Mercurial > hg > Members > kono > Proof > ZF-in-agda
view ordinal.agda @ 224:afc864169325
recover ε-induction
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 10 Aug 2019 12:31:25 +0900 |
| parents | 59771eb07df0 |
| children | e06b76e5b682 |
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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 open import logic open import nat 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 ≡→¬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< 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 ) 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 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 open _∧_ 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 ) → ( (y : Ordinal {suc n} ) → y o< ordinal lx (OSuc lx x) → ψ y ) → 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 ox → 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 (Φ lx) (case1 lt) | case2 lt1 = lemma0 lx (Φ lx) (case1 lt1) lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 lemma2 where lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx ox1) → ψ y lemma2 y lt1 with osuc-≡< lt1 lemma2 y lt1 | case1 refl = lemma lx ox1 (case1 a<sa) lemma2 y lt1 | case2 t = proj2 (TransFinite1 lx ox1) y t lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 lemma2 where lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx1) ∨ (ord y d< OSuc lx1 ox1) → ψ y lemma2 y lt2 with osuc-≡< lt2 lemma2 y lt2 | case1 refl = lemma lx1 ox1 (ordtrans lt2 (case1 lt)) lemma2 y lt2 | case2 (case1 lt3) = proj2 (TransFinite1 lx (Φ lx)) y (case1 (<-trans lt3 lt1 )) lemma2 y lt2 | case2 (case2 lt3) with d<→lv lt3 ... | refl = proj2 (TransFinite1 lx (Φ lx)) y (case1 lt1) TransFinite1 lx (OSuc lx ox) = record { proj1 = caseOSuc lx ox lemma ; proj2 = lemma } where lemma : (y : Ordinal) → y o< ordinal lx (OSuc lx ox) → ψ y lemma y lt with osuc-≡< lt lemma y lt | case1 refl = proj1 ( TransFinite1 lx ox ) lemma y lt | case2 lt1 = proj2 ( TransFinite1 lx ox ) y lt1 -- 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 ) open import Ordinals C-Ordinal : {n : Level} → Ordinals {suc n} C-Ordinal {n} = record { ord = Ordinal {suc n} ; o∅ = o∅ ; osuc = osuc ; _o<_ = _o<_ ; isOrdinal = record { Otrans = ordtrans ; OTri = trio< ; ¬x<0 = ¬x<0 ; <-osuc = <-osuc ; osuc-≡< = osuc-≡< ; TransFinite = TransFinite1 } } where ord1 : Set (suc n) ord1 = Ordinal {suc n} TransFinite1 : { ψ : ord1 → Set (suc (suc n)) } → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) → ∀ (x : ord1) → ψ x TransFinite1 {ψ} lt x = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc x where caseΦ : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → ψ x₁) → ψ (record { lv = lx ; ord = Φ lx }) caseΦ lx prev = lt (ordinal lx (Φ lx) ) prev caseOSuc : (lx : Nat) (x₁ : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x₁) → ψ y) → ψ (record { lv = lx ; ord = OSuc lx x₁ }) caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev module C-Ordinal-with-choice {n : Level} where import OD -- open inOrdinal C-Ordinal open OD (C-Ordinal {n}) open OD.OD -- -- another form of regularity -- ε-induction : {m : Level} { ψ : OD → Set m} → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) → (x : OD ) → ψ x ε-induction {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 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 : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → od→ord z o< record { lv = lx ; ord = ox } lemma z lt with osuc-≡< y<x lemma z lt | case1 refl = o<-subst (c<→o< lt) refl diso lemma z 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 : {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly lemma1 {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< {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 = subst (λ k → ψ k ) oiso ( ε-induction-ord lx (dz oz=lx) {lv (od→ord z)} {ord (od→ord z)} (case2 (dz<dz oz=lx) )) where oz=lx : lv (od→ord z) ≡ lx oz=lx = let open ≡-Reasoning in begin lv (od→ord z) ≡⟨ eq ⟩ lv (od→ord (ord→od (ordinal ly oy))) ≡⟨ cong ( λ k → lv k ) diso ⟩ lv (ordinal ly oy) ≡⟨ sym lx=ly ⟩ lx ∎ dz : lv (od→ord z) ≡ lx → OrdinalD lx dz refl = OSuc lx (ord (od→ord z)) dz<dz : (z=x : lv (od→ord z) ≡ lx ) → ord (od→ord z) d< dz z=x dz<dz refl = s<refl --- --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice --- record choiced ( X : OD) : Set (suc (suc n)) where field a-choice : OD is-in : X ∋ a-choice choice-func' : (X : OD ) → (p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X choice-func' X p∨¬p not = have_to_find where ψ : ( ox : Ordinal {suc n}) → Set (suc (suc n)) ψ ox = (( x : Ordinal {suc n}) → x o< ox → ( ¬ def X x )) ∨ choiced X lemma-ord : ( ox : Ordinal {suc n} ) → ψ ox lemma-ord ox = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc1 ox where ∋-p' : (A x : OD ) → Dec ( A ∋ x ) ∋-p' A x with p∨¬p ( A ∋ x ) ∋-p' A x | case1 t = yes t ∋-p' A x | case2 t = no t ∀-imply-or : {n : Level} {A : Ordinal {suc n} → Set (suc n) } {B : Set (suc (suc n)) } → ((x : Ordinal {suc n}) → A x ∨ B) → ((x : Ordinal {suc n}) → A x) ∨ B ∀-imply-or {n} {A} {B} ∀AB with p∨¬p ((x : Ordinal {suc n}) → A x) ∀-imply-or {n} {A} {B} ∀AB | case1 t = case1 t ∀-imply-or {n} {A} {B} ∀AB | case2 x = case2 (lemma x) where lemma : ¬ ((x : Ordinal {suc n}) → A x) → B lemma not with p∨¬p B lemma not | case1 b = b lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) caseΦ : (lx : Nat) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x) → ψ (ordinal lx (Φ lx) ) caseΦ lx prev with ∋-p' X ( ord→od (ordinal lx (Φ lx) )) caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } ) caseΦ lx prev | no ¬p = lemma where lemma1 : (x : Ordinal) → (((Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X) lemma1 x with trio< x (ordinal lx (Φ lx)) lemma1 x | tri< a ¬b ¬c with prev (osuc x) (lemma2 a) where lemma2 : x o< (ordinal lx (Φ lx)) → osuc x o< ordinal lx (Φ lx) lemma2 (case1 lt) = case1 lt lemma1 x | tri< a ¬b ¬c | case2 found = case2 found lemma1 x | tri< a ¬b ¬c | case1 not_found = case1 ( λ lt df → not_found x <-osuc df ) lemma1 x | tri≈ ¬a refl ¬c = case1 ( λ lt → ⊥-elim (o<¬≡ refl lt )) lemma1 x | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim (o<> lt c )) lemma : ((x : Ordinal) → (Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X lemma = ∀-imply-or lemma1 caseOSuc : (lx : Nat) (x : OrdinalD lx) → ψ ( ordinal lx x ) → ψ ( ordinal lx (OSuc lx x) ) caseOSuc lx x prev with ∋-p' X (ord→od record { lv = lx ; ord = x } ) caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p }) caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where lemma : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx x) → def X y → ⊥ lemma y lt with trio< y (ordinal lx x ) lemma y lt | tri< a ¬b ¬c = not_found y a lemma y lt | tri≈ ¬a refl ¬c = subst (λ k → def X k → ⊥ ) diso ¬p lemma y lt | tri> ¬a ¬b c with osuc-≡< lt lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl ) lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 ) caseOSuc lx x (case2 found) | no ¬p = case2 found caseOSuc1 : (lx : Nat) (x : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x) → ψ y) → ψ (record { lv = lx ; ord = OSuc lx x }) caseOSuc1 lx x lt = caseOSuc lx x (lt ( ordinal lx x ) (case2 s<refl)) have_to_find : choiced X have_to_find with lemma-ord (od→ord X ) have_to_find | t = dont-or t ¬¬X∋x where ¬¬X∋x : ¬ ((x : Ordinal) → (Suc (lv x) ≤ lv (od→ord X)) ∨ (ord x d< ord (od→ord X)) → def X x → ⊥) ¬¬X∋x nn = not record { eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) }