Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/ordinal.agda @ 431:a5f8084b8368
reorganiztion for apkg
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 21 Dec 2020 10:23:37 +0900 |
parents | |
children | b27d92694ed5 |
line wrap: on
line diff
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/ordinal.agda Mon Dec 21 10:23:37 2020 +0900 @@ -0,0 +1,282 @@ +open import Level +module ordinal where + +open import logic +open import nat +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 Relation.Binary.Definitions +open import Data.Nat.Properties +open import Relation.Nullary +open import Relation.Binary.Core + +---- +-- +-- Countable Ordinals +-- + +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∅ : {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) + +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) = ⟪ caseΦ Zero lemma , 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)) = ⟪ caseΦ (Suc lx) (λ x → lemma (lv x) (ord x)) , (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) = ⟪ caseOSuc lx ox lemma , 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 + +-- record CountableOrdinal {n : Level} : Set (suc (suc n)) where +-- field +-- ctl→ : Nat → Ordinal {suc n} +-- ctl← : Ordinal → Nat +-- ctl-iso→ : { x : Ordinal } → ctl→ (ctl← x ) ≡ x +-- ctl-iso← : { x : Nat } → ctl← (ctl→ x ) ≡ x +-- +-- is-C-Ordinal : {n : Level} → CountableOrdinal {n} +-- is-C-Ordinal {n} = record { +-- ctl→ = {!!} +-- ; ctl← = λ x → TransFinite {n} (λ lx lt → Zero ) ctl01 x +-- ; ctl-iso→ = {!!} +-- ; ctl-iso← = {!!} +-- } where +-- ctl01 : (lx : Nat) (x : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x) → Nat) → Nat +-- ctl01 Zero (Φ Zero) prev = Zero +-- ctl01 Zero (OSuc Zero x) prev = Suc ( prev (ordinal Zero x) (ordtrans <-osuc <-osuc )) +-- ctl01 (Suc lx) (Φ (Suc lx)) prev = Suc ( prev (ordinal lx {!!}) {!!}) +-- ctl01 (Suc lx) (OSuc (Suc lx) x) prev = Suc ( prev (ordinal (Suc lx) x) (ordtrans <-osuc <-osuc )) + +open import Ordinals + +C-Ordinal : {n : Level} → Ordinals {suc n} +C-Ordinal {n} = record { + Ordinal = Ordinal {suc n} + ; o∅ = o∅ + ; osuc = osuc + ; _o<_ = _o<_ + ; next = next + ; isOrdinal = record { + ordtrans = ordtrans + ; trio< = trio< + ; ¬x<0 = ¬x<0 + ; <-osuc = <-osuc + ; osuc-≡< = osuc-≡< + ; TransFinite = TransFinite2 + ; Oprev-p = Oprev-p + } ; + isNext = record { + x<nx = x<nx + ; osuc<nx = λ {x} {y} → osuc<nx {x} {y} + ; ¬nx<nx = ¬nx<nx + } + } where + next : Ordinal {suc n} → Ordinal {suc n} + next (ordinal lv ord) = ordinal (Suc lv) (Φ (Suc lv)) + x<nx : { y : Ordinal } → (y o< next y ) + x<nx = case1 a<sa + osuc<nx : { x y : Ordinal } → x o< next y → osuc x o< next y + osuc<nx (case1 lt) = case1 lt + ¬nx<nx : {x y : Ordinal} → y o< x → x o< next y → ¬ ((z : Ordinal) → ¬ (x ≡ osuc z)) + ¬nx<nx {x} {y} = lemma2 x where + 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 + open Oprev + Oprev-p : (x : Ordinal) → Dec ( Oprev (Ordinal {suc n}) osuc x ) + Oprev-p (ordinal lv (Φ lv)) = no (λ not → lemma (oprev not) (oprev=x not) ) where + lemma : (x : Ordinal) → osuc x ≡ (ordinal lv (Φ lv)) → ⊥ + lemma x () + Oprev-p (ordinal lv (OSuc lv ox)) = yes record { oprev = ordinal lv ox ; oprev=x = refl } + ord1 : Set (suc n) + ord1 = Ordinal {suc n} + 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 + +