Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 682:663b34227faf
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 09 Jul 2022 21:51:39 +0900 |
| parents | c5c8e37d9a6c |
| children | 6814fc4e7998 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 70 insertions(+), 61 deletions(-) [+] |
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--- a/src/zorn.agda Sat Jul 09 18:36:23 2022 +0900 +++ b/src/zorn.agda Sat Jul 09 21:51:39 2022 +0900 @@ -465,33 +465,23 @@ ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → (zc0 : (x : Ordinal) → ZChain1 A f mf ay x) → ((z : Ordinal) → z o< x → ZChain A f mf ay zc0 z) → ZChain A f mf ay zc0 x - ind f mf {y} ay x zc0 prev = zc4 where - zc : {z1 : Ordinal} → z1 o< x → ZChain A f mf ay zc0 z1 - zc z1 with Oprev-p x - ... | yes op = ? where - -- - -- we have previous ordinal to use induction - -- - px = Oprev.oprev op - supf : Ordinal → HOD - supf x = ? -- ChainF A f mf ay x zc0 - -- zc : ZChain A f mf ay zc0 (Oprev.oprev op) - -- zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) - px<x : px o< x - px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc - ... | no ¬ox = ? where - supf : Ordinal → HOD - supf x = ? -- Z?Chain1.chain zc0 - uzc : {z : Ordinal} → (u : UChain x {!!} z) → ZChain A f mf ay zc0 (UChain.u u) - uzc {z} u = prev (UChain.u u) (UChain.u<x u) - Uz : HOD - Uz = record { od = record { def = λ z → odef A z ∧ ( UChain z {!!} x ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = {!!} } - + ind f mf {y} ay x zc0 prev with Oprev-p x + ... | yes op = ? where + -- + -- we have previous ordinal to use induction + -- + px = Oprev.oprev op + supf : Ordinal → HOD + supf x = ? -- ChainF A f mf ay x zc0 + zc : ZChain A f mf ay zc0 (Oprev.oprev op) + zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) + 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 z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain (zc z<x )) a → b o< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain (zc z<x) ) ab f ∨ IsSup A (ZChain.chain (zc z<x) ) ab → - * a < * b → odef (ZChain.chain (zc ?) ) b ) → ZChain A f mf ay zc0 x + no-extenion : ( {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain ?) a → b o< osuc x → (ab : odef A b) → + HasPrev A (ZChain.chain zc ) ab f ∨ IsSup A (ZChain.chain ? ) ab → + * a < * b → odef (ZChain.chain zc ) b ) → ZChain A f mf ay zc0 x no-extenion is-max = ? zc4 : ZChain A f mf ay zc0 x @@ -499,29 +489,29 @@ ... | yes x=0 = ? ... | no 0<x with ODC.∋-p O A (* x) ... | no noax = no-extenion zc1 where -- ¬ A ∋ p, just skip - zc1 : {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain (zc z<x) ) a → b o< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain (zc ?) ) ab f ∨ IsSup A (ZChain.chain (zc z<x) ) ab → - * a < * b → odef (ZChain.chain (zc z<x) ) b + zc1 : {a b z : Ordinal} → (z<x : z o< x) → 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} z<x 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 (zc z<x) za ? ab P a<b - ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain (zc ? ) ) ax f ) + ... | case2 lt = ZChain.is-max zc za ? 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 zc0 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 z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain (zc z<x) ) a → b o< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain (zc z<x) ) ab f ∨ IsSup A (ZChain.chain (zc z<x) ) ab → - * a < * b → odef (ZChain.chain (zc z<x) ) b + chain0 = ZChain.chain zc + zc7 : {a b z : Ordinal} → (z<x : z o< x) → 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} z<x za b<ox ab P a<b with osuc-≡< b<ox - ... | case2 lt = ZChain.is-max (zc z<x) za ? 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 z<x) (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 ?) + ... | case2 lt = ZChain.is-max zc za ? 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 = {!!} ; f-total = {!!} ; initial = {!!} ; chain∋init = {!!} ; is-max = {!!