Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff cardinal.agda @ 331:12071f79f3cf
HOD done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 05 Jul 2020 16:56:21 +0900 |
parents | d9d3654baee1 |
children | 6c72bee25653 |
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--- a/cardinal.agda Sun Jul 05 15:49:00 2020 +0900 +++ b/cardinal.agda Sun Jul 05 16:56:21 2020 +0900 @@ -29,49 +29,48 @@ -- we have to work on Ordinal to keep OD Level n -- since we use p∨¬p which works only on Level n - -∋-p : (A x : OD ) → Dec ( A ∋ x ) +∋-p : (A x : HOD ) → Dec ( A ∋ x ) ∋-p A x with ODC.p∨¬p O ( A ∋ x ) ∋-p A x | case1 t = yes t ∋-p A x | case2 t = no t -_⊗_ : (A B : OD) → OD -A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where +_⊗_ : (A B : HOD) → HOD +A ⊗ B = record { od = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } } where checkAB : { p : Ordinal } → def ZFProduct p → Set n - checkAB (pair x y) = def A x ∧ def B y + checkAB (pair x y) = odef A x ∧ odef B y -func→od0 : (f : Ordinal → Ordinal ) → OD -func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where +func→od0 : (f : Ordinal → Ordinal ) → HOD +func→od0 f = record { od = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) }} where checkfunc : { p : Ordinal } → def ZFProduct p → Set n checkfunc (pair x y) = f x ≡ y -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) -Func : ( A B : OD ) → OD -Func A B = record { def = λ x → def (Power (A ⊗ B)) x } +Func : ( A B : HOD ) → HOD +Func A B = record { od = record { def = λ x → odef (Power (A ⊗ B)) x } } -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) -func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD +func→od : (f : Ordinal → Ordinal ) → ( dom : HOD ) → HOD func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) -record Func←cd { dom cod : OD } {f : Ordinal } : Set n where +record Func←cd { dom cod : HOD } {f : Ordinal } : Set n where field func-1 : Ordinal → Ordinal func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom -od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} -od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x {!!} ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where +od→func : { dom cod : HOD } → {f : Ordinal } → odef (Func dom cod ) f → Func←cd {dom} {cod} {f} +od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o {!!} ( λ y lt → lemma x {!!} ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where lemma : Ordinal → Ordinal → Ordinal - lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) + lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → odef (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) lemma x y | p | no n = o∅ lemma x y | p | yes f∋y = lemma2 (proj1 (ODC.double-neg-eilm O ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) lemma2 : {p : Ordinal} → ord-pair p → Ordinal lemma2 (pair x1 y1) with ODC.decp O ( x1 ≡ x) lemma2 (pair x1 y1) | yes p = y1 lemma2 (pair x1 y1) | no ¬p = o∅ - fod : OD - fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) {!!} )) > ) + fod : HOD + fod = Replace dom ( λ x → < x , ord→od (sup-o {!!} ( λ y lt → lemma (od→ord x) {!!} )) > ) open Func←cd @@ -91,18 +90,18 @@ -- X ---------------------------> Y -- ymap <- def Y y -- -record Onto (X Y : OD ) : Set n where +record Onto (X Y : HOD ) : Set n where field xmap : Ordinal ymap : Ordinal - xfunc : def (Func X Y) xmap - yfunc : def (Func Y X) ymap - onto-iso : {y : Ordinal } → (lty : def Y y ) → + xfunc : odef (Func X Y) xmap + yfunc : odef (Func Y X) ymap + onto-iso : {y : Ordinal } → (lty : odef Y y ) → func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y open Onto -onto-restrict : {X Y Z : OD} → Onto X Y → Z ⊆ Y → Onto X Z +onto-restrict : {X Y Z : HOD} → Onto X Y → Z ⊆ Y → Onto X Z onto-restrict {X} {Y} {Z} onto Z⊆Y = record { xmap = xmap1 ; ymap = zmap @@ -114,23 +113,23 @@ xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) zmap : Ordinal zmap = {!!} - xfunc1 : def (Func X Z) xmap1 + xfunc1 : odef (Func X Z) xmap1 xfunc1 = {!!} - zfunc : def (Func Z X) zmap + zfunc : odef (Func Z X) zmap zfunc = {!!} - onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z + onto-iso1 : {z : Ordinal } → (ltz : odef Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z onto-iso1 = {!!} -record Cardinal (X : OD ) : Set n where +record Cardinal (X : HOD ) : Set n where field cardinal : Ordinal conto : Onto X (Ord cardinal) cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) -cardinal : (X : OD ) → Cardinal X +cardinal : (X : HOD ) → Cardinal X cardinal X = record { - cardinal = sup-o ( λ x → proj1 ( cardinal-p {!!}) ) + cardinal = sup-o {!!} ( λ x lt → proj1 ( cardinal-p {!!}) ) ; conto = onto ; cmax = cmax } where @@ -138,24 +137,24 @@ cardinal-p x with ODC.p∨¬p O ( Onto X (Ord x) ) cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } - S = sup-o (λ x → proj1 (cardinal-p {!!})) - lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → - Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) + S = sup-o {!!} (λ x lt → proj1 (cardinal-p {!!})) + lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → (y o< (osuc S) → Onto X (Ord y))) → + (x o< (osuc S) → Onto X (Ord x) ) lemma1 x prev with trio< x (osuc S) lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a - lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) - lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where + lemma1 x prev | tri< a ¬b ¬c | case1 x=S = ( λ lt → {!!} ) + lemma1 x prev | tri< a ¬b ¬c | case2 x<S = ( λ lt → lemma2 ) where lemma2 : Onto X (Ord x) lemma2 with prev {!!} {!!} - ... | lift t = t {!!} - lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) - lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) + ... | t = {!!} + lemma1 x prev | tri≈ ¬a b ¬c = ( λ lt → ⊥-elim ( o<¬≡ b lt )) + lemma1 x prev | tri> ¬a ¬b c = ( λ lt → ⊥-elim ( o<> c lt )) onto : Onto X (Ord S) - onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S - ... | lift t = t <-osuc + onto with TransFinite {λ x → ( x o< osuc S → Onto X (Ord x) ) } lemma1 S + ... | t = t <-osuc cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) - cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} - (sup-o< {λ x → proj1 ( cardinal-p {!!})}{{!!}} ) lemma refl ) where + cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} {!!} lemma refl ) where + -- (sup-o< ? {λ x lt → proj1 ( cardinal-p {!!})}{{!!}} ) lemma refl ) where lemma : proj1 (cardinal-p y) ≡ y lemma with ODC.p∨¬p O ( Onto X (Ord y) ) lemma | case1 x = refl