open import Level module constructible-set where open import zf open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality data OrdinalD {n : Level} : (lv : Nat) → Set n where Φ : (lv : Nat) → OrdinalD lv OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv ℵ_ : (lv : Nat) → OrdinalD (Suc lv) record Ordinal {n : Level} : Set n where field lv : Nat ord : OrdinalD {n} lv data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → Φ (Suc lx) d< (ℵ lx) ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → OSuc (Suc lx) x d< (ℵ lx) open Ordinal _o<_ : {n : Level} ( x y : Ordinal ) → Set n _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core o∅ : {n : Level} → Ordinal {n} o∅ = record { lv = Zero ; ord = Φ Zero } ≡→¬d< : {n : Level} → {lv : Nat} → {x : OrdinalD {n} lv } → x d< x → ⊥ ≡→¬d< {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t trio<> : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → y d< x → x d< y → ⊥ trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ trio<≡ refl = ≡→¬d< trio>≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → y d< x → ⊥ trio>≡ refl = ≡→¬d< triO : {n : Level} → {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly ) triO {n} {lx} {ly} x y = <-cmp lx ly triOrdd : {n : Level} → {lx : Nat} → Trichotomous _≡_ ( _d<_ {n} {lx} {lx} ) triOrdd {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d< triOrdd {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< (ℵΦ< {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) ) triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ< {_} {lv} {Φ (Suc lv)} ) triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} ) ) (λ ()) (ℵ< {_} {lv} {y} ) triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< ) triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) ) triOrdd {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d< triOrdd {_} {lv} (OSuc lv x) (OSuc lv y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c) d<→lv : {n : Level} {x y : Ordinal {n}} → ord x d< ord y → lv x ≡ lv y d<→lv Φ< = refl d<→lv (s< lt) = refl d<→lv ℵΦ< = refl d<→lv ℵ< = refl orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n} lx } → x d< y → y d< z → x d< z orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y ¬a ¬b c = x maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal maxα x y with <-cmp (lv x) (lv y) maxα x y | tri< a ¬b ¬c = x maxα x y | tri> ¬a ¬b c = y maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } _o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n) a o≤ b = (a ≡ b) ∨ ( a o< b ) ordtrans : {n : Level} {x y z : Ordinal {n} } → x o< y → y o< z → x o< z ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ ) ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with d<→lv x₂ ... | refl = case1 x₁ ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂) with d<→lv x₁ ... | refl = case1 x₂ ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂ ... | refl | refl = case2 ( orddtrans x₁ x₂ ) trio< : {n : Level } → Trichotomous {suc n} _≡_ _o<_ trio< a b with <-cmp (lv a) (lv b) trio< a b | tri< a₁ ¬b ¬c = tri< (case1 a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a) lemma1 (case1 x) = ¬c x lemma1 (case2 x) with d<→lv x lemma1 (case2 x) | refl = ¬b refl trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b) lemma1 (case1 x) = ¬a x lemma1 (case2 x) with d<→lv x lemma1 (case2 x) | refl = ¬b refl trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b ) trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b lemma1 refl = refl lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x) lemma2 (case1 x) = ¬a x lemma2 (case2 x) = trio<> x a trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where lemma1 : (record { lv = _ ; ord = x }) ≡ b → x ≡ ord b lemma1 refl = refl lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b) lemma2 (case1 x) = ¬a x lemma2 (case2 x) = trio<> x c trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b) lemma1 (case1 x) = ¬a x lemma1 (case2 x) = ≡→¬d< x OrdTrans : {n : Level} → Transitive {suc n} _o≤_ OrdTrans (case1 refl) (case1 refl) = case1 refl OrdTrans (case1 refl) (case2 lt2) = case2 lt2 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y )) OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n) OrdPreorder {n} = record { Carrier = Ordinal ; _≈_ = _≡_ ; _∼_ = _o≤_ ; isPreorder = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = case1 ; trans = OrdTrans } } TransFinite : {n : Level} → ( ψ : Ordinal {n} → Set n ) → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) ) → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) ) → ∀ (x : Ordinal) → ψ x TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ ( TransFinite ψ caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) TransFinite ψ caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) ' -- Ordinal Definable Set record OD {n : Level} : Set (suc n) where field α : Ordinal {n} def : (x : Ordinal {n} ) → x o< α → Set n open OD open import Data.Unit postulate -- this is proved by Godel numbering of def _c<_ : {n : Level } → (x y : OD {n} ) → Set (suc n) ODpre : {n : Level} → IsPreorder {suc n} {suc n} {suc n} _≡_ _c<_ -- o∋ : {n : Level} → {A : Ordinal {n}} → (OrdinalDefinable {n} A ) → (x : Ordinal {n} ) → (x o< A) → Set n -- o∋ a x x