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

comparison src/zorn.agda @ 920:a2f8d14012aa

fixpoint?

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Sun, 16 Oct 2022 17:26:49 +0900
parents	213f12f27003
children	c0cf2b383064

comparison

equal deleted inserted replaced

-:213f12f27003
+:a2f8d14012aa
 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) →  odef A x → ( * x < * (cf nmx x) ) ∧  odef A (cf nmx x )
 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫
 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) →  ≤-monotonic-f A ( cf nmx )
 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫
-sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y)
-(zc : ZChain A f mf ay x ) → SUP A (ZChain.chain zc)
-sp0 f mf ay zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ztotal where
-ztotal : IsTotalOrderSet (ZChain.chain zc)
-ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where
-uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb))
 --
 -- Second TransFinite Pass for maximality
 --
 SZ1 : ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f)
 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal)   → ZChain1 A f mf ay zc x
 SZ1 f mf {y} ay zc x = TransFinite { λ x →  ZChain1 A f mf ay zc x } zc1 x where
 chain-mono1 :  {a b c : Ordinal} → a o≤ b
 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c
 chain-mono1  {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b
-is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a →
+is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b)
-ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) →
+→ HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f
-HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f →
+→ * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
-* a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
+is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev
-is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev
 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫
 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k )
 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc))  ⟫
 supf = ZChain.supf zc
-csupf-fc : {b s z1 : Ordinal} → b o≤ & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1
+csupf-fc : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1
-csupf-fc {b} {s} {z1} b≤z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05  where
+csupf-fc {b} {s} {z1} b<z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05  where
 s<b : s o< b
 s<b = ZChain.supf-inject zc ss<sb
-s≤<z : s o≤ & A
+s<z : s o< & A
-s≤<z = ordtrans s<b b≤z
+s<z = ordtrans s<b b<z
 zc04 : odef (UnionCF A f mf ay supf (& A))  (supf s)
 zc04 = ZChain.csupf zc (z09 (ZChain.asupf zc))
 zc05 : odef (UnionCF A f mf ay supf b)  (supf s)
 zc05 with zc04
 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc  ⟫
 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫
 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫
 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b)
 order {b} {s} {z1} b<z ss<sb fc = zc04 where
 zc00 : ( z1  ≡  MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1  << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) )
-zc00 = MinSUP.x<sup (ZChain.minsup zc (o<→≤  b<z) ) (subst (λ k →  odef (UnionCF A f mf ay (ZChain.supf zc) b) k ) &iso (csupf-fc (o<→≤ b<z) ss<sb fc ))
+zc00 = MinSUP.x<sup (ZChain.minsup zc (o<→≤  b<z) ) (subst (λ k →  odef (UnionCF A f mf ay (ZChain.supf zc) b) k ) &iso (csupf-fc b<z ss<sb fc ))
 -- supf (supf b) ≡ supf b
 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b)
 zc04 with zc00
 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z))  ) eq )
 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt )
 is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a →
 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) →
 HasPrev A (UnionCF A f mf ay supf x) b f ∨ IsSup A (UnionCF A f mf ay supf x) ab →
 * a < * b → odef (UnionCF A f mf ay supf x) b
 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P
-is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua b<x ab has-prev a<b
+is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup
 = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫   where
 b<A : b o< & A
 b<A = z09 ab
 b<x : b o< x
 is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a →
 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) →
 HasPrev A (UnionCF A f mf ay supf x) b f ∨ IsSup A (UnionCF A f mf ay supf x) ab →
 * a < * b → odef (UnionCF A f mf ay supf x) b
 is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P
-is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua sb<sx ab has-prev a<b
+is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup with IsSup.x<sup (proj2 is-sup) (init-uchain A f mf ay )
 ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl ))  ⟫
 ... | case2 y<b = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl))  ⟫ where
 m09 : b o< & A
 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
 m06 = record { fcy<sup = m07 ;  order = m08 ; supu=u = m05 }
 ---
 --- the maximum chain  has fix point of any ≤-monotonic function
 ---
+sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y)
+(zc : ZChain A f mf ay x ) → SUP A (ZChain.chain zc)
+sp0 f mf {x} ay zc = ZChain.sup zc o≤-refl
 fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f )  (zc : ZChain A f mf as0 (& A) )
 → f (& (SUP.sup (sp0 f mf as0 zc ))) ≡ & (SUP.sup (sp0 f mf as0 zc  ))
 fixpoint f mf zc = z14 where
 chain = ZChain.chain zc
 supf = ZChain.supf zc
 sp1 = sp0 f mf as0 zc
 z10 :  {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b )
 →  HasPrev A chain b f ∨  IsSup A chain {b} ab
 → * a < * b  → odef chain b
 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) )
+z22 : & (SUP.sup sp1) o< & A
+z22 = c<→o< ( SUP.as sp1 )
 z21 : supf (& (SUP.sup sp1)) o< supf (& A)
 z21 = ?
 -- z21 : supf (& (SUP.sup sp1)) o< & A
 -- z21 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k) (sym &iso) (ZChain.asupf zc ) ))
 z12 : odef chain (& (SUP.sup sp1))
 z12 with o≡? (& s) (& (SUP.sup sp1))
 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc )
-... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z21 (SUP.as sp1)
+... | no ne = ZChain1.is-max (SZ1 f mf as0 zc (& A)) {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z21 (SUP.as sp1)
 (case2 z19 ) z13 where
 z13 :  * (& s) < * (& (SUP.sup sp1))
 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc )
 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p )
 ztotal : IsTotalOrderSet (ZChain.chain zc)
 ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where
 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
 uz01 = chain-total A f mf as0 supf ( (proj2 ca)) ( (proj2 cb))
 z14 :  f (& (SUP.sup (sp0 f mf as0 zc ))) ≡ & (SUP.sup (sp0 f mf as0 zc ))
 z14 with ztotal (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12
 ... | tri< a ¬b ¬c = ⊥-elim z16 where
 z16 : ⊥
 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.as sp1 ))

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

comparison src/zorn.agda @ 920:a2f8d14012aa