 k) (sym &iso) zx ))) sc<A {y} (case2 fx) = subst (λ k → k o< (& A)) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fc (& sp) f mf fx )) ) schain : HOD schain = record { od = record { def = λ x → odef chain0 x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } A∋schain : {x : HOD } → schain ∋ x → A ∋ x - A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx + A∋schain (case1 zx ) = ZChain.chain⊆A zc zx A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx s⊆A : schain ⊆' A - s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx + s⊆A {x} (case1 zx) = ZChain.chain⊆A zc zx s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx cmp : {a b : HOD} (za : odef chain0 (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb @@ -534,7 +527,7 @@ a<b : a < b a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) - scmp {a} {b} (case1 za) (case1 zb) = ZChain1.f-total zc1 {px} {px} o≤-refl za zb + scmp {a} {b} (case1 za) (case1 zb) = ZChain.f-total zc {px} {px} o≤-refl za zb scmp {a} {b} (case1 za) (case2 fb) = cmp za fb scmp (case2 fa) (case1 zb) with cmp zb fa ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a @@ -542,24 +535,24 @@ ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a scmp (case2 fa) (case2 fb) = subst₂ (λ a b → Tri (a < b) (a ≡ b) (b < a ) ) *iso *iso (fcn-cmp (& sp) f mf fa fb) scnext : {a : Ordinal} → odef schain a → odef schain (f a) - scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) + scnext {x} (case1 zx) = case1 (ZChain.f-next zc zx) scnext {x} (case2 sx) = case2 ( fsuc x sx ) scinit : {x : Ordinal} → odef schain x → * y ≤ * x - scinit {x} (case1 zx) = ZChain.initial zc0 zx - scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) + scinit {x} (case1 zx) = ZChain.initial zc zx + scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) ( ZChain.chain∋x zc ) ) ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) A∋za : {a : Ordinal } → odef chain0 a → odef A a - A∋za za = ZChain.chain⊆A zc0 za + A∋za za = ZChain.chain⊆A zc za za<sup : {a : Ordinal } → odef chain0 a → ( * a ≡ sp ) ∨ ( * a < sp ) za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) za ) simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) - simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb + simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc za zb simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) - ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb + ... | case2 a<fa with za<sup ( ZChain.f-next zc za ) | s≤fc (& sp) f mf sb ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) ... | case1 fa=sp | case2 sp<b = <-irr (case2 (proj2 p )) (subst (λ k → k < * b) (sym fa=sp) (subst (λ k → k < * b ) *iso sp<b ) ) @@ -580,21 +573,21 @@ s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous 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 ) + case1 (ZChain.is-max zc za (zc-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<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup - z22 : IsSup A (ZChain.chain zc0) ab + s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc za (zc-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup + z22 : IsSup A (ZChain.chain zc) ab z22 = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } s-ismax {a} {b} (case2 sa) b<ox ab (case1 p) a<b | case2 b<x with HasPrev.ay p - ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc0 zy )) -- in previous closure of f + ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc zy )) -- in previous closure of f ... | case2 sy = case2 (subst (λ k → FClosure A f (& (* x)) k ) (sym (HasPrev.x=fy p)) (fsuc (HasPrev.y p) sy )) -- in current closure of f - s-ismax {a} {b} (case2 sa) b<ox ab (case2 p) a<b | case2 b<x = case1 z23 where -- sup o< x is already in zc0 - z24 : IsSup A schain ab → IsSup A (ZChain.chain zc0) ab + s-ismax {a} {b} (case2 sa) b<ox ab (case2 p) a<b | case2 b<x = case1 z23 where -- sup o< x is already in zc + z24 : IsSup A schain ab → IsSup A (ZChain.chain zc) ab z24 p = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } z23 : odef chain0 b - z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) - ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋x zc0 ) - ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b + z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc ) + ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋x zc ) + ... | case2 y<b = ZChain.is-max zc (ZChain.chain∋x zc ) (zc-b<x b b<x) ab (case2 (z24 p)) y<b supf0 : Ordinal → HOD supf0 z with trio< z x ... | tri< a ¬b ¬c = supf z @@ -614,29 +607,27 @@ mono {a} {b} a≤b b<ox = {!!} ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- 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) ab → - * a < * b → odef (ZChain.chain zc0) b + z18 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → + HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → + * a < * b → odef (ZChain.chain zc) 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 + ... | case2 lt = ZChain.is-max zc za (zc-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 record { 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 = record { supf = supf0 ; chain-mono = {!!} ; f-total = u-total - ; zc = record { chain⊆A = {!!} ; f-next = {!!} - ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } } where --- limit ordinal case + ... | no ¬ox = record { supf = supf0 ; chain-mono = u-mono ; f-total = u-total + ; chain⊆A = {!!} ; f-next = {!!} + ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } where --- limit ordinal case record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x field u : Ordinal u<x : u o< x - chain∋z : odef (ZChain.chain (ZChain1.zc (prev u u<x {y} ay ))) z + chain∋z : odef (ZChain.chain (prev u u<x {y} ay )) z Uz⊆A : {z : Ordinal} → UZFChain z → odef A z - Uz⊆A {z} u = ZChain.chain⊆A (ZChain1.zc ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay )) (UZFChain.chain∋z u) - uzc1 : {z : Ordinal} → (u : UZFChain z) → ZChain1 A y f (UZFChain.u u) - uzc1 {z} u = prev (UZFChain.u u) (UZFChain.u<x u) {y} ay - uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (ZChain1.supf (uzc1 u)) (UZFChain.u u) - uzc {z} u = ZChain1.zc (prev (UZFChain.u u) (UZFChain.u<x u) {y} ay) + Uz⊆A {z} u = ZChain.chain⊆A ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay ) (UZFChain.chain∋z u) + uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (UZFChain.u u) + uzc {z} u = prev (UZFChain.u u) (UZFChain.u<x u) {y} ay Uz : HOD Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = λ lt → subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (Uz⊆A lt))) } @@ -645,10 +636,10 @@ u-initial : {z : Ordinal} → odef Uz z → * y ≤ * z u-initial {z} u = ZChain.initial ( uzc u ) (UZFChain.chain∋z u) u-chain∋x : odef Uz y - u-chain∋x = record { u = y ; u<x = {!!} ; chain∋z = ZChain.chain∋x (ZChain1.zc (prev y {!!} ay )) } + u-chain∋x = record { u = y ; u<x = {!!} ; chain∋z = ZChain.chain∋x (prev y {!!} ay ) } supf0 : Ordinal → HOD supf0 z with trio< z x - ... | tri< a ¬b ¬c = ZChain1.supf (prev z a {y} ay) z + ... | tri< a ¬b ¬c = ZChain.supf (prev z a {y} ay) z ... | tri≈ ¬a b ¬c = Uz ... | tri> ¬a ¬b c = Uz seq : Uz ≡ supf0 x @@ -656,29 +647,41 @@ ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = refl - seq<x : {b : Ordinal } → (b<x : b o< x ) → ZChain1.supf (prev b b<x {y} ay) b ≡ supf0 b + seq<x : {b : Ordinal } → (b<x : b o< x ) → ZChain.supf (prev b b<x {y} ay) b ≡ supf0 b seq<x {b} b<x with trio< b x - ... | tri< a ¬b ¬c = cong (λ k → ZChain1.supf (prev b k {y} ay) b) o<-irr -- b<x ≡ a + ... | tri< a ¬b ¬c = cong (λ k → ZChain.supf (prev b k {y} ay) b) o<-irr -- b<x ≡ a ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) - u-mono : {z : Ordinal} {y : Ordinal} → z o≤ y → y o≤ x → supf0 z ⊆' supf0 y - u-mono {z} {y} z≤y y≤x with trio< z x | trio< y x - ... | tri< a ¬b ¬c | t = {!!} - ... | tri≈ ¬a b ¬c | t = {!!} - ... | tri> ¬a ¬b c | t = ⊥-elim ( osuc-< y≤x {!!} ) + ord≤< : {x y z : Ordinal} → x o< z → z o≤ y → x o< y + ord≤< {x} {y} {z} x<z z≤y with osuc-≡< z≤y + ... | case1 z=y = subst (λ k → x o< k ) z=y x<z + ... | case2 z<y = ordtrans x<z z<y + u-mono : {z : Ordinal} {w : Ordinal} → z o≤ w → w o≤ x → supf0 z ⊆' supf0 w + u-mono {z} {w} z≤w w≤x {i} with trio< z x | trio< w x + ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = um00 where -- ZChain.chain-mono (prev w ? ay) ? ? lt + um00 : odef (ZChain.supf (prev z a ay) z) i → odef (ZChain.supf (prev w a₁ ay) w) i + um00 = {!!} + um01 : odef (ZChain.supf (prev z a ay) z) i → odef (ZChain.supf (prev z {!!} ay) w) i + um01 = ZChain.chain-mono (prev z a ay) {!!} {!!} + ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = λ lt → record { u = z ; u<x = a ; chain∋z = lt } + ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = ⊥-elim ( osuc-< w≤x c ) + ... | tri≈ ¬a z=x ¬c | tri< w<x ¬b ¬c₁ = ⊥-elim ( osuc-< z≤w (subst (λ k → w o< k ) (sym z=x) w<x ) ) + ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ = λ lt → lt + ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim ( osuc-< w≤x c ) -- o<> c ( ord≤< w≤x )) -- z≡w x o< w + ... | tri> ¬a ¬b c | t = ⊥-elim ( osuc-< w≤x (ord≤< c z≤w ) ) -- x o< z → x o< w u-total : {z : Ordinal} → z o≤ x → IsTotalOrderSet (supf0 z) u-total {z} z<x ux uy with trio< z x ... | t = {!!} -- with trio< (UZFChain.u ux) (UZFChain.u uy) - -- ... | tri< a ¬b ¬c = ZChain1.f-total (uzc1 uy) {!!} (u-mono (UZFChain.u ux) (UZFChain.u uy) - -- (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc1 ux) (uzc1 uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy) - -- ... | tri≈ ¬a b ¬c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) - -- (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) - -- ... | tri> ¬a ¬b c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) - -- (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) + -- ... | tri< a ¬b ¬c = ZChain.f-total (uzc uy) {!!} (u-mono (UZFChain.u ux) (UZFChain.u uy) + -- (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc ux) (uzc uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy) + -- ... | tri≈ ¬a b ¬c = ZChain.f-total (uzc ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) + -- (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc uy) (uzc ux) (UZFChain.chain∋z uy)) + -- ... | tri> ¬a ¬b c = ZChain.f-total (uzc ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) + -- (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc uy) (uzc ux) (UZFChain.chain∋z uy)) - SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ya : odef A y) → ZChain1 A y f (& A) - SZ f mf = TransFinite {λ z → {y : Ordinal } → (ay : odef A y ) → ZChain1 A y f z } (ind f mf) (& A) + SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ya : odef A y) → ZChain A y f (& A) + SZ f mf = TransFinite {λ z → {y : Ordinal } → (ay : odef A y ) → ZChain A y f z } (ind f mf) (& A) zorn00 : Maximal A zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM @@ -696,7 +699,7 @@ 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) ) ⟫ - zorn04 : ZChain1 A (& s) (cf nmx) (& A) + zorn04 : ZChain A (& s) (cf nmx) (& A) zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) -- usage (see filter.agda )