Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff HOD.agda @ 179:aa89d1b8ce96
fix comments
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 20 Jul 2019 08:21:54 +0900 |
| parents | ecb329ba38ac |
| children | 11490a3170d4 |
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--- a/HOD.agda Sat Jul 20 08:03:54 2019 +0900 +++ b/HOD.agda Sat Jul 20 08:21:54 2019 +0900 @@ -45,7 +45,7 @@ eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m --- Ordinal in OD ( and ZFSet ) +-- Ordinal in OD ( and ZFSet ) Transitive Set Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} Ord {n} a = record { def = λ y → y o< a } @@ -382,8 +382,7 @@ lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x))) lemma = sup-o< - -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A - -- double-neg-eilm {n} {A} notnot = ⊥-elim (notnot (λ A → {!!} )) + -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic -- -- Every set in OD is a subset of Ordinals -- @@ -460,7 +459,7 @@ ≡ od→ord (Union (x , (x , x))) lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso - -- Axiom of choice ( is equivalent to existence of minimul ) + -- Axiom of choice ( is equivalent to the existence of minimul in our case ) -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD choice-func X {x} not X∋x = minimul x not @@ -469,7 +468,6 @@ -- another form of regularity -- - -- {-# TERMINATING #-} ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) → (x : OD {suc n} ) → ψ x @@ -486,6 +484,14 @@ lemma y 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 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly @@ -499,10 +505,6 @@ lemma2 : { lx : Nat } → lx < Suc lx lemma2 {Zero} = s≤s z≤n lemma2 {Suc lx} = s≤s (lemma2 {lx}) - -- lx Suc lx (1) z(a) <lx by case1 - -- ly(1) ly(2) (2) z(b) <lx by case1 - -- z(a) z(b) z(c) z(c) <lx by case2 ( ly==z==x) - -- lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt lemma z lt | case1 lz<ly with <-cmp lx ly @@ -514,12 +516,13 @@ 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 -- z(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 refl ¬c with d<→lv lz=ly -- z(c) - ... | eq = lemma6 {ly} {Φ lx} {oy} refl (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where + ... | 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 -- z(c) + ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → - lx ≡ ly → ly ≡ lv (od→ord z) → ψ z + lx ≡ ly → ly ≡ lv (od→ord z) → ψ z lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)