Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 575:3abf55c8e295
f-next seems bad
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 05 Jun 2022 19:50:18 +0900 |
| parents | 9084a26445a7 |
| children | 59c11152f604 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 28 insertions(+), 23 deletions(-) [+] |
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--- a/src/zorn.agda Sun Jun 05 12:49:48 2022 +0900 +++ b/src/zorn.agda Sun Jun 05 19:50:18 2022 +0900 @@ -248,26 +248,7 @@ 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 ab → * a < * b → odef chain b - fc∨sup : {a : Ordinal } → a o< osuc z → ( ca : odef chain a ) → HasPrev A chain ( chain⊆A ca) f ∨ IsSup A chain ( chain⊆A ca) - -data ZC (A : HOD) (f : Ordinal → Ordinal ) : Ordinal → Set n where - zc-init : ZC A f o∅ - zc-fc : {s x : Ordinal} → ZC A f s → FClosure A f s x → ZC A f x - zc-sup : {s x : Ordinal} → ZC A f s → * s < * x → ZC A f x - -ZC→ZChain : (A : HOD) {x z : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) - → ZC A f z → (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) → ZChain A ax f mf sup z -ZC→ZChain A {x} {z} ax f mf zc sup = record { - chain = ? - ; chain⊆A = ? - ; chain∋x = ? - ; initial = ? - ; f-total = ? - ; f-next = ? - ; f-immediate = ? - ; is-max = ? - ; fc∨sup = ? - } + fc∨sup : {a : Ordinal } → a o< osuc z → ( ca : odef chain a ) → IsSup A chain ( chain⊆A ca) ∨ HasPrev A chain ( chain⊆A ca) f record Maximal ( A : HOD ) : Set (Level.suc n) where field @@ -428,13 +409,20 @@ ... | no noax = -- ¬ A ∋ p, just skip record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 ; 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 ; fc∨sup = {!!} } where -- no extention + ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 ; fc∨sup = zc12 } 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) 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 ) ) ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b + zc12 : {a : Ordinal} → a o< osuc x → (ca : odef (ZChain.chain zc0) a) → + IsSup A (ZChain.chain zc0) (ZChain.chain⊆A zc0 ca) ∨ + HasPrev A (ZChain.chain zc0) (ZChain.chain⊆A zc0 ca) f + zc12 {a} b<ox ca with osuc-≡< b<ox + ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) (ZChain.chain⊆A zc0 ca) ) ) + ... | case2 lt = ZChain.fc∨sup zc0 (zc0-b<x a lt) ca ... | 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 ay f mf supO x ... | 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 @@ -444,13 +432,19 @@ 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 chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) + zc10 : {a : Ordinal} → a o< osuc x → (ca : odef (ZChain.chain zc0) a) → + IsSup A (ZChain.chain zc0) (ZChain.chain⊆A zc0 ca) ∨ HasPrev A (ZChain.chain zc0) (ZChain.chain⊆A zc0 ca) f + zc10 {a} a<x ca with osuc-≡< a<x + ... | case2 lt = ZChain.fc∨sup zc0 (zc0-b<x a lt) ca + ... | case1 refl = case2 record { y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (sym &iso) (HasPrev.x=fy pr) } 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 -- no extention - ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 ; fc∨sup = {!!}} + ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 + ; fc∨sup = zc10 } ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) ... | case1 is-sup = -- x is a sup of zc0 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 = s-ismax ; fc∨sup = {!!}} where + ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax ; fc∨sup = zc19 } where sup0 : SUP A (ZChain.chain zc0) sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) @@ -473,6 +467,17 @@ s⊆A : schain ⊆' A s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx + zc19 : {a : Ordinal} → a o< osuc x → (sa : odef schain a) → + IsSup A schain (s⊆A sa) ∨ HasPrev A schain (s⊆A sa) f + zc19 a<x (case2 (init asp)) = case1 {!!} + zc19 a<x (case2 (fsuc y fc)) = case2 record { y = y ; ay = case2 fc ; x=fy = refl } + zc19 {a} a<x (case1 ca) with osuc-≡< a<x + ... | case1 refl = case1 record { x<sup = {!!} } -- a ≡ x + ... | case2 lt with ZChain.fc∨sup zc0 (zc0-b<x a lt) ca + ... | case2 c1 = case2 record { y = HasPrev.y c1 ; ay = case1 (HasPrev.ay c1) ; x=fy = HasPrev.x=fy c1 } + ... | case1 c2 = case1 record { x<sup = {!!} } where -- a < x + zc190 : {y : Ordinal} → odef chain y → (y ≡ a) ∨ (y << a) + zc190 = IsSup.x<sup c2 cmp : {a b : HOD} (za : odef chain (& 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 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where