Mercurial > hg > Members > kono > Proof > ZF-in-agda
view ordinal.agda @ 210:2c7d45734e3b
axiom of choice from exclusive middle done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 01 Aug 2019 12:23:07 +0900 |
parents | d4802eb159ff |
children | 22d435172d1a |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level module ordinal 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 data OrdinalD {n : Level} : (lv : Nat) → Set n where Φ : (lv : Nat) → OrdinalD lv OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv record Ordinal {n : Level} : Set n where constructor ordinal 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 open Ordinal _o<_ : {n : Level} ( x y : Ordinal ) → Set n _o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x s<refl {n} {lv} {Φ lv} = Φ< s<refl {n} {lv} {OSuc lv x} = s< s<refl 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} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< () 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 o<-subst : {n : Level } {Z X z x : Ordinal {n}} → Z o< X → Z ≡ z → X ≡ x → z o< x o<-subst df refl refl = df open import Data.Nat.Properties open import Data.Unit using ( ⊤ ) 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 } open import Relation.Binary.HeterogeneousEquality using (_≅_;refl) ordinal-cong : {n : Level} {x y : Ordinal {n}} → lv x ≡ lv y → ord x ≅ ord y → x ≡ y ordinal-cong refl refl = refl ordinal-lv : {n : Level} {x y : Ordinal {n}} → x ≡ y → lv x ≡ lv y ordinal-lv refl = refl ordinal-d : {n : Level} {x y : Ordinal {n}} → x ≡ y → ord x ≅ ord y ordinal-d refl = refl ≡→¬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 } → 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 {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< 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) osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n} osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox } <-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x <-osuc {n} {record { lv = lx ; ord = Φ .lx }} = case2 Φ< <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl ) osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x ) osuc-lveq {n} = refl osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox → osuc oy o< osuc ox osucc {n} {ox} {oy} (case1 x) = case1 x osucc {n} {ox} {oy} (case2 x) with d<→lv x ... | refl = case2 (s< x) nat-<> : { x y : Nat } → x < y → y < x → ⊥ nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x nat-<≡ : { x : Nat } → x < x → ⊥ nat-<≡ (s≤s lt) = nat-<≡ lt nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥ nat-≡< refl lt = nat-<≡ lt ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥ ¬a≤a (s≤s lt) = ¬a≤a lt a<sa : {la : Nat} → la < Suc la a<sa {Zero} = s≤s z≤n a<sa {Suc la} = s≤s a<sa =→¬< : {x : Nat } → ¬ ( x < x ) =→¬< {Zero} () =→¬< {Suc x} (s≤s lt) = =→¬< lt <-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) ) <-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl <-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n) <-∨ {Suc x} {Zero} (s≤s ()) <-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt <-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq) <-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1) case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥ case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2 ... | refl = nat-≡< refl lt1 case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥ case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2 ... | refl = nat-≡< refl lt1 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy → ox o< oy → ⊥ o<¬≡ {_} {ox} {ox} refl (case1 lt) = =→¬< lt o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt ¬x<0 : {n : Level} → { x : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} ) ¬x<0 {n} {x} (case1 ()) ¬x<0 {n} {x} (case2 ()) o<> : {n : Level} → {x y : Ordinal {n} } → y o< x → x o< y → ⊥ o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<> x₁ x₂ o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁ o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂ o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ()) o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = o<> (case2 y<x) (case2 x<y) 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<z) = Φ< orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a → (x ≡ a ) ∨ (x o< a) osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt) osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<) osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ())) osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt ) ... | case1 refl = case1 refl ... | case2 (case2 x) = case2 (case2( s< x) ) ... | case2 (case1 x) = ⊥-elim (¬a≤a x) where osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x → x o< y → ⊥ osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁) osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂) osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d< x₁ osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁ osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂ osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x max : (x y : Nat) → Nat max Zero Zero = Zero max Zero (Suc x) = (Suc x) max (Suc x) Zero = (Suc x) max (Suc x) (Suc y) = Suc ( max x y ) maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx maxαd x y with triOrdd x y maxαd x y | tri< a ¬b ¬c = y maxαd x y | tri≈ ¬a b ¬c = x maxαd x y | tri> ¬a ¬b c = x minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx → OrdinalD lx → OrdinalD lx minαd x y with triOrdd x y minαd x y | tri< a ¬b ¬c = x minαd x y | tri≈ ¬a b ¬c = y minαd x y | tri> ¬a ¬b c = x _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) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁ ) 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) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c ) 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 xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa → ox o< ob ) → oa o< osuc ob xo<ab {n} {oa} {ob} a→b with trio< oa ob xo<ab {n} {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc xo<ab {n} {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc xo<ab {n} {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) maxα : {n : Level} → Ordinal {suc n} → 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 : {n : Level} → Ordinal {suc n} → Ordinal → Ordinal minα {n} x y with trio< {n} 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 : {n : Level} → {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y -- -- max ( osuc x , osuc y ) -- omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n} omax {n} x y with trio< x y omax {n} x y | tri< a ¬b ¬c = osuc y omax {n} x y | tri> ¬a ¬b c = osuc x omax {n} x y | tri≈ ¬a refl ¬c = osuc x omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y omax< {n} x y lt with trio< x y omax< {n} x y lt | tri< a ¬b ¬c = refl omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y omax≡ {n} x y eq with trio< x y omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y omax-x {n} x y with trio< x y omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc omax-x {n} x y | tri> ¬a ¬b c = <-osuc omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y omax-y {n} x y with trio< x y omax-y {n} x y | tri< a ¬b ¬c = <-osuc omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x omxx {n} x with trio< x x omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) omxx {n} x | tri≈ ¬a refl ¬c = refl omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x) omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) open _∧_ osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y) proj1 (osuc2 {n} x y) (case1 lt) = case1 lt proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt proj2 (osuc2 {n} x y) (case1 lt) = case1 lt proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt ... | refl = case2 (s< lt) 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 x) (case2 y) = case2 (ordtrans 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 m : Level} → { ψ : Ordinal {suc n} → Set m } → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) → 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 {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x ) ) TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x lemma x (case1 ()) lemma x (case2 ()) lemma1 : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x lemma1 x (case1 ()) lemma1 x (case2 ()) TransFinite1 (Suc lx) (Φ (Suc lx)) = record { proj1 = caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) ; proj2 = (λ x → lemma (lv x) (ord x)) } where lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal lx (Φ lx) → ψ (ordinal ly oy) lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy) lemma lx1 ox1 (case1 lt) with <-∨ lt lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) ) lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa)) lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0 lx (Φ lx) (case1 (s≤s lt1)) lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa ))) TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )} -- we cannot prove this in intutionistic logic -- (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p ) → p -- postulate -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) -- → (exists : ¬ (∀ y → ¬ ( ψ y ) )) -- → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → p ) -- → p -- -- Instead we prove -- TransFiniteExists : {n m l : Level} → ( ψ : Ordinal {n} → Set m ) → {p : Set l} ( P : { y : Ordinal {n} } → ψ y → ¬ p ) → (exists : ¬ (∀ y → ¬ ( ψ y ) )) → ¬ p TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )