Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff ordinal.agda @ 29:fce60b99dc55
posturate OD is isomorphic to Ordinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 20 May 2019 18:18:43 +0900 |
parents | constructible-set.agda@f36e40d5d2c3 |
children | 3b0fdb95618e |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ordinal.agda Mon May 20 18:18:43 2019 +0900 @@ -0,0 +1,179 @@ +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 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 + ℵ_ : (lv : Nat) → OrdinalD (Suc lv) + +record Ordinal {n : Level} : Set n where + 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 + ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → Φ (Suc lx) d< (ℵ lx) + ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → OSuc (Suc lx) x d< (ℵ lx) + +open Ordinal + +_o<_ : {n : Level} ( x y : Ordinal ) → Set n +_o<_ x y = (lv x < lv y ) ∨ ( ord x d< ord y ) + +open import Data.Nat.Properties +open import Data.Empty +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 } + + +≡→¬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 } → y d< x → x d< y → ⊥ +trio<> {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = + trio<> s 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 {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d< +triOrdd {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< ) +triOrdd {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri< (ℵΦ< {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) ) +triOrdd {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ< {_} {lv} {Φ (Suc lv)} ) +triOrdd {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} ) ) (λ ()) (ℵ< {_} {lv} {y} ) +triOrdd {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ< +triOrdd {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< ) +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) + +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 +d<→lv ℵΦ< = refl +d<→lv ℵ< = refl + +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 {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< ℵ< = ℵΦ< {_} {lx} {y} +orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z ) +orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ< +orddtrans {_} {Suc lx} {Φ (Suc lx)} {.(ℵ _)} {z} ℵΦ< () +orddtrans {_} {Suc lx} {OSuc (Suc lx) _} {.(ℵ _)} {z} ℵ< () + +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 + +maxα : {n : Level} → Ordinal {n} → Ordinal → Ordinal +maxα x y with <-cmp (lv x) (lv y) +maxα x y | tri< a ¬b ¬c = x +maxα x y | tri> ¬a ¬b c = y +maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) } + +_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) with d<→lv x + lemma1 (case2 x) | refl = ¬b refl +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) with d<→lv x + lemma1 (case2 x) | refl = ¬b refl +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 + +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 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) ) +OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y +OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x ) +OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x +OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y) +OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y +OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans 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 : Level} → ( ψ : Ordinal {n} → Set n ) + → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) + → ( ∀ (lx : Nat ) → 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 ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv +TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁ + ( TransFinite ψ caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } )) +TransFinite ψ caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁ +