Mercurial > hg > Members > kono > Proof > ZF-in-agda
view ordinal-definable.agda @ 180:11490a3170d4
new ordinal-definable
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 20 Jul 2019 14:05:32 +0900 |
| parents | ea0e7927637a |
| children | 7012158bf2d9 |
line wrap: on
line source
{-# OPTIONS --allow-unsolved-metas #-} open import Level module ordinal-definable where open import zf open import ordinal open import HOD open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core -- Ordinal Definable Set open OD open import Data.Unit open Ordinal open _==_ postulate -- a property of supermum required in Power Set Axiom sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) ... | t = lemma0 (lemma t) where lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x) lemma xo<z = o<→c< xo<z lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x) lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl ) ... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) ... | () o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl ) ... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx ) ... | () ==→o≡o : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y ==→o≡o {n} {x} {y} eq with trio< {n} x y ==→o≡o {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) ==→o≡o {n} {x} {y} eq | tri≈ ¬a b ¬c = b ==→o≡o {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) ≡-def {n} {x} = ==→o≡o {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where lemma : ord→od x == record { def = λ z → z o< x } eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) ==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y ==→o≡1 eq = ==→o≡o (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq ) ==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y ==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡o eq) z>x ==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z ==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where t : def (ord→od (od→ord a)) (od→ord x) t = o<→c< {suc n} {od→ord x} {od→ord a} lt o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) o<→¬== {n} {x} {y} lt eq = o<→o> eq lt tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a ) tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso refl) c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ c<> {n} {x} {y} x<y y<x with tri-c< x y c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y is-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) is-∋ {n} x y with tri-c< x y is-∋ {n} x y | tri< a ¬b ¬c = no ¬c is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c is-∋ {n} x y | tri> ¬a ¬b c = yes c open _∧_ -- -- This menas OD is Ordinal here -- ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} lemma ox ne with is-o∅ ox lemma ox ne | yes refl with ne ( ord→== lemma1 ) where lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ lemma o∅ ne | yes refl | () lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) ) -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) csuc : {n : Level} → OD {suc n} → OD {suc n} csuc x = Ord ( osuc ( od→ord x )) Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n} Ord→ZF {n} = record { ZFSet = OD {suc n} ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) ; isZF = isZF } where Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } _,_ : OD {suc n} → OD {suc n} → OD {suc n} x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } _∩_ : ( A B : OD {suc n} ) → OD A ∩ B = record { def = λ x → def A x ∧ def B x } Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x } Union : OD {suc n} → OD {suc n} Union U = record { def = λ y → osuc y o< (od→ord U) } -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) Power : OD {suc n} → OD {suc n} Power A = Def A ZFSet = OD {suc n} _∈_ : ( A B : ZFSet ) → Set (suc n) A ∈ B = B ∋ A _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) _⊆_ A B {x} = A ∋ x → B ∋ x -- _∪_ : ( A B : ZFSet ) → ZFSet -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) infixr 200 _∈_ -- infixr 230 _∩_ _∪_ infixr 220 _⊆_ isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } ; pair = pair ; union→ = union→ ; union← = union← ; empty = empty ; power→ = power→ ; power← = power← ; extensionality = extensionality ; minimul = minimul ; regularity = regularity ; infinity∅ = infinity∅ ; infinity = infinity ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} ; replacement← = replacement← ; replacement→ = replacement→ } where pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) empty x (case1 ()) empty x (case2 ()) --- --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A -- -- if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity -- power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) power→ A t P∋t {x} t∋x = double-neg (proj1 lemma-s) where minsup : OD minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) lemma-t : csuc minsup ∋ t lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x lemma-s with osuc-≡< ( o<-subst (c<→o< {!!} ) refl diso ) lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) t∋x -- -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t -- Power A is a sup of ZFSubset A t, so Power A ∋ t -- power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} {!!} refl lemma1 where lemma-eq : ZFSubset A t == t eq→ lemma-eq {z} w = proj2 w eq← lemma-eq {z} w = record { proj2 = w ; proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq)) lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) lemma = sup-o< union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z union-lemma-u {X} {z} U>z = {!!} where -- lemma <-osuc where lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} (o<→c< lt) refl refl union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) union→ X y u xx | tri< a ¬b ¬c | () union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX lemma refl lt = lt union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) -- (X ∋ csuc z) ∧ (csuc z ∋ z ) union← X z X∋z not = not (csuc z) record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym {!!}) ; proj2 = union-lemma-u X∋z } ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) selection {ψ} {X} {y} = record { proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } } replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where lemma : def (in-codomain X ψ) (od→ord (ψ x)) lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ) ) replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) replacement→ {ψ} X x lt = contra-position lemma (lemma2 {!!}) where lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (Ord y)))) → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y))) lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (Ord y))) → (ord→od (od→ord x) == ψ (Ord y)) lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)) ) lemma not y not2 = not (Ord y) (subst (λ k → k == ψ (Ord y)) oiso ( proj2 not2 )) minimul-o : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} minimul-o x not = od∅ regularity : (x : OD) (not : ¬ (x == od∅)) → (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) proj1 (regularity x not ) = {!!} proj2 (regularity x not ) = record { eq→ = reg ; eq← = lemma } where lemma : {ox : Ordinal} → def od∅ ox → def (Select (minimul x not) (λ y → (minimul x not ∋ y) ∧ (x ∋ y))) ox lemma (case1 ()) lemma (case2 ()) reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y reg {y} t = ⊥-elim ( ¬x<0 {!!} ) extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) lemma1 {x} = c<→o< ( proj1 (pair x x ) ) lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡o (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) omega = record { lv = Suc Zero ; ord = Φ 1 } infinite : OD {suc n} infinite = ord→od ( omega ) infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} (o<→c< ( case1 (s≤s z≤n ))) refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) o∅≡od∅ )) infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where t : od→ord x o< od→ord (ord→od (omega)) t = ∋→o< {n} {infinite} {x} lt infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) infinite∋uxxx x lt = o<∋→ t where t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt ) infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl -- Axiom of choice ( is equivalent to the existence of minimul in our case ) -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD choice-func X {x} not X∋x = od∅ {suc n} choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A choice X {A} X∋A not = ¬∅=→∅∈ not -- another form of regularity -- ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) → (x : OD {suc n} ) → ψ x ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) ε-induction-ord Zero (Φ 0) (case1 ()) ε-induction-ord Zero (Φ 0) (case2 ()) ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } lemma y lt with osuc-≡< y<x lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where -- -- if lv of z if less than x Ok -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma -- -- lx Suc lx (1) lz(a) <lx by case1 -- ly(1) ly(2) (2) lz(b) <lx by case1 -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) -- lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡⟨ cong ( λ k → lv k ) diso ⟩ lv (record { lv = ly ; ord = oy }) ≡⟨⟩ ly ∎ lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt lemma z lt | case1 lz<ly with <-cmp lx ly lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) lemma z lt | case2 lz=ly with <-cmp lx ly lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → lx ≡ ly → ly ≡ lv (od→ord z) → ψ z lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)