Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff OD.agda @ 339:feb0fcc430a9
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 12 Jul 2020 19:55:37 +0900 |
parents | bca043423554 |
children | 639fbb6284d8 |
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--- a/OD.agda Sun Jul 12 12:32:42 2020 +0900 +++ b/OD.agda Sun Jul 12 19:55:37 2020 +0900 @@ -103,12 +103,13 @@ sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ --- another form of infinite --- pair-ord< : {x : Ordinal } → od→ord ( ord→od x , ord→od x ) o< next (od→ord x) - postulate odAxiom : ODAxiom open ODAxiom odAxiom +-- possible restriction +hod-ord< : {x : HOD } → Set n +hod-ord< {x} = od→ord x o< next (odmax x) + -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD ¬OD-order : ( od→ord : OD → Ordinal ) → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥ ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) @@ -221,6 +222,9 @@ lemma {t} (case1 refl) = omax-x _ _ lemma {t} (case2 refl) = omax-y _ _ +-- another form of infinite +pair-ord< : {x : Ordinal } → Set n +pair-ord< {x} = od→ord ( ord→od x , ord→od x ) o< next x -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) @@ -370,15 +374,23 @@ infinite : HOD infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } - -- infinite' : HOD - -- infinite' = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where - -- u : (y : Ordinal ) → HOD - -- u y = Union (ord→od y , (ord→od y , ord→od y)) - -- lemma : {y : Ordinal} → infinite-d y → y o< next o∅ - -- lemma {o∅} iφ = {!!} - -- lemma (isuc {y} x) = {!!} where - -- lemma1 : od→ord (Union (ord→od y , (ord→od y , ord→od y))) o< od→ord (Union (u y , (u y , u y ))) - -- lemma1 = {!!} + infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD + infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where + u : (y : Ordinal ) → HOD + u y = Union (ord→od y , (ord→od y , ord→od y)) + lemma : {y : Ordinal} → infinite-d y → y o< next o∅ + lemma {o∅} iφ = proj1 next-limit + lemma (isuc {y} x) = lemma2 where + lemma0 : y o< next o∅ + lemma0 = lemma x + lemma3 : odef (u y ) y + lemma3 = FExists _ (λ {z} t not → not (od→ord (ord→od y , ord→od y)) record { proj1 = case2 refl ; proj2 = lemma4 }) (λ not → not y (infinite-d y)) where + lemma4 : def (od (ord→od (od→ord (ord→od y , ord→od y)))) y + lemma4 = subst₂ ( λ j k → def (od j) k ) (sym oiso) diso (case1 refl) + lemma1 : od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) + lemma1 = ho< + lemma2 : od→ord (u y) o< next o∅ + lemma2 = {!!} _=h=_ : (x y : HOD) → Set n