Mercurial > hg > Members > kono > Proof > ZF-in-agda
view ordinal.agda @ 347:cfecd05a4061
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 13 Jul 2020 22:40:37 +0900 |
parents | feb0fcc430a9 |
children | adc3c3a37308 811152bf2f47 |
line wrap: on
line source
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 open import Ordinals C-Ordinal : {n : Level} → Ordinals {suc n} C-Ordinal {n} = record { ord = Ordinal {suc n} ; o∅ = o∅ ; osuc = osuc ; _o<_ = _o<_ ; next = next ; isOrdinal = record { Otrans = ordtrans ; OTri = trio< ; ¬x<0 = ¬x<0 ; <-osuc = <-osuc ; osuc-≡< = osuc-≡< ; TransFinite = TransFinite1 ; TransFinite1 = TransFinite2 ; not-limit = not-limit ; next-limit = next-limit } } where next : Ordinal {suc n} → Ordinal {suc n} next (ordinal lv ord) = ordinal (Suc lv) (Φ (Suc lv)) next-limit : {y : Ordinal} → (y o< next y) ∧ ((x : Ordinal) → x o< next y → osuc x o< next y) ∧ ( (x : Ordinal) → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ (x ≡ osuc z) )) next-limit {y} = record { proj1 = case1 a<sa ; proj2 = record { proj1 = lemma ; proj2 = lemma2 } } where lemma : (x : Ordinal) → x o< next y → osuc x o< next y lemma x (case1 lt) = case1 lt lemma2 : (x : Ordinal) → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ x ≡ osuc z) lemma2 (ordinal Zero (Φ 0)) (case2 ()) (case1 (s≤s z≤n)) not lemma2 (ordinal Zero (OSuc 0 dx)) (case2 Φ<) (case1 (s≤s z≤n)) not = not _ refl lemma2 (ordinal Zero (OSuc 0 dx)) (case2 (s< x)) (case1 (s≤s z≤n)) not = not _ refl lemma2 (ordinal (Suc lx) (OSuc (Suc lx) ox)) y<x (case1 (s≤s (s≤s lt))) not = not _ refl lemma2 (ordinal (Suc lx) (Φ (Suc lx))) (case1 x) (case1 (s≤s (s≤s lt))) not = lemma3 x lt where lemma3 : {n l : Nat} → (Suc (Suc n) ≤ Suc l) → l ≤ n → ⊥ lemma3 (s≤s sn≤l) (s≤s l≤n) = lemma3 sn≤l l≤n not-limit : (x : Ordinal) → Dec (¬ ((y : Ordinal) → ¬ (x ≡ osuc y))) not-limit (ordinal lv (Φ lv)) = no (λ not → not (λ y () )) not-limit (ordinal lv (OSuc lv ox)) = yes (λ not → not (ordinal lv ox) refl ) ord1 : Set (suc n) ord1 = Ordinal {suc n} TransFinite1 : { ψ : ord1 → Set (suc n) } → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) → ∀ (x : ord1) → ψ x TransFinite1 {ψ} lt x = TransFinite {n} {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 TransFinite2 : { ψ : ord1 → Set (suc (suc n)) } → ( (x : ord1) → ( (y : ord1 ) → y o< x → ψ y ) → ψ x ) → ∀ (x : ord1) → ψ x TransFinite2 {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