Members/kono/Proof/ZF-in-agda: src/zorn.agda comparison

comparison src/zorn.agda @ 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

comparison

equal deleted inserted replaced

-:d041941a8866
+:71a1ed72cd21
 --
 ≤-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
 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx )
 ... | 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 )
 fcn-inject : {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  ≡ 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 _) (init x₁) 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 ) )
+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 refl ) cy i=x i=y )
+... | 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 refl ) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) )
+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 refl ) i=x i=y )
+... | 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 ) )
 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy  i=x j=y )
 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where
 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where
 fc05 : * x ≡ * y1
 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
 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y
 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) )
 fc10 : * x ≡ * y
 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b
 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12  where
 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
 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx )
 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy )
 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy
 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)) )
 ... | case1 _ = case1 refl
 ... | case2 _ = case2 refl
 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx
 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx
 ... | 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)
 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b)  → Tri (a < b) (a ≡ b) (b < a )
 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x)     : Set n where
 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
 chain : HOD
 chain⊆A : chain ⊆' A
 zc4 : ZChain A y f x
 zc4 with ODC.∋-p O A (* x)
 ... | 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
 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 ) )
 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))
 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)
 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)  )
 A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx
 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx
 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
 eq : a ≡ b
 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b )
 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p
 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b)
 → 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 } =
 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) 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) )
 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 )
 ... | case1 y=b  = subst (λ k → odef chain k )  y=b ( ZChain.chain∋x zc0 )
 ... | 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 →
 * a < * b → odef (ZChain.chain zc0) b
 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x
 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy)  } )
 ... | no ¬ox with trio< x y
 ... | 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
 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x
 field
 FC : Fc∨sup A (chain⊆A za (chain∋x za)) f (& (chain za)) i
 FC = fc∨sup za zai
 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 )
 u-total : IsTotalOrderSet Uz
 u-total {x} {y} ux uy  with trio< (UZFChain.u ux) (UZFChain.u uy)

Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison src/zorn.agda @ 600:71a1ed72cd21