Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 759:944f50265914
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 24 Jul 2022 19:01:24 +0900 |
| parents | a2947dfff80d |
| children | 0dc7999b1d50 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 12 insertions(+), 28 deletions(-) [+] |
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--- a/src/zorn.agda Sun Jul 24 16:40:35 2022 +0900 +++ b/src/zorn.agda Sun Jul 24 19:01:24 2022 +0900 @@ -451,8 +451,8 @@ px = Oprev.oprev op 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 - fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u - fcy<sup {u} {w} u≤x fc = ? + -- fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u + -- fcy<sup {u} {w} u≤x fc = ? is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → b o< x → (ab : odef A b) → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → @@ -477,7 +477,7 @@ m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b m04 = ZChain1.is-max (prev px px<x) m03 b<px ab (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono2 x ( o<→≤ (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )) o≤-refl lt) } ) a<b - ... | tri≈ ¬a b=px ¬c = ? -- b = px case, u = px u< osuc x + ... | tri≈ ¬a b=px ¬c = ⟪ ab , ch-is-sup b (o<→≤ b<x) ? (subst (λ k → FClosure A f k b) ? (init ab)) ⟫ -- b = px case, u = px u< osuc x ... | no lim = record { is-max = is-max } where is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → b o< x → (ab : odef A b) → @@ -488,12 +488,14 @@ ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA (chain-mono2 (osuc x) (o<→≤ <-osuc ) o≤-refl ua )) ) (subst (λ k → * a < * k ) (sym b=y) a<b ) ) ... | case2 y<b = chain-mono2 x (o<→≤ (ob<x lim b<x) ) o≤-refl m04 where + m09 : b o< & A + m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) m07 : {z : Ordinal} → FClosure A f y z → z << ZChain.supf zc b - m07 {z} fc = ZChain.fcy<sup zc ? fc + m07 {z} fc = ZChain.fcy<sup zc m09 fc m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b - m08 {sup1} {z1} s<b fc = ZChain.order zc ? ? fc + m08 {sup1} {z1} s<b fc = ZChain.order zc m09 (o<→≤ s<b) fc m05 : b ≡ ZChain.supf zc b - m05 = sym (ZChain.sup=u zc {_} {ab} ? + m05 = sym (ZChain.sup=u zc {_} {ab} m09 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono2 x (o<→≤ (ob<x lim b<x)) o≤-refl lt )} ) -- ZChain on x m06 : ChainP A f mf ay (ZChain.supf zc) b b m06 = record { fcy<sup = m07 ; csupz = subst (λ k → FClosure A f k b ) m05 (init ab) ; order = m08 @@ -585,14 +587,16 @@ isupf z = & (SUP.sup (ysup f mf ay)) cy : odef (UnionCF A f mf ay isupf o∅) y cy = ⟪ ay , ch-init (init ay) ⟫ + y<sup : * y ≤ SUP.sup (ysup f mf ay) + y<sup = SUP.x<sup (ysup f mf ay) (subst (λ k → FClosure A f y k ) (sym &iso) (init ay)) isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z isy {z} ⟪ az , uz ⟫ with uz ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u u≤x is-sup fc = ⊥-elim ( <-irr (case1 refl) ? ) + ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (subst (λ k → * y ≤ k) (sym *iso) y<sup) (s≤fc (& (SUP.sup (ysup f mf ay))) f mf fc) inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) inext {a} ua with (proj2 ua) ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-init (fsuc _ fc ) ⟫ - ... | ch-is-sup u u≤x is-sup fc = ⊥-elim ( <-irr (case1 refl) ? ) + ... | ch-is-sup u u≤x is-sup fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-is-sup u u≤x (ChainP-next A f mf ay isupf is-sup) (fsuc _ fc) ⟫ itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) itotal {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) ) @@ -792,26 +796,6 @@ ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z ... | tri≈ ¬a b ¬c = x ... | tri> ¬a ¬b c = x - is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay psupf1 x) a ) → b o< x → (ab : odef A b) - → HasPrev A (UnionCF A f mf ay psupf1 x) ab f ∨ IsSup A (UnionCF A f mf ay psupf1 x) ab - → * a < * b → odef ((UnionCF A f mf ay psupf1 x)) b - is-max {a} {b} ua b<x ab (case1 hasp) a<b = is-max-hp psupf1 x ua b<x ab hasp a<b - is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay ) - ... | case1 b=y = ⊥-elim ( <-irr ? - (subst (λ k → * a < * k ) (sym b=y) a<b ) ) - ... | case2 y<b = m04 where - m07 : {z : Ordinal} → FClosure A f y z → z << psupf1 b - m07 {z} fc = ? - m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (psupf1 sup1) z1 → z1 << psupf1 b - m08 {sup1} {z1} s<b fc = ? - m05 : b ≡ psupf1 b - m05 = ? - m06 : ChainP A f mf ay psupf1 b b - m06 = record { fcy<sup = m07 ; csupz = subst (λ k → FClosure A f k b ) m05 (init ab) ; order = m08 ; supfu=u = sym m05 } - m04 : odef (UnionCF A f mf ay psupf1 x) b - m04 = ⟪ ab , ch-is-sup b ? m06 (subst (λ k → FClosure A f k b) m05 (init ab)) ⟫ - - ... | case2 ¬x=sup = no-extension -- x is not f y' nor sup of former ZChain from y -- no extention SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A)