Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 600:71a1ed72cd21
not yet ...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 14 Jun 2022 06:17:24 +0900 |
| parents | d041941a8866 |
| children | 8b2a4af84e25 |
| files | src/zorn.agda |
| diffstat | 1 files changed, 32 insertions(+), 27 deletions(-) [+] |
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--- a/src/zorn.agda Tue Jun 14 02:10:15 2022 +0900 +++ b/src/zorn.agda Tue Jun 14 06:17:24 2022 +0900 @@ -89,19 +89,16 @@ ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) --- immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n --- immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) - data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where - init : {x : Ordinal} → odef A s → x ≡ s → FClosure A f s x + init : odef A s → FClosure A f s s fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y -A∋fc {A} s f mf (init as refl ) = as +A∋fc {A} s f mf (init as) = as A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y -s≤fc {A} s {.s} f mf (init x refl ) = case1 refl +s≤fc {A} s {.s} f mf (init x) = case1 refl s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) ... | case2 x<fx with s≤fc {A} s f mf fcy @@ -109,7 +106,7 @@ ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ -fcn s mf (init as refl ) = zero +fcn s mf (init as) = zero fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) ... | case1 eq = fcn s mf p ... | case2 y<fy = suc (fcn s mf p ) @@ -118,11 +115,11 @@ → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y - fc00 zero zero refl (init _ refl ) (init x₁ refl ) i=x i=y = refl - fc00 zero zero refl (init as refl ) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) - ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as refl ) cy i=x i=y ) - fc00 zero zero refl (fsuc x cx) (init as refl ) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) - ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as refl ) i=x i=y ) + fc00 zero zero refl (init _) (init x₁) i=x i=y = refl + fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) + ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) + fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) + ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) @@ -143,6 +140,7 @@ fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) + fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where @@ -171,6 +169,7 @@ fc12 : * y < * x fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c + fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where @@ -193,7 +192,7 @@ cxx : FClosure A f s (f x) cxx = fsuc x cx fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) - fc16 x (init as refl ) with proj1 (mf s as ) + fc16 x (init as) with proj1 (mf s as ) ... | case1 _ = case1 refl ... | case2 _ = case2 refl fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) @@ -208,7 +207,6 @@ ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) - -- open import Relation.Binary.Properties.Poset as Poset IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) @@ -235,19 +233,22 @@ field x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) -y1 : (A : HOD) → (y : Ordinal) → odef A y → * (& (* y , * y)) ⊆' A -y1 A y ay {x} lt with subst (λ k → odef k x) *iso lt -... | case1 eq = subst (λ k → odef A k ) (sym (trans eq &iso)) ay -... | case2 eq = subst (λ k → odef A k ) (sym (trans eq &iso)) ay - record FChain ( A : HOD ) ( f : Ordinal → Ordinal ) (p c : Ordinal) ( x : Ordinal ) : Set n where field fc∨sup : FClosure A f p x chain∋p : odef (* c) p +record FSup ( A : HOD ) ( f : Ordinal → Ordinal ) (p c : Ordinal) ( x : Ordinal ) : Set n where + field + sup : (z : Ordinal) → FClosure A f p z → * z < * x + min : ( x1 : Ordinal) → ((z : Ordinal) → FClosure A f p z → * z < * x1 ) → ( x ≡ x1 ) ∨ ( * x < * x1 ) + chain∋x : odef (* c) x + chain∋p : odef (* c) p + data Fc∨sup (A : HOD) {y : Ordinal} (ay : odef A y) ( f : Ordinal → Ordinal ) (c : Ordinal) : (x : Ordinal) → Set n where Finit : {z : Ordinal} → z ≡ y → Fc∨sup A ay f c z - Fc : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FChain A f p c x → Fc∨sup A ay f c x + Fsup : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FSup A f p c x → Fc∨sup A ay f c x + Fc : {p x : Ordinal} → p o< x → Fc∨sup A ay f c p → FChain A f p c x → Fc∨sup A ay f c x record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where field @@ -423,7 +424,7 @@ ... | 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 = ZChain.fc∨sup zc0 } 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 @@ -441,11 +442,11 @@ ... | 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)) zc9 : ZChain A y f 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 = ZChain.fc∨sup zc0 } ... | 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 = s-fc∨sup} 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) @@ -468,6 +469,9 @@ 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 + s-fc∨sup : {c : Ordinal} → odef schain c → Fc∨sup A (s⊆A (case1 (ZChain.chain∋x zc0))) f (& schain) c + s-fc∨sup {c} (case1 cx) = {!!} + s-fc∨sup {c} (case2 fc) = {!!} 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 @@ -525,7 +529,7 @@ → HasPrev A schain ab f ∨ IsSup A schain ab → * a < * b → odef schain b s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? - ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) refl )) + ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) 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 } = @@ -546,7 +550,7 @@ ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 - ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = {!!} } where + ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = ZChain.fc∨sup zc0 } where -- no extention z18 : {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 → @@ -561,7 +565,7 @@ ... | tri< a ¬b ¬c = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } ... | tri≈ ¬a b ¬c = {!!} - ... | tri> ¬a ¬b y<x = {!!} where + ... | tri> ¬a ¬b y<x = UnionZ where UnionZ : ZChain A y f x UnionZ = record { chain = Uz ; chain⊆A = Uz⊆A ; f-total = u-total ; f-next = u-next ; initial = u-initial ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } where --- limit ordinal case @@ -595,10 +599,11 @@ um01 : odef (chain zb) i um01 with FC ... | Finit i=y = subst (λ k → odef (chain zb) k ) (sym i=y) ( chain∋x zb ) + ... | Fsup {p} {i} p<i pFC sup = ? ... | Fc {p} {i} p<i pFC FC with (FChain.fc∨sup FC) ... | fc = um02 i fc where um02 : (i2 : Ordinal) → FClosure A f p i2 → odef (chain zb) i2 - um02 i2 (init ap i2=p ) = subst (λ k → odef (chain zb) k ) (sym i2=p) (previ p p<i um04 ) where + um02 i2 (init ap ) = previ p p<i um04 where um04 : odef (chain za) p um04 = subst (λ k → odef k p ) *iso ( FChain.chain∋p FC ) um02 i (fsuc j fc) = f-next zb ( um02 j fc )