open import Level module Ordinals where open import zf open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Data.Empty open import Relation.Binary.PropositionalEquality open import logic open import nat open import Data.Unit using ( ⊤ ) open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord ) (osuc : ord → ord ) (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where field Otrans : {x y z : ord } → x o< y → y o< z → x o< z OTri : Trichotomous {n} _≡_ _o<_ ¬x<0 : { x : ord } → ¬ ( x o< o∅ ) <-osuc : { x : ord } → x o< osuc x osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a) TransFinite : { ψ : ord → Set (suc n) } → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x ) → ∀ (x : ord) → ψ x record Ordinals {n : Level} : Set (suc (suc n)) where field ord : Set n o∅ : ord osuc : ord → ord _o<_ : ord → ord → Set n isOrdinal : IsOrdinals ord o∅ osuc _o<_ module inOrdinal {n : Level} (O : Ordinals {n} ) where Ordinal : Set n Ordinal = Ordinals.ord O _o<_ : Ordinal → Ordinal → Set n _o<_ = Ordinals._o<_ O osuc : Ordinal → Ordinal osuc = Ordinals.osuc O o∅ : Ordinal o∅ = Ordinals.o∅ O ¬x<0 = IsOrdinals.¬x<0 (Ordinals.isOrdinal O) osuc-≡< = IsOrdinals.osuc-≡< (Ordinals.isOrdinal O) <-osuc = IsOrdinals.<-osuc (Ordinals.isOrdinal O) TransFinite = IsOrdinals.TransFinite (Ordinals.isOrdinal O) o<-dom : { x y : Ordinal } → x o< y → Ordinal o<-dom {x} _ = x o<-cod : { x y : Ordinal } → x o< y → Ordinal o<-cod {_} {y} _ = y o<-subst : {Z X z x : Ordinal } → Z o< X → Z ≡ z → X ≡ x → z o< x o<-subst df refl refl = df ordtrans : {x y z : Ordinal } → x o< y → y o< z → x o< z ordtrans = IsOrdinals.Otrans (Ordinals.isOrdinal O) trio< : Trichotomous _≡_ _o<_ trio< = IsOrdinals.OTri (Ordinals.isOrdinal O) o<¬≡ : { ox oy : Ordinal } → ox ≡ oy → ox o< oy → ⊥ o<¬≡ {ox} {oy} eq lt with trio< ox oy o<¬≡ {ox} {oy} eq lt | tri< a ¬b ¬c = ¬b eq o<¬≡ {ox} {oy} eq lt | tri≈ ¬a b ¬c = ¬a lt o<¬≡ {ox} {oy} eq lt | tri> ¬a ¬b c = ¬b eq o<> : {x y : Ordinal } → y o< x → x o< y → ⊥ o<> {ox} {oy} lt tl with trio< ox oy o<> {ox} {oy} lt tl | tri< a ¬b ¬c = ¬c lt o<> {ox} {oy} lt tl | tri≈ ¬a b ¬c = ¬a tl o<> {ox} {oy} lt tl | tri> ¬a ¬b c = ¬a tl osuc-< : { x y : Ordinal } → y o< osuc x → x o< y → ⊥ osuc-< {x} {y} y x x < osuc x -> y = x ∨ x < y → ⊥ osucc {ox} {oy} oy ¬a ¬b c with osuc-≡< c osucc {ox} {oy} oy ¬a ¬b c | case1 eq = ⊥-elim (o<¬≡ (sym eq) oy ¬a ¬b c | case2 lt = ⊥-elim (o<> lt oy ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) maxα : Ordinal → Ordinal → Ordinal maxα x y with trio< x y maxα x y | tri< a ¬b ¬c = y maxα x y | tri> ¬a ¬b c = x maxα x y | tri≈ ¬a refl ¬c = x minα : Ordinal → Ordinal → Ordinal minα x y with trio< x y minα x y | tri< a ¬b ¬c = x minα x y | tri> ¬a ¬b c = y minα x y | tri≈ ¬a refl ¬c = x min1 : {x y z : Ordinal } → z o< x → z o< y → z o< minα x y min1 {x} {y} {z} z ¬a ¬b c = z ¬a ¬b c = osuc x omax x y | tri≈ ¬a refl ¬c = osuc x omax< : ( x y : Ordinal ) → x o< y → osuc y ≡ omax x y omax< x y lt with trio< x y omax< x y lt | tri< a ¬b ¬c = refl omax< x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) omax< x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) omax≡ : ( x y : Ordinal ) → x ≡ y → osuc y ≡ omax x y omax≡ x y eq with trio< x y omax≡ x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) omax≡ x y eq | tri≈ ¬a refl ¬c = refl omax≡ x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) omax-x : ( x y : Ordinal ) → x o< omax x y omax-x x y with trio< x y omax-x x y | tri< a ¬b ¬c = ordtrans a <-osuc omax-x x y | tri> ¬a ¬b c = <-osuc omax-x x y | tri≈ ¬a refl ¬c = <-osuc omax-y : ( x y : Ordinal ) → y o< omax x y omax-y x y with trio< x y omax-y x y | tri< a ¬b ¬c = <-osuc omax-y x y | tri> ¬a ¬b c = ordtrans c <-osuc omax-y x y | tri≈ ¬a refl ¬c = <-osuc omxx : ( x : Ordinal ) → omax x x ≡ osuc x omxx x with trio< x x omxx x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) omxx x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) omxx x | tri≈ ¬a refl ¬c = refl omxxx : ( x : Ordinal ) → omax x (omax x x ) ≡ osuc (osuc x) omxxx x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) open _∧_ OrdTrans : Transitive _o≤_ OrdTrans (case1 refl) (case1 refl) = case1 refl OrdTrans (case1 refl) (case2 lt2) = case2 lt2 OrdTrans (case2 lt1) (case1 refl) = case2 lt1 OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y) OrdPreorder : Preorder n n n OrdPreorder = record { Carrier = Ordinal ; _≈_ = _≡_ ; _∼_ = _o≤_ ; isPreorder = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = case1 ; trans = OrdTrans } } FExists : {m l : Level} → ( ψ : Ordinal → Set m ) → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p ) → (exists : ¬ (∀ y → ¬ ( ψ y ) )) → ¬ p FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )