Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison HOD.agda @ 172:8c4d1423d7c4

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non terminateing on ε-induction
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Jul 2019 14:59:28 +0900
parents 729b80df8a8a
children e6e1bdbda450
comparison
equal deleted inserted replaced
171:729b80df8a8a 172:8c4d1423d7c4
236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α 236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α
237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x 237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x
238 238
239 -- another form of regularity 239 -- another form of regularity
240 -- 240 --
241 -- {-# TERMINATING #-} 241 {-# TERMINATING #-}
242 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} 242 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
243 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) 243 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x )
244 → (x : OD {suc n} ) → ψ x 244 → (x : OD {suc n} ) → ψ x
245 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc) where 245 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where
246 ε-induction-ord : ( ox : Ordinal {suc n} ) {oy : Ordinal {suc n} } → oy o< ox → ψ (ord→od oy) 246 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
247 ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case1 ()) 247 → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) )
248 ε-induction-ord record { lv = Zero ; ord = (Φ 0) } (case2 ()) 248 ε-induction-ord Zero (Φ 0) (case1 ())
249 ε-induction-ord record { lv = lx ; ord = (OSuc lx ox) } {oy} y<x = 249 ε-induction-ord Zero (Φ 0) (case2 ())
250 ind {ord→od oy} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord (record { lv = lx ; ord = ox} ) (lemma y lt ))) where 250 ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x =
251 lemma : (y : OD) → ord→od oy ∋ y → od→ord y o< record { lv = lx ; ord = ox } 251 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where
252 lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox }
252 lemma y lt with osuc-≡< y<x 253 lemma y lt with osuc-≡< y<x
253 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso 254 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
254 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 255 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1
255 ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {oy} = 256 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) =
256 TransFinite {suc n} {suc n ⊔ m} {λ x → x o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od x)} lemma1 lemma2 oy where 257 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where
257 lemma1 : (ly : Nat) → 258 lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥
258 record { lv = ly ; ord = Φ ly } o< record { lv = Suc lx ; ord = Φ (Suc lx) } → 259 lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2
259 ψ (ord→od (record { lv = ly ; ord = Φ ly })) 260 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
260 lemma1 ly lt = ind {!!} 261 lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin
261 lemma2 : (ly : Nat) (oy : OrdinalD ly) → 262 lv (od→ord (ord→od (record { lv = ly ; ord = oy })))
262 (record { lv = ly ; ord = oy } o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od (record { lv = ly ; ord = oy }))) → 263 ≡⟨ cong ( λ k → lv k ) diso ⟩
263 record { lv = ly ; ord = OSuc ly oy } o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od (record { lv = ly ; ord = OSuc ly oy })) 264 lv (record { lv = ly ; ord = oy })
264 lemma2 ly oy p lt = ind {!!} 265 ≡⟨⟩
265 266 ly
267 ∎
268 lemma2 : { lx : Nat } → lx < Suc lx
269 lemma2 {Zero} = s≤s z≤n
270 lemma2 {Suc lx} = s≤s (lemma2 {lx})
271 -- lx Suc lx (1) z(a) <lx by case1
272 -- ly(1) ly(2) (2) z(b) <lx by case1
273 -- z(a) z(b) z(c) z(c) <lx by case2 ( ly==z==x)
274 --
275 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z
276 lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt
277 lemma z lt | case1 lz<ly with <-cmp lx ly
278 lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
279 lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- (1)
280 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) ))
281 lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- z(a)
282 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c))))
283 lemma z lt | case2 lz=ly with <-cmp lx ly
284 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
285 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- z(b)
286 ... | 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 )))
287 lemma z lt | case2 lz=ly | tri≈ ¬a refl ¬c with d<→lv lz=ly -- z(c)
288 ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord (Suc lx) (Φ (Suc lx)) {_} {ord (od→ord z)}
289 (case1 (subst (λ k → k < Suc lx) (trans (sym lemma1) (sym eq)) lemma2 )))
290
266 291
267 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} 292 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
268 OD→ZF {n} = record { 293 OD→ZF {n} = record {
269 ZFSet = OD {suc n} 294 ZFSet = OD {suc n}
270 ; _∋_ = _∋_ 295 ; _∋_ = _∋_