Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 938:93a49ffa9183
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 28 Oct 2022 18:37:05 +0900 |
| parents | 3a511519bd10 |
| children | 187594116449 |
| files | src/zorn.agda src/zorn1.agda |
| diffstat | 2 files changed, 45 insertions(+), 818 deletions(-) [+] |
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--- a/src/zorn.agda Mon Oct 24 19:11:19 2022 +0900 +++ b/src/zorn.agda Fri Oct 28 18:37:05 2022 +0900 @@ -278,7 +278,7 @@ chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca ... | case1 eq with s≤fc (supf ub) f mf fcb ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b @@ -286,14 +286,14 @@ ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct01 : * a < * b ct01 = subst (λ k → * k < * b ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct00 : * a < * (supf ub) ct00 = lt ct01 : * a < * b ct01 with s≤fc (supf ub) f mf fcb ... | case1 eq = subst (λ k → * a < k ) eq ct00 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt - ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb + ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb ... | case1 eq with s≤fc (supf ua) f mf fca ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b @@ -301,7 +301,7 @@ ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct01 : * b < * a ct01 = subst (λ k → * k < * a ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where + ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct00 : * b < * (supf ua) ct00 = lt ct01 : * b < * a @@ -424,11 +424,11 @@ chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ - f-next {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ + f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ initial : {z : Ordinal } → odef chain z → * y ≤ * z initial {a} ⟪ aa , ua ⟫ with ua ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where + ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where zc7 : y <= supf u zc7 = ChainP.fcy<sup is-sup (init ay refl) f-total : IsTotalOrderSet chain @@ -590,8 +590,8 @@ → * 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 ... | ⟪ 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)) ⟫ + ... | ⟪ 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 @@ -606,7 +606,7 @@ zc05 : odef (UnionCF A f mf ay supf b) (supf s) zc05 with zc04 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ - ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where + ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s zc07 = fc zc06 : supf u ≡ u @@ -617,7 +617,7 @@ zc04 : odef (UnionCF A f mf ay supf b) (f x) zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb 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) ⟫ + ... | ⟪ 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) ) ) @@ -889,7 +889,7 @@ zc21 : {z1 : Ordinal } → FClosure A f (supf0 u) z1 → odef (UnionCF A f mf ay supf1 x) z1 zc21 {z1} (fsuc z2 fc ) with zc21 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ zc21 (init asp refl ) with trio< u px | inspect supf1 u ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u ? @@ -928,7 +928,7 @@ zc21 : {z1 : Ordinal } → FClosure A f (supf0 px) z1 → odef (UnionCF A f mf ay supf1 x) z1 zc21 {z1} (fsuc z2 fc ) with zc21 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ zc21 (init asp refl ) with osuc-≡< ( subst (λ k → supf0 px o< k ) (sym (Oprev.oprev=x op)) sfpx<x ) ... | case1 sfpx=px = ⟪ asp , ch-is-sup px ? -- (pxo<x op) record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where @@ -956,7 +956,7 @@ ... | case2 sfp<px with ZChain.csupf zc sfp<px -- odef (UnionCF A f mf ay supf0 px) (supf0 px) ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ ua1 , ch-init fc₁ ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u ? + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u ? record { fcy<sup = z10 ; order = z11 ; supu=u = z12 } (fcpu fc₁ ? ) ⟫ where z10 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) z10 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.fcy<sup is-sup fc) @@ -967,7 +967,7 @@ s<u : s o< u s<u = supf-inject0 supf1-mono lt s≤px : s o≤ px - s≤px = ordtrans s<u ? -- (o<→≤ u≤x) + s≤px = ordtrans s<u ? -- (o<→≤ u<x) lt0 : supf0 s o< supf0 u lt0 = subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 ? ) lt z12 : supf1 u ≡ u @@ -1111,7 +1111,7 @@ zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) x<sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) x<sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? - x<sup {w} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u≤x) ? )) + x<sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where zc14 : u ≡ osuc px zc14 = begin @@ -1235,9 +1235,9 @@ * a < * b → odef (UnionCF A f mf ay supf x) b is-max-hp supf 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 , + ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , -- subst (λ k → UChain A f mf ay supf x k ) - -- (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ + -- (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ zc70 : HasPrev A pchain x f → ¬ xSUP pchain x zc70 pr xsup = ? @@ -1251,23 +1251,23 @@ pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where + zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z zc12 (fsuc x fc) with zc12 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ - zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where + zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z zc13 (fsuc x fc) with zc13 fc ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ - ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ + ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ zc13 (init asu su=z ) with trio< u x ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ ... | tri≈ ¬a b ¬c = ? - ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u≤x c ) + ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) sup {z} z≤x with trio< z x ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) @@ -1469,20 +1469,38 @@ → supf mc << MinSUP.sup spd sc<<d {mc} {asc} spd = z25 where d1 : Ordinal - d1 = MinSUP.sup spd + d1 = MinSUP.sup spd -- supf d1 ≡ d z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) z24 = MinSUP.x<sup spd (init asc refl) - z26 : odef (ZChain.chain zc) (supf mc) - z26 = ? z28 : supf mc o< & A z28 = z09 (ZChain.asupf zc) z25 : supf mc << d1 z25 with z24 ... | case2 lt = lt - ... | case1 eq = ? where + ... | case1 eq = ⊥-elim ( <-irr z29 (proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) ) ) where + -- supf mc ≡ d1 z27 : odef (ZChain.chain zc) (cf nmx d1) z27 = ZChain.f-next zc (subst (λ k → odef (ZChain.chain zc) k ) eq (ZChain.csupf zc z28)) - + z31 : {z w : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z + → (* w ≡ * z) ∨ (* w < * z) + → (* w ≡ * d1) ∨ (* w < * d1) + z31 {z} uz (case1 w=z) with MinSUP.x<sup spd uz + ... | case1 eq = case1 (trans w=z (cong (*) eq) ) + ... | case2 lt = case2 (subst (λ k → k < * d1 ) (sym w=z) lt ) + z31 {z} {w} uz (case2 w<z) with MinSUP.x<sup spd uz + ... | case1 eq = case2 (subst (λ k → * w < k ) (cong (*) eq) w<z ) + ... | case2 lt = case2 ( IsStrictPartialOrder.trans PO w<z lt) + z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) + z29 with z27 + ... | ⟪ aa , ch-init fc ⟫ = ? where + z30 : FClosure A (cf nmx) (& s) (cf nmx d1) + z30 = fc + ... | ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ with trio< u (supf mc) -- u<x : supf u o< supf (& A) + ... | tri< a ¬b ¬c = ? -- u o< supf mc + ... | tri> ¬a ¬b c = ? -- supf mc o< u + ... | tri≈ ¬a b ¬c with MinSUP.x<sup spd ( subst₂ (λ j k → FClosure A (cf nmx) j k ) (trans (ChainP.supu=u is-sup) b) refl fc ) + ... | case1 eq = case1 (cong (*) eq) + ... | case2 lt = case2 lt sc<sd : {mc d : Ordinal } → supf mc << supf d → supf mc o< supf d sc<sd {mc} {d} sc<<sd with osuc-≡< ( ZChain.supf-<= zc (case2 sc<<sd ) ) ... | case1 eq = ⊥-elim ( <-irr (case1 (cong (*) (sym eq) )) sc<<sd )
--- a/src/zorn1.agda Mon Oct 24 19:11:19 2022 +0900 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,791 +0,0 @@ -{-# OPTIONS --allow-unsolved-metas #-} -open import Level hiding ( suc ; zero ) -open import Ordinals -open import Relation.Binary -open import Relation.Binary.Core -open import Relation.Binary.PropositionalEquality -import OD -module zorn1 {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where - --- --- Zorn-lemma : { A : HOD } --- → o∅ o< & A --- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition --- → Maximal A --- - -open import zf -open import logic --- open import partfunc {n} O - -open import Relation.Nullary -open import Data.Empty -import BAlgbra - -open import Data.Nat hiding ( _<_ ; _≤_ ) -open import Data.Nat.Properties -open import nat - - -open inOrdinal O -open OD O -open OD.OD -open ODAxiom odAxiom -import OrdUtil -import ODUtil -open Ordinals.Ordinals O -open Ordinals.IsOrdinals isOrdinal -open Ordinals.IsNext isNext -open OrdUtil O -open ODUtil O - - -import ODC - -open _∧_ -open _∨_ -open Bool - -open HOD - --- --- Partial Order on HOD ( possibly limited in A ) --- - -_<<_ : (x y : Ordinal ) → Set n -x << y = * x < * y - -_<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain -x <= y = (x ≡ y ) ∨ ( * x < * y ) - -POO : IsStrictPartialOrder _≡_ _<<_ -POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } - ; trans = IsStrictPartialOrder.trans PO - ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y - ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } - -_≤_ : (x y : HOD) → Set (Level.suc n) -x ≤ y = ( x ≡ y ) ∨ ( x < y ) - -≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z -≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl -≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z -≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y -≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) - -<-ftrans : {x y z : Ordinal } → x <= y → y <= z → x <= z -<-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl -<-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z -<-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y -<-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) - -<=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y -<=to≤ (case1 eq) = case1 (cong (*) eq) -<=to≤ (case2 lt) = case2 lt - -≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y -≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) -≤to<= (case2 lt) = case2 lt - -<-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ -<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a -<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl - (IsStrictPartialOrder.trans PO b<a a<b) - -ptrans = IsStrictPartialOrder.trans PO - -open _==_ -open _⊆_ - --- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A --- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x --- <-TransFinite = ? - --- --- Closure of ≤-monotonic function f has total order --- - -≤-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 ) - -data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where - init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 - 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 (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) - -A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s -A∋fcs {A} s f mf (init as refl) = as -A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {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 {.