Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 1091:63c1167b2343
fix comments
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 20 Dec 2022 11:20:52 +0900 |
| parents | 2cf182f0f583 |
| children | 87c2da3811c3 08b6aa6870d9 |
| files | src/OD.agda src/zorn.agda |
| diffstat | 2 files changed, 180 insertions(+), 176 deletions(-) [+] |
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--- a/src/OD.agda Mon Dec 19 09:50:51 2022 +0900 +++ b/src/OD.agda Tue Dec 20 11:20:52 2022 +0900 @@ -1,12 +1,12 @@ -{-# OPTIONS --allow-unsolved-metas #-} +{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module OD {n : Level } (O : Ordinals {n} ) where open import zf -open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) +open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality hiding ( [_] ) -open import Data.Nat.Properties +open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary hiding (_⇔_) @@ -17,8 +17,8 @@ open import nat open Ordinals.Ordinals O -open Ordinals.IsOrdinals isOrdinal -open Ordinals.IsNext isNext +open Ordinals.IsOrdinals isOrdinal +open Ordinals.IsNext isNext open OrdUtil O -- Ordinal Definable Set @@ -35,24 +35,24 @@ record _==_ ( a b : OD ) : Set n where field - eq→ : ∀ { x : Ordinal } → def a x → def b x - eq← : ∀ { x : Ordinal } → def b x → def a x + eq→ : ∀ { x : Ordinal } → def a x → def b x + eq← : ∀ { x : Ordinal } → def b x → def a x ==-refl : { x : OD } → x == x ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x } -open _==_ +open _==_ ==-trans : { x y z : OD } → x == y → y == z → x == z ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } -==-sym : { x y : OD } → x == y → y == x +==-sym : { x y : OD } → x == y → y == x ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } -⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y -eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m -eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m +⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y +eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m +eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m -- next assumptions are our axiom -- @@ -62,7 +62,7 @@ -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. -- bbounded ODs are ZF Set. Unbounded ODs are classes. -- --- In classical Set Theory, HOD is used, as a subset of OD, +-- In classical Set Theory, HOD is used, as a subset of OD, -- HOD = { x | TC x ⊆ OD } -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. @@ -89,18 +89,18 @@ open HOD -record ODAxiom : Set (suc n) where +record ODAxiom : Set (suc n) where field -- HOD is isomorphic to Ordinal (by means of Goedel number) - & : HOD → Ordinal - * : Ordinal → HOD + & : HOD → Ordinal + * : Ordinal → HOD c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) *iso : {x : HOD } → * ( & x ) ≡ x &iso : {x : Ordinal } → & ( * x ) ≡ x ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y - sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal - sup-o≤ : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤ sup-o A ψ + sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal + sup-o≤ : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o≤ sup-o A ψ -- possible order restriction ho< : {x : HOD} → & x o< next (odmax x) @@ -112,7 +112,7 @@ -- -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. -- We can calculate the minimum using sup but it is tedius. --- Only Select has non minimum odmax. +-- Only Select has non minimum odmax. -- We have the same problem on 'def' itself, but we leave it. odmaxmin : Set (suc n) @@ -122,20 +122,14 @@ ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥ ¬OD-order & * c<→o< = o≤> <-osuc (c<→o< {Ords} OneObj ) --- Open supreme upper bound leads a contradition, so we use domain restriction on sup -¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ -¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where - next-ord : Ordinal → Ordinal - next-ord x = osuc x - -- Ordinal in OD ( and ZFSet ) Transitive Set -Ord : ( a : Ordinal ) → HOD +Ord : ( a : Ordinal ) → HOD Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where lemma : {x : Ordinal} → x o< a → x o< a lemma {x} lt = lt -od∅ : HOD -od∅ = Ord o∅ +od∅ : HOD +od∅ = Ord o∅ odef : HOD → Ordinal → Set n odef A x = def ( od A ) x @@ -144,7 +138,7 @@ _∋_ a x = odef a ( & x ) -- _c<_ : ( x a : HOD ) → Set n --- x c< a = a ∋ x +-- x c< a = a ∋ x d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x) d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt @@ -169,11 +163,11 @@ ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y ==-iso {x} {y} eq = record { - eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ; - eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) } + eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ; + eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) } where lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z - lemma {x} {z} d = subst (λ k → odef k z) (*iso) d + lemma {x} {z} d = subst (λ k → odef k z) (*iso) d =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y) =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso) @@ -192,7 +186,7 @@ &≡&→≡ : { x y : HOD } → & x ≡ & y → x ≡ y &≡&→≡ eq = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) eq ) -o∅≡od∅ : * (o∅ ) ≡ od∅ +o∅≡od∅ : * (o∅ ) ≡ od∅ o∅≡od∅ = ==→o≡ lemma where lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt) @@ -202,20 +196,20 @@ lemma : od (* o∅) == od od∅ lemma = record { eq→ = lemma0 ; eq← = lemma1 } -ord-od∅ : & (od∅ ) ≡ o∅ +ord-od∅ : & (od∅ ) ≡ o∅ ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) ) ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅ ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt)))) - ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} + ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} -=od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅ +=od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅ =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅ ≡od∅→=od∅ : {x : HOD} → x ≡ od∅ → od x == od od∅ ≡od∅→=od∅ {x} eq = ≡o∅→=od∅ (subst (λ k → & x ≡ k ) ord-od∅ ( cong & eq ) ) -∅0 : record { def = λ x → Lift n ⊥ } == od od∅ +∅0 : record { def = λ x → Lift n ⊥ } == od od∅ eq→ ∅0 {w} (lift ()) eq← ∅0 {w} lt = lift (¬x<0 lt) @@ -241,8 +235,14 @@ is-o∅ x | tri≈ ¬a b ¬c = yes b is-o∅ x | tri> ¬a ¬b c = no ¬b +odef< : {b : Ordinal } { A : HOD } → odef A b → b o< & A +odef< {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) + +odef∧< : {A : HOD } {y : Ordinal} {n : Level } → {P : Set n} → odef A y ∧ P → y o< & A +odef∧< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) + -- the pair -_,_ : HOD → HOD → HOD +_,_ : HOD → HOD → HOD x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y) lemma {t} (case1 refl) = omax-x _ _ @@ -258,15 +258,15 @@ odmax<& : { x y : HOD } → x ∋ y → Set n odmax<& {x} {y} x∋y = odmax x o< & x -in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD +in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) } -_∩_ : ( A B : HOD ) → HOD +_∩_ : ( A B : HOD ) → HOD A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} record _⊆_ ( A B : HOD ) : Set (suc n) where - field + field incl : { x : HOD } → A ∋ x → B ∋ x open _⊆_ @@ -275,7 +275,7 @@ -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o< ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) ) → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) ) - → {x y : HOD } → def (od y) ( & x ) → & x o< & y + → {x y : HOD } → def (od y) ( & x ) → & x o< & y ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y) ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x ))) @@ -285,36 +285,64 @@ lemma (case1 refl) = refl lemma (case2 refl) = refl y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z - y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x + y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x lemma1 : osuc (& y) o< & (x , x) - lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c ) + lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c ) ε-induction : { ψ : HOD → Set (suc n)} → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) → (x : HOD ) → ψ x ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox) - induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso ))) + induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso ))) ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy) ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy -Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD +record OSUP (A x : Ordinal ) ( ψ : ( x : Ordinal ) → odef (* A) x → Ordinal ) : Set n where + field + sup-o' : Ordinal + sup-o≤' : {z : Ordinal } → z o< x → (lt : odef (* A) z ) → ψ z lt o≤ sup-o' + +-- o<-sup : ( A x : Ordinal) { ψ : ( x : Ordinal ) → odef (* A) x → Ordinal } → OSUP A x ψ +-- o<-sup A x {ψ} = ? where +-- m00 : (x : Ordinal ) → OSUP A x ψ +-- m00 x = Ordinals.inOrdinal.TransFinite0 O ind x where +-- ind : (x : Ordinal) → ((z : Ordinal) → z o< x → OSUP A z ψ ) → OSUP A x ψ +-- ind x prev = ? where +-- if has prev , od01 is empty use prev (cannot be odef (* A) x) +-- not empty, take larger +-- limit ordinal, take address of Union +-- +-- od01 : HOD +-- od01 = record { od = record { def = λ z → odef A z ∧ ( z ≡ & x ) } ; odmax = & A ; <odmax = odef∧< } +-- m01 : OSUP A x ψ +-- m01 with is-o∅ (& od01 ) +-- ... | case1 t=0 = record { sup-o' = prevjjk +-- ... | case2 t>0 = ? + +-- Open supreme upper bound leads a contradition, so we use domain restriction on sup +¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ +¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where + next-ord : Ordinal → Ordinal + next-ord x = osuc x + +Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) } -Replace : HOD → (HOD → HOD) → HOD +Replace : HOD → (HOD → HOD) → HOD Replace X ψ = record { od = record { def = λ x → (x o≤ sup-o X (λ y X∋y → & (ψ (* y)))) ∧ def (in-codomain X ψ) x } - ; odmax = rmax ; <odmax = rmax<} where + ; odmax = rmax ; <odmax = rmax<} where rmax : Ordinal rmax = osuc ( sup-o X (λ y X∋y → & (ψ (* y)))) rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax rmax< lt = proj1 lt -- --- If we have LEM, Replace' is equivalent to Replace +-- If we have LEM, Replace' is equivalent to Replace -- -in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD +in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ & (ψ (* y ) (d→∋ X lt) )))) } -Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD +Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x } ; odmax = rmax ; <odmax = rmax< } where rmax : Ordinal @@ -322,7 +350,7 @@ rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax rmax< lt = proj1 lt -Union : HOD → HOD +Union : HOD → HOD Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x))) } ; odmax = osuc (& U) ; <odmax = umax< } where umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (* u)) y) → y o< osuc (& U) @@ -330,7 +358,7 @@ lemma0 : {x : Ordinal} → def (od (* x)) y → y o< x lemma0 {x} x<y = subst₂ (λ j k → j o< k ) &iso &iso (c<→o< (d→∋ (* x) x<y )) lemma2 : {x : Ordinal} → def (od U) x → x o< & U - lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U)) + lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U)) lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (* x)) y → ¬ (& U o< y) lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) lemma : ¬ ((& U) o< y ) → y o< osuc (& U) @@ -341,10 +369,10 @@ _∈_ : ( A B : HOD ) → Set n A ∈ B = B ∋ A -OPwr : (A : HOD ) → HOD +OPwr : (A : HOD ) → HOD OPwr A = Ord ( osuc ( sup-o (Ord (osuc (& A))) ( λ x A∋x → & ( A ∩ (* x)) ) ) ) -Power : HOD → HOD +Power : HOD → HOD Power A = Replace (OPwr (Ord (& A))) ( λ x → A ∩ x ) -- ｛_｝ : ZFSet → ZFSet -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly @@ -364,25 +392,25 @@ -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. -- We simply assumes infinite-d y has a maximum. --- +-- -- This means that many of OD may not be HODs because of the & mapping divergence. -- We should have some axioms to prevent this such as & x o< next (odmax x). --- +-- -- postulate -- ωmax : Ordinal -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax --- --- infinite : HOD --- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } +-- +-- infinite : HOD +-- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } -infinite : HOD +infinite : HOD infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where u : (y : Ordinal ) → HOD u y = Union (* y , (* y , * y)) -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y)) lemma8 = ho< - --- (x,y) < next (omax x y) < next (osuc y) = next y + --- (x,y) < next (omax x y) < next (osuc y) = next y lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y) lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< ) lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y)) @@ -399,24 +427,24 @@ lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1)) empty : (x : HOD ) → ¬ (od∅ ∋ x) -empty x = ¬x<0 +empty x = ¬x<0 _=h=_ : (x y : HOD) → Set n x =h= y = od x == od y -pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) +pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x )) pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y )) -pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t +pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x)) pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y)) -o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) -o<→c< lt = record { incl = λ z → ordtrans z lt } +o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) +o<→c< lt = record { incl = λ z → ordtrans z lt } ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y -⊆→o< {x} {y} lt with trio< x y +⊆→o< {x} {y} lt with trio< x y ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym &iso) refl ) @@ -430,14 +458,14 @@ , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ ) ⟫ -selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y +selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt) sup-c≤ : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → & (ψ x) o≤ (sup-o X (λ y X∋y → & (ψ (* y)))) sup-c≤ ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o≤ X lt ) replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x -replacement← {ψ} X x lt = ⟪ sup-c≤ ψ {X} {x} lt , lemma ⟫ where +replacement← {ψ} X x lt = ⟪ sup-c≤ ψ {X} {x} lt , lemma ⟫ where lemma : def (in-codomain X ψ) (& (ψ x)) lemma not = ⊥-elim ( not ( & x ) ⟪ lt , cong (λ k → & (ψ k)) (sym *iso)⟫ ) replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) @@ -445,7 +473,7 @@ lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((& x) ≡ & (ψ (* y)))) → ¬ ((y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y))) lemma2 not not2 = not ( λ y d → not2 y ⟪ proj1 d , lemma3 (proj2 d)⟫) where - lemma3 : {y : Ordinal } → (& x ≡ & (ψ (* y))) → (* (& x) =h= ψ (* y)) + lemma3 : {y : Ordinal } → (& x ≡ & (ψ (* y))) → (* (& x) =h= ψ (* y)) lemma3 {y} eq = subst (λ k → * (& x) =h= k ) *iso (o≡→== eq ) lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)) ) lemma not y not2 = not (* y) (subst (λ k → k =h= ψ (* y)) *iso ( proj2 not2 )) @@ -462,29 +490,29 @@ ∩-≡ {a} {b} inc = record { eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ; eq← = λ {x} x<a∩b → proj2 x<a∩b } --- +-- -- Transitive Set case -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t --- OPwr A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) ) --- +-- OPwr A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) ) +-- ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t -ord-power← a t t→A = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where +ord-power← a t t→A = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where lemma-eq : ((Ord a) ∩ t) =h= t - eq→ lemma-eq {z} w = proj2 w - eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫ + eq→ lemma-eq {z} w = proj2 w + eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫ lemma1 : {a : Ordinal } { t : HOD } → (eq : ((Ord a) ∩ t) =h= t) → & ((Ord a) ∩ (* (& t))) ≡ & t lemma1 {a} {t} eq = subst (λ k → & ((Ord a) ∩ k) ≡ & t ) (sym *iso) (cong (λ k → & k ) (==→o≡ eq )) lemma2 : (& t) o< (osuc (& (Ord a))) - lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t))) + lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t))) lemma : & ((Ord a) ∩ (* (& t)) ) o≤ sup-o (Ord (osuc (& (Ord a)))) (λ x lt → & ((Ord a) ∩ (* x))) lemma = sup-o≤ _ lemma2 --- +-- -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (& A)) first -- then replace of all elements of the Power set by A ∩ y --- +-- -- Power A = Replace (OPwr (Ord (& A))) ( λ y → A ∩ y ) -- we have oly double negation form because of the replacement axiom @@ -502,7 +530,7 @@ lemma5 {y} eq not = (lemma3 (* y) eq) not power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t -power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where +power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where a = & A lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x lemma0 {x} t∋x = c<→o< (t→A t∋x) @@ -519,17 +547,17 @@ sup1 : Ordinal sup1 = sup-o (Ord (osuc (& (Ord (& A))))) (λ x A∋x → & ((Ord (& A)) ∩ (* x))) lemma9 : def (od (Ord (Ordinals.osuc O (& (Ord (& A)))))) (& (Ord (& A))) - lemma9 = <-osuc + lemma9 = <-osuc lemmab : & ((Ord (& A)) ∩ (* (& (Ord (& A) )))) o≤ sup1 - lemmab = sup-o≤ (Ord (osuc (& (Ord (& A))))) lemma9 + lemmab = sup-o≤ (Ord (osuc (& (Ord (& A))))) lemma9 lemmad : Ord (osuc (& A)) ∋ t - lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt))) + lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt))) lemmac : ((Ord (& A)) ∩ (* (& (Ord (& A) )))) =h= Ord (& A) lemmac = record { eq→ = lemmaf ; eq← = lemmag } where lemmaf : {x : Ordinal} → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x → def (od (Ord (& A))) x lemmaf {x} lt = proj1 lt lemmag : {x : Ordinal} → def (od (Ord (& A))) x → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x - lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫ + lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫ lemmae : & ((Ord (& A)) ∩ (* (& (Ord (& A))))) ≡ & (Ord (& A)) lemmae = cong (λ k → & k ) ( ==→o≡ lemmac) lemma7 : def (od (OPwr (Ord (& A)))) (& t) @@ -540,18 +568,18 @@ lemmah {x} lt = subst (λ k → def (od k) x ) (sym *iso) (subst (λ k → k o< (& t)) &iso (c<→o< (subst₂ (λ j k → def (od j) k) *iso (sym &iso) lt ))) - lemma7 | case1 eq | case1 eq1 = subst (λ k → k o≤ sup1) (trans lemmae lemmai) lemmab where + lemma7 | case1 eq | case1 eq1 = subst (λ k → k o≤ sup1) (trans lemmae lemmai) lemmab where lemmai : & (Ord (& A)) ≡ & t lemmai = let open ≡-Reasoning in begin - & (Ord (& A)) + & (Ord (& A)) ≡⟨ sym (cong (λ k → & (Ord k)) eq) ⟩ - & (Ord (& t)) + & (Ord (& t)) ≡⟨ sym &iso ⟩ & (* (& (Ord (& t)))) ≡⟨ sym eq1 ⟩ & (* (& t)) ≡⟨ &iso ⟩ - & t + & t ∎ lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o≤ sup1) lemmae lemmab ) where lemmak : & (* (& (Ord (& t)))) ≡ & (Ord (& A)) @@ -563,26 +591,26 @@ & (Ord (& A)) ∎ lemmaj : & t o< & (Ord (& A)) - lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt + lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt lemma1 : & t o≤ sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x))) lemma1 = subst (λ k → & k o≤ sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x)))) lemma4 (sup-o≤ (OPwr (Ord (& A))) lemma7 ) lemma2 : def (in-codomain (OPwr (Ord (& A))) (_∩_ A)) (& t) lemma2 not = ⊥-elim ( not (& t) ⟪ lemma3 , lemma6 ⟫ ) where - lemma6 : & t ≡ & (A ∩ * (& t)) + lemma6 : & t ≡ & (A ∩ * (& t)) lemma6 = cong ( λ k → & k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym *iso) ( ∩-≡ {t} {A} t→A ))) extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B -eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d -eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d +eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d +eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d -proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d +proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d -infinity∅ : infinite ∋ od∅ -infinity∅ = subst (λ k → odef infinite k ) lemma iφ where +infinity∅ : infinite ∋ od∅ +infinity∅ = subst (λ k → odef infinite k ) lemma iφ where lemma : o∅ ≡ & od∅ lemma = let open ≡-Reasoning in begin o∅ @@ -595,7 +623,7 @@ infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where lemma : & (Union (* (& x) , (* (& x) , * (& x)))) ≡ & (Union (x , (x , x))) - lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso + lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite isZF = record { @@ -605,22 +633,22 @@ ; union→ = union→ ; union← = union← ; empty = empty - ; power→ = power→ - ; power← = power← - ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} + ; power→ = power→ + ; power← = power← + ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} ; ε-induction = ε-induction ; infinity∅ = infinity∅ ; infinity = infinity ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} ; replacement← = replacement← ; replacement→ = λ {ψ} → replacement→ {ψ} - } + } -HOD→ZF : ZF -HOD→ZF = record { - ZFSet = HOD - ; _∋_ = _∋_ - ; _≈_ = _=h=_ +HOD→ZF : ZF +HOD→ZF = record { + ZFSet = HOD + ; _∋_ = _∋_ + ; _≈_ = _=h=_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union @@ -628,7 +656,7 @@ ; Select = Select ; Replace = Replace ; infinite = infinite - ; isZF = isZF - } - + ; isZF = isZF + } +
