Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 833:3fa321cbc337
... dead end
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 23 Aug 2022 10:33:47 +0900 |
| parents | e61cbf28ec31 |
| children | 6bf0899a6150 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 39 insertions(+), 175 deletions(-) [+] |
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--- a/src/zorn.agda Mon Aug 22 22:07:02 2022 +0900 +++ b/src/zorn.agda Tue Aug 23 10:33:47 2022 +0900 @@ -649,6 +649,11 @@ SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } + record xSUP (B : HOD) (x : Ordinal) : Set n where + field + ax : odef A x + is-sup : IsSup A B ax + -- -- create all ZChains under o< x -- @@ -690,166 +695,47 @@ supf0 = ZChain.supf zc - sup1 : SUP A (UnionCF A f mf ay supf0 px) - sup1 = supP pchain pchain⊆A ptotal - sp1 = & (SUP.sup sup1 ) - supf1 : Ordinal → Ordinal - supf1 z with trio< z px - ... | tri< a ¬b ¬c = ZChain.supf zc z - ... | tri≈ ¬a b ¬c = sp1 - ... | tri> ¬a ¬b c = sp1 - - pchain1 : HOD - pchain1 = UnionCF A f mf ay supf1 x - pcy1 : odef pchain1 y - pcy1 = ⟪ ay , ch-init (init ay refl) ⟫ - pinit1 : {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁ - pinit1 {a} ⟪ aa , ua ⟫ with ua - ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where - zc7 : y <= supf1 u - zc7 = ChainP.fcy<sup is-sup (init ay refl) - pnext1 : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a) - pnext1 {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ - pnext1 {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ - ptotal1 : IsTotalOrderSet pchain1 - ptotal1 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where - uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) - - zc64 : {z : Ordinal } → supf0 z o< supf0 px → odef (UnionCF A f mf ay supf0 px) (supf0 z) - zc64 {z} sz<spx = zc73 where - z<px = ZChain.supf-inject zc sz<spx - zc70 : odef (UnionCF A f mf ay supf0 (supf0 z) ) (supf0 z) - zc70 = ZChain.csupf zc (o<→≤ z<px ) - zc73 : odef (UnionCF A f mf ay supf0 px ) (supf0 z) - zc73 with osuc-≡< (ZChain.supf-mono zc (o<→≤ z<px)) - ... | case1 eq2 = ⊥-elim ( o<¬≡ eq2 sz<spx ) - ... | case2 lt = subst (λ k → odef (UnionCF A f mf ay supf0 px ) k ) &iso ( ZChain.csupf-fc zc o≤-refl lt (init (proj1 zc70) refl) ) - - supf1<sp1 : {z : Ordinal } → supf1 z o≤ sp1 - supf1<sp1 {z} = ? where - zc50 : supf0 px ≡ sp1 - zc50 = ? -- ZChain.sup=u zc ? ? ? - zc53 : SUP A ( UnionCF A f mf ay supf0 px ) - zc53 = ZChain.sup zc o≤-refl - zc52 : supf0 px ≡ ? - zc52 = ? -- ZChain.sup=u zc ? ? ? - zc51 : supf0 sp1 ≡ sp1 - zc51 = ZChain.sup=u zc ? ? ? - -- if previous chain satisfies maximality, we caan reuse it -- -- supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x - record xSUP : Set n where - field - ax : odef A x - is-sup : IsSup A (UnionCF A f mf ay supf0 px) ax - - UnionCF⊆ : {z0 z1 : Ordinal} → (z0≤1 : z0 o≤ z1 ) → (z0≤px : z0 o< px ) → UnionCF A f mf ay supf0 z0 ⊆' UnionCF A f mf ay supf1 z1 - UnionCF⊆ {z0} {z1} z0≤1 z0<px ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ - UnionCF⊆ {z0} {z1} z0≤1 z0<px ⟪ au , ch-is-sup u1 {w} u1≤x u1-is-sup fc ⟫ = zc60 fc where - zc60 : {w : Ordinal } → FClosure A f (supf0 u1) w → odef (UnionCF A f mf ay supf1 z1 ) w - zc60 (init asp refl) with trio< u1 px | inspect supf1 u1 - ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) - record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 (ChainP.supu=u u1-is-sup) } (init (subst (λ k → odef A k ) (sym eq1) asp) eq1 ) ⟫ where - fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 ) - fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) ( ChainP.fcy<sup u1-is-sup fc ) - order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 → - (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1) - order {s} {z2} s<u1 fc with trio< s px - ... | tri< a ¬b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup zc61 fc ) where - zc61 : supf0 s o< supf0 u1 - zc61 = subst (λ k → supf0 s o< k ) eq1 s<u1 - ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> supf1<sp1 s<u1 ) - ... | tri> ¬a ¬b px<s = ⊥-elim ( o<¬≡ refl (ordtrans px<s {!!