(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 {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 - fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) - fc06 {x} {y} refl {j} not = fc08 not where - fc08 : {j : ℕ} → ¬ suc j ≡ 0 - fc08 () - fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x - fc07 {x} (init as refl) eq = refl - fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) - ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) - -- ... | case2 x<fx = ? - 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 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) - fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) - 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 (fc07 cy i=y) -- ( 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 (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) 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 - fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) - fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) - fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) - ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x - ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where - fc04 : * x1 ≡ * y - fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) - ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where - fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 - fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) - fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) - ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq - ... | 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 - fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) - fc06 {x} {y} refl {j} not = fc08 not where - fc08 : {j : ℕ} → ¬ suc j ≡ 0 - fc08 () - 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) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) - fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) - ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) - ... | case2 y<fy with <-cmp (fcn s mf cx ) i - ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) - ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy - ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where - fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy - fc03 eq = cong pred eq - fc02 : * x < * y1 - fc02 = fc01 i cx cy (fc03 i=y ) a - - -fcn-cmp : {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 ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) -fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) -... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where - fc11 : * x < * y - fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a -... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where - 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 - - - --- 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 ) - -⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B -⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) - -_⊆'_ : ( A B : HOD ) → Set n -_⊆'_ A B = {x : Ordinal } → odef A x → odef B x - --- --- inductive maxmum tree from x --- tree structure --- - -record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where - field - ax : odef A x - y : Ordinal - ay : odef B y - x=fy : x ≡ f y - -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 ) - -record SUP ( A B : HOD ) : Set (Level.suc n) where - field - sup : HOD - as : A ∋ sup - x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive - --- --- sup and its fclosure is in a chain HOD --- chain HOD is sorted by sup as Ordinal and <-ordered --- whole chain is a union of separated Chain --- minimum index is sup of y not ϕ --- - -record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where - field - fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) - order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) - supu=u : supf u ≡ u - -data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) - (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where - ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z - ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) - ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z - --- --- f (f ( ... (sup y))) f (f ( ... (sup z1))) --- / | / | --- / | / | --- sup y < sup z1 < sup z2 --- o< o< --- data UChain is total - -chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) - {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) -chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where - ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) - ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca - ... | case1 eq with s≤fc (supf ub) f mf fcb - ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * a ≡ * b - ct00 = trans (cong (*) eq) eq1 - ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where - ct01 : * a < * b - ct01 = subst (λ k → * k < * b ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where - ct00 : * a < * (supf ub) - ct00 = lt - ct01 : * a < * b - ct01 with s≤fc (supf ub) f mf fcb - ... | case1 eq = subst (λ k → * a < k ) eq ct00 - ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt - ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb - ... | case1 eq with s≤fc (supf ua) f mf fca - ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * a ≡ * b - ct00 = sym (trans (cong (*) eq) eq1 ) - ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where - ct01 : * b < * a - ct01 = subst (λ k → * k < * a ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where - ct00 : * b < * (supf ua) - ct00 = lt - ct01 : * b < * a - ct01 with s≤fc (supf ua) f mf fca - ... | case1 eq = subst (λ k → * b < k ) eq ct00 - ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt - ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub - ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca - ... | case1 eq with s≤fc (supf ub) f mf fcb - ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * a ≡ * b - ct00 = trans (cong (*) eq) eq1 - ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where - ct02 : * a < * b - ct02 = subst (λ k → * k < * b ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where - ct03 : * a < * (supf ub) - ct03 = lt - ct02 : * a < * b - ct02 with s≤fc (supf ub) f mf fcb - ... | case1 eq = subst (λ k → * a < k ) eq ct03 - ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt - ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c - = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) - ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb - ... | case1 eq with s≤fc (supf ua) f mf fca - ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where - ct00 : * a ≡ * b - ct00 = sym (trans (cong (*) eq) eq1) - ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where - ct02 : * b < * a - ct02 = subst (λ k → * k < * a ) (sym eq) lt - ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where - ct05 : * b < * (supf ua) - ct05 = lt - ct04 : * b < * a - ct04 with s≤fc (supf ua) f mf fca - ... | case1 eq = subst (λ k → * b < k ) eq ct05 - ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt - -∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A -∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) - --- Union of supf z which o< x --- -UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) - ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD -UnionCF A f mf ay supf x - = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } - -supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) - → supf x o< supf y → x o< y -supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y -... | tri< a ¬b ¬c = a -... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) -... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) -... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) -... | case2 lt = ⊥-elim ( o<> sx<sy lt ) - -record MinSUP ( A B : HOD ) : Set n where - field - sup : Ordinal - asm : odef A sup - x<sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) - minsup : { sup1 : Ordinal } → odef A sup1 - → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 - -z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A -z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) - -M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } - → (supf : Ordinal → Ordinal ) - → MinSUP A (UnionCF A f mf ay supf x) - → SUP A (UnionCF A f mf ay supf x) -M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) - ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x<sup = ms00 } where - msup = MinSUP.sup ms - ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) - ms00 {z} uz with MinSUP.x<sup ms uz - ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) - ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) - - -chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) - (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b - → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c -chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = - ⟪ ua , ch-init fc ⟫ -chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = - ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ - -record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) - {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where - field - supf : Ordinal → Ordinal - sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z - → IsSup A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) b f) → supf b ≡ b - - asupf : {x : Ordinal } → odef A (supf x) - supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y - supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y - supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z - - minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) - supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) - csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain - - chain : HOD - chain = UnionCF A f mf ay supf z - chain⊆A : chain ⊆' A - chain⊆A = λ lt → proj1 lt - - sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) - sup {x} x≤z = M→S supf (minsup x≤z) - - s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) - s=ms {x} x≤z = &iso - - chain∋init : odef chain y - chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ - f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) - f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ - f-next {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ - initial : {z : Ordinal } → odef chain z → * y ≤ * z - initial {a} ⟪ aa , ua ⟫ with ua - ... | ch-init fc = s≤fc y f mf fc - ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where - zc7 : y <= supf u - zc7 = ChainP.fcy<sup is-sup (init ay refl) - f-total : IsTotalOrderSet chain - f-total {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 supf ( (proj2 ca)) ( (proj2 cb)) - - supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y - supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy - supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) - ... | tri< a ¬b ¬c = o<→≤ a - ... | tri≈ ¬a b ¬c = o≤-refl0 b - ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) - - supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y - supf-inject {x} {y} sx<sy with trio< x y - ... | tri< a ¬b ¬c = a - ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) - ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) - ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) - ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) - - fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf - fcy<sup {u} {w} u≤z fc with MinSUP.x<sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) - , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ - ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) - ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) - - -- ordering is not proved here but in ZChain1 - -record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) - {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where - supf = ZChain.supf zc - field - is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) - → HasPrev A (UnionCF A f mf ay supf z) b f ∨ IsSup A (UnionCF A f mf ay supf z) ab - → * a < * b → odef ((UnionCF A f mf ay supf z)) b - -record Maximal ( A : HOD ) : Set (Level.suc n) where - field - maximal : HOD - as : A ∋ maximal - ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative - -init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) - { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y -init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ - -Zorn-lemma : { A : HOD } - → o∅ o< & A - → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition - → Maximal A -Zorn-lemma {A} 0<A supP = zorn00 where - <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ - <-irr0 {a} {b} A∋a A∋b = <-irr - z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A - z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) - s : HOD - s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) - as : A ∋ * ( & s ) - as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) - as0 : odef A (& s ) - as0 = subst (λ k → odef A k ) &iso as - s<A : & s o< & A - s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) - HasMaximal : HOD - HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } - no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ - no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) - Gtx : { x : HOD} → A ∋ x → HOD - Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } - z08 : ¬ Maximal A → HasMaximal =h= od∅ - z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) - ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} - x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) - x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where - ¬x<m : ¬ (* x < * m) - ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) - - minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B - minsupP B B⊆A total = m02 where - xsup : (sup : Ordinal ) → Set n - xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) - ∀-imply-or : {A : Ordinal → Set n } {B : Set n } - → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B - ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM - ∀-imply-or {A} {B} ∀AB | case1 t = case1 t - ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where - lemma : ¬ ((x : Ordinal ) → A x) → B - lemma not with ODC.p∨¬p O B - lemma not | case1 b = b - lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) - m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B - m00 x = TransFinite0 ind x where - ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) - → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B - ind x prev = ∀-imply-or m01 where - m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B - m01 z with trio< z x - ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) - ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) - ... | tri< a ¬b ¬c with prev z a - ... | case2 mins = case2 mins - ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) - ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x<sup = proj2 mins ; minsup = m04 } where - m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 - m04 {s} as lt with trio< z s - ... | tri< a ¬b ¬c = o<→≤ a - ... | tri≈ ¬a b ¬c = o≤-refl0 b - ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) - ... | case2 notz = case1 (λ _ → notz ) - m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) - m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where - S : SUP A B - S = supP B B⊆A total - s1 = & (SUP.sup S) - m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) - m05 {w} bw with SUP.x<sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) - ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) - ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) - m02 : MinSUP A B - m02 = dont-or (m00 (& A)) m03 - - -- Uncountable ascending chain by axiom of choice - cf : ¬ Maximal A → Ordinal → Ordinal - cf nmx x with ODC.∋-p O A (* x) - ... | no _ = o∅ - ... | yes ax with is-o∅ (& ( Gtx ax )) - ... | yes nogt = -- no larger element, so it is maximal - ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) - ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) - is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) - is-cf nmx {x} ax with ODC.∋-p O A (* x) - ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) - ... | yes ax with is-o∅ (& ( Gtx ax )) - ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) - ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) - - --- - --- infintie ascention sequence of f - --- - 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 ) ⟫ - - -- - -- 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 = ? - - uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD - uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = - λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) } - - utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) - → IsTotalOrderSet (uchain f mf ay) - utotal f mf {y} ay {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 = fcn-cmp y f mf ca cb - - ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) - → MinSUP A (uchain f mf ay) - ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) - - SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B - SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } - - record xSUP (B : HOD) (x : Ordinal) : Set n where - field - ax : odef A x - is-sup : IsSup A B ax - - -- - -- create all ZChains under o< x - -- - - ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) - → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x - ind f mf {y} ay x prev = ? - - --- - --- the maximum chain has fix point of any ≤-monotonic function - --- - - SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x - SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x - - data ZChainP ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - ( supf : Ordinal → Ordinal ) (z : Ordinal) : Set n where - zchain : (uz : Ordinal ) → odef (UnionCF A f mf ay supf uz) z → ZChainP f mf ay supf z - - auzc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → odef A x - auzc f mf {y} ay supf {x} (zchain uz ucf) = proj1 ucf - - zp-uz : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → Ordinal - zp-uz f mf ay supf (zchain uz _) = uz - - uzc⊆zc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - (supf : Ordinal → Ordinal ) → {x : Ordinal } → (zp : ZChainP f mf ay supf x ) → UChain A f mf ay supf (zp-uz f mf ay supf zp) x - uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-init fc ⟫) = ch-init fc - uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-is-sup u u<x is-sup fc ⟫) with ChainP.supu=u is-sup - ... | eq = ch-is-sup u u<x is-sup fc - - UnionZF : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - (supf : Ordinal → Ordinal ) → HOD - UnionZF f mf {y} ay supf = record { od = record { def = λ x → ZChainP f mf ay supf x } - ; odmax = & A ; <odmax = λ lt → ∈∧P→o< ⟪ auzc f mf ay supf lt , lift true ⟫ } - - uzctotal : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) - → ( supf : Ordinal → Ordinal ) - → IsTotalOrderSet (UnionZF f mf ay supf ) - uzctotal f mf ay supf {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (uz01 ca cb) where - uz01 : {ua ub : Ordinal } → ZChainP f mf ay supf ua → ZChainP f mf ay supf ub - → Tri (* ua < * ub) (* ua ≡ * ub) (* ub < * ua ) - uz01 {ua} {ub} (zchain uza uca) (zchain uzb ucb) = chain-total A f mf ay supf (proj2 uca) (proj2 ucb) - - msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) - → (zc : ZChain A f mf ay x ) - → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) - msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (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)) - - 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 (UnionCF A f mf ay (ZChain.supf zc) x) - sp0 f mf ay zc = M→S (ZChain.supf zc) (msp0 f mf ay zc ) - - fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) - → ZChain.supf zc (& (SUP.sup (sp0 f mf as0 zc))) o< ZChain.supf zc (& A) - → f (& (SUP.sup (sp0 f mf as0 zc ))) ≡ & (SUP.sup (sp0 f mf as0 zc )) - fixpoint f mf zc ss<sa = ? - - - -- ZChain contradicts ¬ Maximal - -- - -- ZChain forces fix point on any ≤-monotonic function (fixpoint) - -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain - -- - - z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ - z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) - (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) - (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ss<sa ))) -- x ≡ f x ̄ - (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x - supf = ZChain.supf zc - msp1 : MinSUP A (ZChain.chain zc) - msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc - sp1 : SUP A (ZChain.chain zc) - sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc - c : Ordinal - c = & ( SUP.sup sp1 ) - mc = MinSUP.sup msp1 - c=mc : c ≡ mc - c=mc = &iso - z20 : mc << cf nmx mc - z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) - asc : odef A (supf mc) - asc = ZChain.asupf zc - spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) - spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc - d = MinSUP.sup spd - d<A : d o< & A - d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ - msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) - msup = ZChain.minsup zc (o<→≤ d<A) - sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) - sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) - -- z26 : {x : Ordinal } → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) x - -- → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) c) x ∨ odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x - -- z26 = ? - is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd) - is-sup = record { x<sup = z22 } where - z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) - z23 lt = MinSUP.x<sup spd lt - z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → - (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) - z22 {a} ⟪ aa , ch-init fc ⟫ = ? - z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ? - -- u<x : ZChain.supf zc u o< ZChain.supf zc d - -- supf u o< spuf c → order - not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx) - not-hasprev hp = ? where - y : Ordinal - y = HasPrev.y hp - z24 : y << d - z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) )) - -- z26 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ d ) ∨ (x << d ) - -- z26 lt with MinSUP.x<sup spd (subst (λ k → odef _ k ) ? lt) - -- ... | case1 eq = ? - -- ... | case2 lt = ? - -- z25 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ y ) ∨ (x << y ) - -- z25 {x} (init au eq ) = ? -- sup c = x, cf y ≡ d, sup c =< d - -- z25 (fsuc x lt) = ? -- cf (sup c) - - sd=d : supf d ≡ d - sd=d = ZChain.sup=u zc (MinSUP.asm spd) (o<→≤ d<A) ⟪ is-sup , not-hasprev ⟫ - - sc<<d : {mc : Ordinal } → {asc : odef A (supf mc)} → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) - → supf mc << MinSUP.sup spd - sc<<d {mc} {asc} spd = z25 where - d1 : Ordinal - d1 = MinSUP.sup spd - z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) - z24 = MinSUP.x<sup spd (init asc refl) - z25 : supf mc << d1 - z25 with z24 - ... | case2 lt = lt - ... | case1 eq = ? - - sc<sd : {mc d : Ordinal } → supf mc << supf d → supf mc o< supf d - sc<sd {mc} {d} sc<<sd with osuc-≡< ( ZChain.supf-<= zc (case2 sc<<sd ) ) - ... | case1 eq = ⊥-elim ( <-irr (case1 (cong (*) (sym eq) )) sc<<sd ) - ... | case2 lt = lt - - sms<sa : supf mc o< supf (& A) - sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) - ... | case2 lt = lt - ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} {asc} spd)) ) - ( ZChain.supf-mono zc (o<→≤ d<A )))) - - ss<sa : supf c o< supf (& A) - ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa - - zorn00 : Maximal A - zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM - ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where - -- yes we have the maximal - zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) - zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice - zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) - zorn01 = proj1 zorn03 - zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) - zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) - ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where - -- if we have no maximal, make ZChain, which contradict SUP condition - nmx : ¬ Maximal A - nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where - zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) - zc5 = ⟪ Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ - --- usage (see filter.agda ) --- --- _⊆'_ : ( A B : HOD ) → Set n --- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x - --- MaximumSubset : {L P : HOD} --- → o∅ o< & L → o∅ o< & P → P ⊆ L --- → IsPartialOrderSet P _⊆'_ --- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) --- → Maximal P (_⊆'_) --- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP