Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff generic-filter.agda @ 388:19687f3304c9
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 25 Jul 2020 12:54:28 +0900 |
parents | 8b0715e28b33 |
children | cb183674facf |
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--- a/generic-filter.agda Sat Jul 25 09:09:00 2020 +0900 +++ b/generic-filter.agda Sat Jul 25 12:54:28 2020 +0900 @@ -60,11 +60,6 @@ extend : {x : Nat} → (fr : dom f (lift x) ) → dom g (lift x ) feq : {x : Nat} → {fr : dom f (lift x) } → pmap f (lift x) fr ≡ pmap g (lift x) (extend fr) -record Gf (f : Nat → Two) (p : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where - field - gn : Nat - f<n : (f ↑ gn) f⊆ p - record FiniteF (p : PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where field f-max : Nat @@ -84,60 +79,9 @@ -- ; dense-p = λ {p} d → {!!} -- } -open Gf open _f⊆_ open import Data.Nat.Properties -GF : (Nat → Two ) → F-Filter (PFunc (Lift n Nat) (Lift n Two)) (λ x → Lift (suc n) (One {n}) ) _f⊆_ _f∩_ -GF f = record { - filter = λ p → Gf f p - ; f⊆P = lift OneObj - ; filter1 = λ {p} {q} _ fp p⊆q → record { gn = gn fp ; f<n = f1 fp p⊆q } - ; filter2 = λ {p} {q} fp fq → record { gn = min (gn fp) (gn fq) ; f<n = f2 fp fq } - } where - f1 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → (f ↑ gn fp) f⊆ q - f1 {p} {q} fp p⊆q = record { extend = λ {x} x<g → extend p⊆q (extend (f<n fp ) x<g) ; feq = λ {x} {fr} → lemma {x} {fr} } where - lemma : {x : Nat} {fr : Lift n (x ≤ gn fp)} → pmap (f ↑ gn fp) (lift x) fr ≡ pmap q (lift x) (extend p⊆q (extend (f<n fp) fr)) - lemma {x} {fr} = begin - pmap (f ↑ gn fp) (lift x) fr - ≡⟨ feq (f<n fp ) ⟩ - pmap p (lift x) (extend (f<n fp) fr) - ≡⟨ feq p⊆q ⟩ - pmap q (lift x) (extend p⊆q (extend (f<n fp) fr)) - ∎ where open ≡-Reasoning - f2 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf f p ) → (fq : Gf f q ) → (f ↑ (min (gn fp) (gn fq))) f⊆ (p f∩ q) - f2 {p} {q} fp fq = record { extend = λ {x} x<g → lemma2 x<g ; feq = λ {x} {fr} → lemma3 fr } where - fmin : PFunc (Lift n Nat) (Lift n Two) - fmin = f ↑ (min (gn fp) (gn fq)) - lemma1 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → (fr : dom p (lift x)) (gr : dom q (lift x)) → pmap p (lift x) fr ≡ pmap q (lift x) gr - lemma1 {x} x<g fr gr = begin - pmap p (lift x) fr - ≡⟨ meq p ⟩ - pmap p (lift x) (extend (f<n fp) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _)))) - ≡⟨ sym (feq (f<n fp)) ⟩ - pmap (f ↑ (min (gn fp) (gn fq))) (lift x) x<g - ≡⟨ feq (f<n fq) ⟩ - pmap q (lift x) (extend (f<n fq) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _)))) - ≡⟨ meq q ⟩ - pmap q (lift x) gr - ∎ where open ≡-Reasoning - lemma2 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → dom (p f∩ q) (lift x) - lemma2 x<g = record { proj1 = extend (f<n fp ) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))) ; - proj2 = record {proj1 = extend (f<n fq ) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))) ; proj2 = lemma1 x<g }} - f∩→⊆ : (p q : PFunc (Lift n Nat) (Lift n Two) ) → (p f∩ q ) f⊆ q - f∩→⊆ p q = record { - extend = λ {x} pq → proj1 (proj2 pq) - ; feq = λ {x} {fr} → (proj2 (proj2 fr)) (proj1 fr) (proj1 (proj2 fr)) - } - lemma3 : {x : Nat} → ( fr : Lift n (x ≤ min (gn fp) (gn fq))) → pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr ≡ pmap (p f∩ q) (lift x) (lemma2 fr) - lemma3 {x} fr = - pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr - ≡⟨ feq (f<n fq) ⟩ - pmap q (lift x) (extend (f<n fq) ( lift (≤-trans (lower fr) (m⊓n≤n _ _)) )) - ≡⟨ sym (feq (f∩→⊆ p q ) {x} {lemma2 fr} ) ⟩ - pmap (p f∩ q) (lift x) (lemma2 fr) - ∎ where open ≡-Reasoning - ODSuc : (y : HOD) → infinite ∋ y → HOD ODSuc y lt = Union (y , (y , y)) @@ -200,35 +144,6 @@ ↑n : (f n : HOD) → ((ω→2 ∋ f ) ∧ (infinite ∋ n)) → HOD ↑n f n lt = 3→Hω2 ( ω→2f f (proj1 lt) 3↑ (ω→nat n (proj2 lt) )) -record Gfo (x : Ordinal) : Set n where - field - gfunc : Ordinal - gmax : Ordinal - gcond : (odef ω→2 gfunc) ∧ (odef infinite gmax) - gfdef : {!!} -- ( ↑n (ord→od gfunc) (ord→od gmax) (subst₂ ? ? ? gcond) ) ⊆ ord→od x - pcond : odef HODω2 x - -open Gfo - -HODGf : HOD -HODGf = record { od = record { def = λ x → Gfo x } ; odmax = next o∅ ; <odmax = {!!} } - -G : (Nat → Two) → Filter HODω2 -G f = record { - filter = HODGf - ; f⊆PL = {!!} - ; filter1 = {!!} - ; filter2 = {!!} - } where - filter0 : HOD - filter0 = {!!} - f⊆PL1 : filter0 ⊆ Power HODω2 - f⊆PL1 = {!!} - filter11 : { p q : HOD } → q ⊆ HODω2 → filter0 ∋ p → p ⊆ q → filter0 ∋ q - filter11 = {!!} - filter12 : { p q : HOD } → filter0 ∋ p → filter0 ∋ q → filter0 ∋ (p ∩ q) - filter12 = {!!} - -- the set of finite partial functions from ω to 2 Hω2f : Set (suc n) @@ -244,26 +159,62 @@ ctl← : Ordinal → Nat ctl-iso→ : { x : Ordinal } → ctl→ (ctl← x ) ≡ x ctl-iso← : { x : Nat } → ctl← (ctl→ x ) ≡ x + +record CountableHOD : Set (suc (suc n)) where + field + phod : HOD + ptl→ : Nat → Ordinal + ptl→∈P : (i : Nat) → odef phod (ptl→ i) + ptl← : (x : Ordinal) → odef phod x → Nat + ptl-iso→ : { x : Ordinal } → (lt : odef phod x ) → ptl→ (ptl← x lt ) ≡ x + ptl-iso← : { x : Nat } → ptl← (ptl→ x ) (ptl→∈P x) ≡ x + open CountableOrdinal +open CountableHOD -PGOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Set n -PGOD i C p q = ¬ ( odef (ord→od (ctl→ C i)) q ∧ ( (x : Ordinal ) → odef (ord→od p) x → odef (ord→od q) x )) +PGHOD : (P : CountableHOD ) (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → HOD +PGHOD P i C p = record { od = record { def = λ x → odef (ord→od (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (ord→od p) y → odef (ord→od x) y ) } + ; odmax = ctl→ C i ; <odmax = λ {y} lt → odefo→o< (proj1 lt)} + +next-p : (P : CountableHOD ) (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → Ordinal +next-p P i C p with ODC.decp O ( PGHOD P i C p =h= od∅ ) +next-p P i C p | yes y = p +next-p P i C p | no not = od→ord (ODC.minimal O (PGHOD P i C p ) not) -PGHOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → HOD -PGHOD i C p = record { od = record { def = λ x → PGOD i C {!!} {!!} } ; odmax = {!!} ; <odmax = {!!} } +data PD (P : CountableHOD ) (C : CountableOrdinal) : (x : Ordinal) (i : Nat) → Set n where + pd0 : PD P C o∅ 0 + pdsuc : {p : Ordinal } {i : Nat} → PD P C p i → PD P C (next-p P i C p) (Suc i) + +record PDN (P : CountableHOD ) (C : CountableOrdinal) (x : Ordinal) : Set n where + field + px∈ω : odef (phod P) x + pdod : PD P C x (ptl← P x px∈ω) + +open PDN -ord-compare : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Ordinal -ord-compare i C p q with ODC.p∨¬p O ( (q : Ordinal ) → PGOD i C p q ) -ord-compare i C p q | case1 y = p -ord-compare i C p q | case2 n = od→ord (ODC.minimal O (PGHOD i C p ) (∅< (subst₂ (λ j k → odef j {!!} ) refl {!!} n)) ) +PDHOD : (P : CountableHOD ) → (C : CountableOrdinal) → HOD +PDHOD P C = record { od = record { def = λ x → PDN P C x } + ; odmax = odmax (phod P) ; <odmax = λ {y} lt → <odmax (phod P) (px∈ω lt) } where + +-- +-- p 0 ≡ ∅ +-- p (suc n) = if ∃ q ∈ ord→od ( ctl→ n ) ∧ p n ⊆ q → q +--- else p n -data PD (P : HOD) (C : CountableOrdinal) : (x : Ordinal) (i : Nat) → Set (suc n) where - pd0 : PD P C o∅ 0 - -- pdq : {q pnx : Ordinal } {n : Nat} → (pn : PD P C pnx n ) → odef (ctl→ C n) q → ord→od p0x ⊆ ord→od q → PD P C q (suc n) +Gω2r : (x : Ordinal) → (lt : infinite ∋ ord→od x ) → Hω2 (ω→nat (ord→od x) lt ) x +Gω2r x = subst (λ k → (lt : odef infinite (od→ord (ord→od x))) → Hω2 (ω→nat (ord→od x) lt ) k ) diso ( ε-induction {ψ} {!!} (ord→od x)) where + ψ : HOD → Set n + ψ y = (lt : odef infinite (od→ord y) ) → Hω2 (ω→nat y lt ) (od→ord y ) + ind : {x : HOD} → ({y : HOD} → OD.def (od x) (od→ord y) → + (lt : infinite-d (od→ord y)) → Hω2 (ω→nat y lt) (od→ord y)) → + (lt : infinite-d (od→ord x)) → Hω2 (ω→nat x lt) (od→ord x) + ind {x} prev lt with ω→nat x lt + ... | Zero = subst (λ k → Hω2 Zero k) ? hφ + ... | Suc t = {!!} -P-GenericFilter : {P : HOD} → (C : CountableOrdinal) → GenericFilter P +P-GenericFilter : {P : CountableHOD } → (C : CountableOrdinal) → GenericFilter (phod P) P-GenericFilter {P} C = record { - genf = record { filter = {!!} ; f⊆PL = {!!} ; filter1 = {!!} ; filter2 = {!!} } + genf = record { filter = PDHOD P C ; f⊆PL = {!!} ; filter1 = {!!} ; filter2 = {!!} } ; generic = λ D → {!!} }