--- a/src/zorn.agda Mon Dec 19 09:50:51 2022 +0900 +++ b/src/zorn.agda Tue Dec 20 11:20:52 2022 +0900 @@ -4,7 +4,7 @@ open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality -import OD +import OD hiding ( _⊆_ ) module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where -- @@ -14,7 +14,7 @@ -- → Maximal A -- -open import zf +open import zf -- hiding ( _⊆_ ) open import logic -- open import partfunc {n} O @@ -90,10 +90,12 @@ ptrans = IsStrictPartialOrder.trans PO open _==_ -open _⊆_ +-- open _⊆_ -- we use different definition +-- We cannot prove this without Well foundedness of A +-- -- <-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 +-- → ({y : HOD} → A ∋ y → y < x → P y ) → P x -- <-TransFinite = ? -- @@ -216,16 +218,11 @@ 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 -_⊆'_ : ( A B : HOD ) → Set n -_⊆'_ A B = {x : Ordinal } → odef A x → odef B x - --- --- inductive masum tree from x --- tree structure --- +⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B +⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (B⊆A ax) (B⊆A ay) record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where field @@ -247,6 +244,7 @@ x≤sup = IsSUP.x≤sup isSUP -- +-- Our Proof strategy of the Zorn Lemma -- -- f (f ( ... (supf y))) f (f ( ... (supf z1))) -- / | / | @@ -339,7 +337,7 @@ chain : HOD chain = UnionCF A f ay supf z - chain⊆A : chain ⊆' A + chain⊆A : chain ⊆ A chain⊆A = λ lt → proj1 lt chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y @@ -382,14 +380,6 @@ initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc) - supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b - supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b) - ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( - IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) - ... | tri≈ ¬a b ¬c = b - ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( - IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) - sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z → IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b sup=u {b} ab b≤z is-sup = z50 where @@ -403,6 +393,14 @@ ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) + supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b + supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b) + ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( + IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) + ... | tri≈ ¬a b ¬c = b + ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( + IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) + union-max : {a b : Ordinal } → b o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b union-max {a} {b} b≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w @@ -553,7 +551,7 @@ Zorn-lemma : { A : HOD } → o∅ o< & A - → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition + → ( ( 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 → ⊥ @@ -585,7 +583,7 @@ -- -- we have minsup using LEM, this is similar to the proof of the axiom of choice -- - minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B + 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 ) @@ -835,7 +833,7 @@ lt1 = subst₂ (λ j k → j < k ) *iso *iso lt ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf (proj1 a) (proj1 b)) - pcha : pchainpx ⊆' A + pcha : pchainpx ⊆ A pcha (case1 lt) = proj1 lt pcha (case2 fc) = A∋fc _ f mf (proj1 fc) @@ -1175,23 +1173,9 @@ uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia uz03 = sa<x - chain⊆pchainU : {z w : Ordinal } → (oz<x : osuc z o< x) → odef (ZChain.chain (pzc oz<x)) w → odef pchainU w - chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ - chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ - = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where - u<x : u o< x - u<x = ordtrans u<oz oz<x - su=su : ZChain.supf (pzc oz<x) u ≡ supfz u<x - su=su = sym ( zeq _ _ (osucc u<oz) (o<→≤ <-osuc) ) - su≡u : supfz u<x ≡ u - su≡u = begin - ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ - ZChain.supf (pzc oz<x) u ≡⟨ su=u ⟩ - u ∎ where open ≡-Reasoning - - chain⊆pchainU1 : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w - chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ - chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ + chain⊆pchainU : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w + chain⊆pchainU {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ + chain⊆pchainU {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where u<x : u o< x u<x = ordtrans u<oz z<x @@ -1203,20 +1187,6 @@ ZChain.supf (pzc (ob<x lim z<x)) u ≡⟨ su=u ⟩ u ∎ where open ≡-Reasoning - ichain-inject : {a b : Ordinal } {ia : IChain ay supfz a } {ib : IChain ay supfz b } - → ZChain.supf (pzc (pic<x ia)) (IChain-i ia) o< ZChain.supf (pzc (pic<x ib)) (IChain-i ib) - → IChain-i ia o< IChain-i ib - ichain-inject {a} {b} {ia} {ib} sa<sb = uz11 where - uz11 : IChain-i ia o< IChain-i ib - uz11 with trio< (IChain-i ia ) (IChain-i ib) - ... | tri< a ¬b ¬c = a - ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (trans (zeq _ _ (o≤-refl0 (cong osuc b)) (o<→≤ <-osuc) ) - ( cong (ZChain.supf (pzc (pic<x ib))) b )) sa<sb ) - ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( begin - ZChain.supf (pzc (pic<x ib)) (IChain-i ib) ≡⟨ zeq _ _ (o<→≤ (osucc c)) (o<→≤ <-osuc) ⟩ - ZChain.supf (pzc (pic<x ia)) (IChain-i ib) ≤⟨ ZChain.supf-mono (pzc (pic<x ia)) (o<→≤ c) ⟩ - ZChain.supf (pzc (pic<x ia)) (IChain-i ia) ∎ ) sa<sb ) where open o≤-Reasoning O - IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b ) → IChain-i ia o< IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a IC⊆ {a} {b} (ic-init fc ) ib ia<ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫ @@ -1302,7 +1272,7 @@ lt1 = subst₂ (λ j k → j < k ) *iso *iso lt ptotalS (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp spu0 f mf (proj1 a) (proj1 b)) - S⊆A : pchainS ⊆' A + S⊆A : pchainS ⊆ A S⊆A (case1 lt) = proj1 lt S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc) @@ -1368,12 +1338,12 @@ zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y) ... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x)) ... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) - (MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU1 z<x uw)) ) + (MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU z<x uw)) ) ... | tri> ¬a ¬b c = zc01 where -- supf1 z o≤ sps zc01 : supf1 z o≤ sps zc01 with trio< z x ... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) - (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU1 z<x uw)) ) + (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU z<x uw)) ) ... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) ) ... | tri> ¬a ¬b c = o≤-refl -- (sf1=sps c) @@ -1422,7 +1392,6 @@ ⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫ - cfcs : {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x ... | case1 b=x with trio< a x @@ -1509,7 +1478,7 @@ z70 : odef (UnionCF A f ay supf1 z) (supf1 spu) z70 = cfcs spu<x o≤-refl ssp<x (init asupf refl ) z73 : IsSUP A (UnionCF A f ay (ZChain.supf (pzc (ob<x lim spu<x))) spu) spu - z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU1 spu<x uw ) } + z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU spu<x uw ) } z49 : supfz spu<x ≡ spu z49 = begin supfz spu<x ≡⟨ ZChain.sup=u (pzc (ob<x lim spu<x)) (MinSUP.as usup) (o<→≤ <-osuc) z73 ⟩ @@ -1622,12 +1591,19 @@ -- usage (see filter.agda ) -- --- _⊆'_ : ( A B : HOD ) → Set n --- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x +-- import OD hiding ( _⊆_ ) +-- _⊆_ : ( A B : HOD ) → Set n +-- _⊆_ A B = {x : Ordinal } → odef A x → odef B x +-- +-- import zorn +-- open zorn O _⊆_ -- Zorn on Set inclusion order +-- +-- open import Relation.Binary.Structures -- 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 +-- → 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 +--