} } where - sup0 : SUP A (ZChain.chain (zc ?) ) + sup0 : SUP A (ZChain.chain zc ) sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where - x21 : {y : HOD} → ZChain.chain (zc ?) ∋ 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 ) @@ -529,9 +519,9 @@ sp = SUP.sup sup0 x=sup : x ≡ & sp x=sup = sym &iso - chain0 = ZChain.chain (zc ?) + 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 (zc ?) (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 } @@ -541,10 +531,10 @@ ... | tri≈ ¬a b ¬c = schain ... | tri> ¬a ¬b c = schain A∋schain : {x : HOD } → schain ∋ x → A ∋ x - A∋schain (case1 zx ) = ZChain.chain⊆A (zc ?) 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 (zc ?) 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 @@ -561,7 +551,7 @@ 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) = {!!} -- ZChain.f-total (zc ?) {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 @@ -569,17 +559,17 @@ ... | 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 (zc ?) 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 (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∋init (zc ?) ) ) + 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∋init 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 (zc ?) 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 ) s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) @@ -590,21 +580,21 @@ 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 (zc ?) za ? ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) + case1 (ZChain.is-max zc za ? 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 (zc ?) za ? ab (case2 z22) a<b ) where -- previous sup - z22 : IsSup A (ZChain.chain (zc ?) ) ab + s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc za ? 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 (zc ?) 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 zc - z24 : IsSup A schain ab → IsSup A (ZChain.chain (zc ?) ) ab + 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∋init (zc ?) ) - ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋init (zc ?) ) - ... | case2 y<b = ZChain.is-max (zc ?) (ZChain.chain∋init (zc ?) ) ? ab (case2 (z24 p)) y<b + z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋init zc ) + ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋init zc ) + ... | case2 y<b = ZChain.is-max zc (ZChain.chain∋init zc ) ? ab (case2 (z24 p)) y<b seq : schain ≡ supf0 x seq with trio< x x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) @@ -617,15 +607,34 @@ ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention - z18 : {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain (zc z<x) ) a → b o< osuc x → (ab : odef A b) → - HasPrev A (ZChain.chain (zc ?) ) ab f ∨ IsSup A (ZChain.chain (zc z<x) ) ab → - * a < * b → odef (ZChain.chain (zc z<x) ) b + z18 : {a b z : Ordinal} → (z<x : z o< x) → 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} z<x za b<x ab p a<b with osuc-≡< b<x - ... | case2 lt = ZChain.is-max (zc z<x) za ? ab p a<b + ... | case2 lt = ZChain.is-max zc za ? ab p a<b ... | case1 b=x with p ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = ? ; 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 ? ) } ) + ... | no op = zc5 where + supf : (z : Ordinal ) → z o< x → HOD + supf x lt = ZChain1.chainf ( zc0 (& A) ) x + uzc : {z : Ordinal} → (u : UChain x supf z) → ZChain A f mf ay zc0 (UChain.u u) + uzc {z} u = prev (UChain.u u) (UChain.u<x u) + Uz : HOD + Uz = record { od = record { def = λ z → odef A z ∧ ( UChain x supf z ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = {!!} } + zc5 : ZChain A f mf ay zc0 x + zc5 with o≤? x o∅ + ... | yes x=0 = ? + ... | no 0<x with ODC.∋-p O A (* x) + ... | no noax = ? where -- ¬ A ∋ p, just skip + ... | yes ax with ODC.p∨¬p O ( HasPrev A Uz ax f ) + -- we have to check adding x preserve is-max ZChain A y f mf zc0 x + ... | case1 pr = ? where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next + ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A Uz ax ) + ... | case1 is-sup = ? -- x is a sup of (zc ?) + ... | case2 ¬x=sup = ? -- no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention + SZ0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain1 A f mf ay x SZ0 f mf ay x = TransFinite {λ z → ZChain1 A f mf ay z} (sind f mf ay ) x