} )) -- px o< s < u1 < px - ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) - record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 {!!} } (init (subst (λ k → odef A k ) (sym eq1) {!!} ) {!!} ) ⟫ where - fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 ) - fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) {!!} -- ( ChainP.fcy<sup u1-is-sup fc ) - order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 → - (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1) - order {s} {z2} s<u1 fc with trio< s px - ... | tri< a ¬b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) {!!} - ... | tri≈ ¬a b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) {!!} -- ( ChainP.order u1-is-sup s<u1 fc ) - ... | tri> ¬a ¬b px<s = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k) b {!!} ) )) -- px o< s < u1 = px - ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with osuc-≡< (OrdTrans u1≤x (o<→≤ z0<px)) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) - ... | case2 lt = ⊥-elim ( o<> lt px<u1 ) - zc60 (fsuc w1 fc) with zc60 fc - ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ - no-extension : ¬ xSUP → ZChain A f mf ay x - no-extension ¬sp=x = record { supf = supf1 ; sup = sup ; supf-mono = {!!} - ; initial = pinit1 ; chain∋init = pcy1 ; sup=u = sup=u ; supf-is-sup = sis ; csupf = {!!} - ; chain⊆A = λ lt → proj1 lt ; f-next = pnext1 ; f-total = ptotal1 } where - UnionCFR⊆ : {z0 z1 : Ordinal} → z0 o≤ z1 → z0 o< x → UnionCF A f mf ay supf1 z0 ⊆' UnionCF A f mf ay supf0 z1 - UnionCFR⊆ {z0} {z1} z0≤1 z0<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ - UnionCFR⊆ {z0} {z1} z0≤1 z0<x ⟪ au , ch-is-sup u1 {w} u1≤x u1-is-sup fc ⟫ = zc60 fc where - zc60 : {w : Ordinal } → FClosure A f (supf1 u1) w → odef (UnionCF A f mf ay supf0 z1 ) w - zc60 {w} (init asp refl) with trio< u1 px | inspect supf1 u1 - ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) - record { fcy<sup = fcy<sup ; order = {!!} ; supu=u = trans (sym eq1) (ChainP.supu=u u1-is-sup) } (init asp refl ) ⟫ where - fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 ) - fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) eq1 ( ChainP.fcy<sup u1-is-sup fc ) - order : {s : Ordinal} {z2 : Ordinal} → s o< u1 → FClosure A f (supf0 s) z2 → - (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1) - order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s - ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) fc )) - ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) {!!} )) - ... | tri> ¬a ¬b px<s | record { eq = eq2 } = ⊥-elim ( o<¬≡ refl (ordtrans px<s (ordtrans s<u1 a) )) -- px o< s < u1 < px - ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) - record { fcy<sup = fcy<sup ; order = {!!} ; supu=u = trans (sym {!!} ) (ChainP.supu=u u1-is-sup) } (init {!!} {!!} ) ⟫ where - fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 ) - fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) {!!} ( ChainP.fcy<sup u1-is-sup fc ) - order : {s : Ordinal} {z2 : Ordinal} → s o< u1 → FClosure A f (supf0 s) z2 → - (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1) - order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s - ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) {!!}( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) fc )) - ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) {!!} ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) {!!} )) - ... | tri> ¬a ¬b px<s | _ = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k) b s<u1 ) )) -- px o< s < u1 = px - ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with trio< z0 px - ... | tri< a ¬b ¬c with osuc-≡< (OrdTrans u1≤x (o<→≤ a) ) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) - ... | case2 lt = ⊥-elim ( o<> lt px<u1 ) - zc60 (init asp refl) | tri> ¬a ¬b px<u1 | record { eq = eq1} | tri≈ ¬a' b ¬c with osuc-≡< (OrdTrans u1≤x (o≤-refl0 b) ) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) - ... | case2 lt = ⊥-elim ( o<> lt px<u1 ) - zc60 (init asp refl) | tri> ¬a ¬b px<u1 | record { eq = eq1} | tri> ¬a' ¬b' px<z0 = ⊥-elim (¬p<x<op ⟪ px<z0 , subst (λ k → z0 o< k ) (sym (Oprev.oprev=x op)) z0<x ⟫ ) - zc60 (fsuc w1 fc) with zc60 fc - ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ - sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) + no-extension : (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A pchain ? f → ZChain A f mf ay x + no-extension ¬sp=x = record { supf = supf0 ; sup = ? ; supf-mono = {!!} + ; initial = ? ; chain∋init = ? ; sup=u = ? ; supf-is-sup = ? ; csupf = {!!} + ; chain⊆A = λ lt → proj1 lt ; f-next = ? ; f-total = ? } where + pchain0=1 : ? + pchain0=1 = ? + sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf0 z) sup {z} z≤x with trio< z px - ... | tri< a ¬b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl ? ) ( ZChain.sup zc (o<→≤ a) ) - ... | tri≈ ¬a b ¬c = record { sup = SUP.sup sup1 ; as = SUP.as sup1 ; x<sup = zc61 } where - zc61 : {w : HOD} → UnionCF A f mf ay supf1 z ∋ w → (w ≡ SUP.sup sup1) ∨ (w < SUP.sup sup1) + ... | tri< a ¬b ¬c = SUP⊆ ? ( ZChain.sup zc (o<→≤ a) ) + ... | tri≈ ¬a b ¬c = record { sup = SUP.sup ? ; as = SUP.as ? ; x<sup = ? } where + zc61 : {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup ?) ∨ (w < SUP.sup ? ) zc61 {w} lt = ? -- SUP.x<sup sup1 (UnionCFR⊆ (o<→≤ z<x) z<x lt ) ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ? ⟫ ) sup=u : {b : Ordinal} (ab : odef A b) → - b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b + b o≤ x → IsSup A (UnionCF A f mf ay supf0 b) ab → supf0 b ≡ b sup=u {b} ab b≤x is-sup with trio< b px - ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl a lt) } + ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup ? } ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u zc ab (o≤-refl0 b) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl (o≤-refl0 b) lt) } - ... | tri> ¬a ¬b px<b = ⊥-elim (¬sp=x zcsup ) where + ... | tri> ¬a ¬b px<b = ? where -- ⊥-elim (¬sp=x zcsup ) where zc30 : x ≡ b zc30 with osuc-≡< b≤x ... | case1 eq = sym (eq) ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) - zcsup : xSUP - zcsup with zc30 - ... | refl = record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ (pxo≤x op) {!!} lt) } } - csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf1 b) (supf1 b) - csupf {b} b<x with trio< b px | inspect supf1 b - ... | tri< a ¬b ¬c | _ = UnionCF⊆ o≤-refl a {!!} + zcsup : ? + zcsup = ? -- with zc30 + -- ... | refl = case1 record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup ? } } + csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf0 b) (supf0 b) + csupf {b} b<x with trio< b px | inspect supf0 b + ... | tri< a ¬b ¬c | _ = ? -- UnionCF⊆ o≤-refl a {!!} ... | tri≈ ¬a refl ¬c | _ = {!!} -- UnionCF⊆ o≤-refl o≤-refl ( ZChain.csupf zc o≤-refl ) ... | tri> ¬a ¬b px<b | record { eq = eq1 } = {!!} -- UnionCF⊆ (o<→≤ px<b) o≤-refl ( ZChain.csupf zc o≤-refl ) - sis : {z : Ordinal} (z≤x : z o≤ x) → supf1 z ≡ & (SUP.sup (sup z≤x)) + sis : {z : Ordinal} (z≤x : z o≤ x) → supf0 z ≡ & (SUP.sup (sup z≤x)) sis {z} z<x with trio< z px ... | tri< a ¬b ¬c = ZChain.supf-is-sup zc (o<→≤ a ) ... | tri≈ ¬a b ¬c = {!!} ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ? ⟫ ) + zc4 : ZChain A f mf ay x zc4 with ODC.∋-p O A (* x) ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip @@ -887,36 +773,19 @@ pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z pzc z z<x = prev z z<x - ysp = & (SUP.sup (ysup f mf ay)) - record SupE ( z : Ordinal ) : Set n where + record Usupf : Set n where field - z<x : z o< x - z=supfz : z ≡ ZChain.supf (pzc z z<x) z - - psupf0 : (z : Ordinal) → Ordinal - psupf0 z with trio< z x - ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z - ... | tri≈ ¬a b ¬c = ysp - ... | tri> ¬a ¬b c = ysp - - pchain0 : HOD - pchain0 = UnionCF A f mf ay psupf0 x - - ptotal0 : IsTotalOrderSet pchain0 - ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where - uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) - uz01 = chain-total A f mf ay psupf0 ( (proj2 ca)) ( (proj2 cb)) - - usup : SUP A pchain0 - usup = supP pchain0 (λ lt → proj1 lt) ptotal0 - spu = & (SUP.sup usup) + umax : Ordinal → Ordinal + umax<x : {z : Ordinal } → umax z o< x + supf : Ordinal → Ordinal + supf z = ZChain.supf (pzc (osuc (umax z)) (ob<x lim umax<x )) z + field + supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y supf1 : Ordinal → Ordinal - supf1 z with trio< z x - ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z - ... | tri≈ ¬a b ¬c = spu - ... | tri> ¬a ¬b c = spu + supf1 z = ? + -- ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z pchain : HOD pchain = UnionCF A f mf ay supf1 x @@ -949,12 +818,7 @@ subst (λ k → UChain A f mf ay supf x k ) (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ - record xSUP : Set n where - field - ax : odef A x - not-sup : IsSup A (UnionCF A f mf ay psupf0 x) ax - - no-extension : ¬ xSUP → ZChain A f mf ay x + no-extension : ¬ ( xSUP (UnionCF A f mf ay supf1 x) x ) ∨ HasPrev A pchain ? f → ZChain A f mf ay x no-extension ¬sp=x = record { initial = pinit ; chain∋init = pcy ; supf = supf1 ; sup=u = sup=u ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; csupf = {!!} ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal } where @@ -967,7 +831,7 @@ -- UnionCF⊆ {u} u<x ⟪ A∋fc _ f mf fcu1 , ch-is-sup u1 u1≤x u1-is-sup fcu1 ⟫ -- ... | ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-init (fsuc _ fc) ⟫ -- ... | ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ - UnionCFR⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay psupf0 x + UnionCFR⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay supf1 x UnionCFR⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ UnionCFR⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init au1 refl) ⟫ = ⟪ au , ch-is-sup u1 {!!} {!!} (init {!!} {!!}) ⟫ UnionCFR⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ = {!!} -- with @@ -987,12 +851,12 @@ ... | tri> ¬a ¬b c = {!!} sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b sup=u {b} ab b<x is-sup with trio< b x - ... | tri< a ¬b ¬c = ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x<sup = {!!} } + ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x<sup = {!!} } ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x<sup = {!!} } ... | tri> ¬a ¬b c = {!!} csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 z) (supf1 z) csupf {z} z≤x with trio< z x - ... | tri< a ¬b ¬c = zc9 where + ... | tri< a ¬b ¬c = ? where zc9 : odef (UnionCF A f mf ay supf1 z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) zc9 = {!!} zc8 : odef (UnionCF A f mf ay (supfu a) z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z)