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

comparison partfunc.agda @ 387:8b0715e28b33

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 25 Jul 2020 09:09:00 +0900
parents filter.agda@24a66a246316
children 19687f3304c9
comparison
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386:24a66a246316 387:8b0715e28b33
1 {-# OPTIONS --allow-unsolved-metas #-}
2 open import Level
3 open import Relation.Nullary
4 open import Relation.Binary.PropositionalEquality
5 open import Ordinals
6 module partfunc {n : Level } (O : Ordinals {n}) where
7
8 open import logic
9 open import Relation.Binary
10 open import Data.Empty
11 open import Data.List
12 open import Data.Maybe
13 open import Relation.Binary
14 open import Relation.Binary.Core
15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
16 open import filter O
17
18 open _∧_
19 open _∨_
20 open Bool
21
22
23 record PFunc (Dom : Set n) (Cod : Set n) : Set (suc n) where
24 field
25 dom : Dom → Set n
26 pmap : (x : Dom ) → dom x → Cod
27 meq : {x : Dom } → { p q : dom x } → pmap x p ≡ pmap x q
28
29
30 data Findp : {Cod : Set n} → List (Maybe Cod) → (x : Nat) → Set where
31 v0 : {Cod : Set n} → {f : List (Maybe Cod)} → ( v : Cod ) → Findp ( just v ∷ f ) Zero
32 vn : {Cod : Set n} → {f : List (Maybe Cod)} {d : Maybe Cod} → {x : Nat} → Findp f x → Findp (d ∷ f) (Suc x)
33
34 open PFunc
35
36 find : {Cod : Set n} → (f : List (Maybe Cod) ) → (x : Nat) → Findp f x → Cod
37 find (just v ∷ _) 0 (v0 v) = v
38 find (_ ∷ n) (Suc i) (vn p) = find n i p
39
40 findpeq : {Cod : Set n} → (f : List (Maybe Cod)) → {x : Nat} {p q : Findp f x } → find f x p ≡ find f x q
41 findpeq n {0} {v0 _} {v0 _} = refl
42 findpeq [] {Suc x} {()}
43 findpeq (just x₁ ∷ n) {Suc x} {vn p} {vn q} = findpeq n {x} {p} {q}
44 findpeq (nothing ∷ n) {Suc x} {vn p} {vn q} = findpeq n {x} {p} {q}
45
46 List→PFunc : {Cod : Set n} → List (Maybe Cod) → PFunc (Lift n Nat) Cod
47 List→PFunc fp = record { dom = λ x → Lift n (Findp fp (lower x))
48 ; pmap = λ x y → find fp (lower x) (lower y)
49 ; meq = λ {x} {p} {q} → findpeq fp {lower x} {lower p} {lower q}
50 }
51
52 _3⊆b_ : (f g : List (Maybe Two)) → Bool
53 [] 3⊆b [] = true
54 [] 3⊆b (nothing ∷ g) = [] 3⊆b g
55 [] 3⊆b (_ ∷ g) = true
56 (nothing ∷ f) 3⊆b [] = f 3⊆b []
57 (nothing ∷ f) 3⊆b (_ ∷ g) = f 3⊆b g
58 (just i0 ∷ f) 3⊆b (just i0 ∷ g) = f 3⊆b g
59 (just i1 ∷ f) 3⊆b (just i1 ∷ g) = f 3⊆b g
60 _ 3⊆b _ = false
61
62 _3⊆_ : (f g : List (Maybe Two)) → Set
63 f 3⊆ g = (f 3⊆b g) ≡ true
64
65 _3∩_ : (f g : List (Maybe Two)) → List (Maybe Two)
66 [] 3∩ (nothing ∷ g) = nothing ∷ ([] 3∩ g)
67 [] 3∩ g = []
68 (nothing ∷ f) 3∩ [] = nothing ∷ f 3∩ []
69 f 3∩ [] = []
70 (just i0 ∷ f) 3∩ (just i0 ∷ g) = just i0 ∷ ( f 3∩ g )
71 (just i1 ∷ f) 3∩ (just i1 ∷ g) = just i1 ∷ ( f 3∩ g )
72 (_ ∷ f) 3∩ (_ ∷ g) = nothing ∷ ( f 3∩ g )
73
74 3∩⊆f : { f g : List (Maybe Two) } → (f 3∩ g ) 3⊆ f
75 3∩⊆f {[]} {[]} = refl
76 3∩⊆f {[]} {just _ ∷ g} = refl
77 3∩⊆f {[]} {nothing ∷ g} = 3∩⊆f {[]} {g}
78 3∩⊆f {just _ ∷ f} {[]} = refl
79 3∩⊆f {nothing ∷ f} {[]} = 3∩⊆f {f} {[]}
80 3∩⊆f {just i0 ∷ f} {just i0 ∷ g} = 3∩⊆f {f} {g}
81 3∩⊆f {just i1 ∷ f} {just i1 ∷ g} = 3∩⊆f {f} {g}
82 3∩⊆f {just i0 ∷ f} {just i1 ∷ g} = 3∩⊆f {f} {g}
83 3∩⊆f {just i1 ∷ f} {just i0 ∷ g} = 3∩⊆f {f} {g}
84 3∩⊆f {nothing ∷ f} {just _ ∷ g} = 3∩⊆f {f} {g}
85 3∩⊆f {just i0 ∷ f} {nothing ∷ g} = 3∩⊆f {f} {g}
86 3∩⊆f {just i1 ∷ f} {nothing ∷ g} = 3∩⊆f {f} {g}
87 3∩⊆f {nothing ∷ f} {nothing ∷ g} = 3∩⊆f {f} {g}
88
89 3∩sym : { f g : List (Maybe Two) } → (f 3∩ g ) ≡ (g 3∩ f )
90 3∩sym {[]} {[]} = refl
91 3∩sym {[]} {just _ ∷ g} = refl
92 3∩sym {[]} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {[]} {g})
93 3∩sym {just _ ∷ f} {[]} = refl
94 3∩sym {nothing ∷ f} {[]} = cong (λ k → nothing ∷ k) (3∩sym {f} {[]})
95 3∩sym {just i0 ∷ f} {just i0 ∷ g} = cong (λ k → just i0 ∷ k) (3∩sym {f} {g})
96 3∩sym {just i0 ∷ f} {just i1 ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
97 3∩sym {just i1 ∷ f} {just i0 ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
98 3∩sym {just i1 ∷ f} {just i1 ∷ g} = cong (λ k → just i1 ∷ k) (3∩sym {f} {g})
99 3∩sym {just i0 ∷ f} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
100 3∩sym {just i1 ∷ f} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
101 3∩sym {nothing ∷ f} {just i0 ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
102 3∩sym {nothing ∷ f} {just i1 ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
103 3∩sym {nothing ∷ f} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
104
105 3⊆-[] : { h : List (Maybe Two) } → [] 3⊆ h
106 3⊆-[] {[]} = refl
107 3⊆-[] {just _ ∷ h} = refl
108 3⊆-[] {nothing ∷ h} = 3⊆-[] {h}
109
110 3⊆trans : { f g h : List (Maybe Two) } → f 3⊆ g → g 3⊆ h → f 3⊆ h
111 3⊆trans {[]} {[]} {[]} f<g g<h = refl
112 3⊆trans {[]} {[]} {just _ ∷ h} f<g g<h = refl
113 3⊆trans {[]} {[]} {nothing ∷ h} f<g g<h = 3⊆trans {[]} {[]} {h} refl g<h
114 3⊆trans {[]} {nothing ∷ g} {[]} f<g g<h = refl
115 3⊆trans {[]} {just _ ∷ g} {just _ ∷ h} f<g g<h = refl
116 3⊆trans {[]} {nothing ∷ g} {just _ ∷ h} f<g g<h = refl
117 3⊆trans {[]} {nothing ∷ g} {nothing ∷ h} f<g g<h = 3⊆trans {[]} {g} {h} f<g g<h
118 3⊆trans {nothing ∷ f} {[]} {[]} f<g g<h = f<g
119 3⊆trans {nothing ∷ f} {[]} {just _ ∷ h} f<g g<h = 3⊆trans {f} {[]} {h} f<g (3⊆-[] {h})
120 3⊆trans {nothing ∷ f} {[]} {nothing ∷ h} f<g g<h = 3⊆trans {f} {[]} {h} f<g g<h
121 3⊆trans {nothing ∷ f} {nothing ∷ g} {[]} f<g g<h = 3⊆trans {f} {g} {[]} f<g g<h
122 3⊆trans {nothing ∷ f} {nothing ∷ g} {just _ ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
123 3⊆trans {nothing ∷ f} {nothing ∷ g} {nothing ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
124 3⊆trans {[]} {just i0 ∷ g} {[]} f<g ()
125 3⊆trans {[]} {just i1 ∷ g} {[]} f<g ()
126 3⊆trans {[]} {just x ∷ g} {nothing ∷ h} f<g g<h = 3⊆-[] {h}
127 3⊆trans {just i0 ∷ f} {[]} {h} () g<h
128 3⊆trans {just i1 ∷ f} {[]} {h} () g<h
129 3⊆trans {just x ∷ f} {just i0 ∷ g} {[]} f<g ()
130 3⊆trans {just x ∷ f} {just i1 ∷ g} {[]} f<g ()
131 3⊆trans {just i0 ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
132 3⊆trans {just i1 ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
133 3⊆trans {just x ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g ()
134 3⊆trans {just x ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g ()
135 3⊆trans {just i0 ∷ f} {nothing ∷ g} {_} () g<h
136 3⊆trans {just i1 ∷ f} {nothing ∷ g} {_} () g<h
137 3⊆trans {nothing ∷ f} {just i0 ∷ g} {[]} f<g ()
138 3⊆trans {nothing ∷ f} {just i1 ∷ g} {[]} f<g ()
139 3⊆trans {nothing ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
140 3⊆trans {nothing ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
141 3⊆trans {nothing ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g ()
142 3⊆trans {nothing ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g ()
143
144 3⊆∩f : { f g h : List (Maybe Two) } → f 3⊆ g → f 3⊆ h → f 3⊆ (g 3∩ h )
145 3⊆∩f {[]} {[]} {[]} f<g f<h = refl
146 3⊆∩f {[]} {[]} {x ∷ h} f<g f<h = 3⊆-[] {[] 3∩ (x ∷ h)}
147 3⊆∩f {[]} {x ∷ g} {h} f<g f<h = 3⊆-[] {(x ∷ g) 3∩ h}
148 3⊆∩f {nothing ∷ f} {[]} {[]} f<g f<h = 3⊆∩f {f} {[]} {[]} f<g f<h
149 3⊆∩f {nothing ∷ f} {[]} {just _ ∷ h} f<g f<h = f<g
150 3⊆∩f {nothing ∷ f} {[]} {nothing ∷ h} f<g f<h = 3⊆∩f {f} {[]} {h} f<g f<h
151 3⊆∩f {just i0 ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
152 3⊆∩f {just i1 ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
153 3⊆∩f {nothing ∷ f} {just _ ∷ g} {[]} f<g f<h = f<h
154 3⊆∩f {nothing ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
155 3⊆∩f {nothing ∷ f} {just i0 ∷ g} {just i1 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
156 3⊆∩f {nothing ∷ f} {just i1 ∷ g} {just i0 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
157 3⊆∩f {nothing ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
158 3⊆∩f {nothing ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
159 3⊆∩f {nothing ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
160 3⊆∩f {nothing ∷ f} {nothing ∷ g} {[]} f<g f<h = 3⊆∩f {f} {g} {[]} f<g f<h
161 3⊆∩f {nothing ∷ f} {nothing ∷ g} {just _ ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
162 3⊆∩f {nothing ∷ f} {nothing ∷ g} {nothing ∷ h} f<g f<h = 3⊆∩f {f} {g} {h} f<g f<h
163
164 3↑22 : (f : Nat → Two) (i j : Nat) → List (Maybe Two)
165 3↑22 f Zero j = []
166 3↑22 f (Suc i) j = just (f j) ∷ 3↑22 f i (Suc j)
167
168 _3↑_ : (Nat → Two) → Nat → List (Maybe Two)
169 _3↑_ f i = 3↑22 f i 0
170
171 3↑< : {f : Nat → Two} → { x y : Nat } → x ≤ y → (_3↑_ f x) 3⊆ (_3↑_ f y)
172 3↑< {f} {x} {y} x<y = lemma x y 0 x<y where
173 lemma : (x y i : Nat) → x ≤ y → (3↑22 f x i ) 3⊆ (3↑22 f y i )
174 lemma 0 y i z≤n with f i
175 lemma Zero Zero i z≤n | i0 = refl
176 lemma Zero (Suc y) i z≤n | i0 = 3⊆-[] {3↑22 f (Suc y) i}
177 lemma Zero Zero i z≤n | i1 = refl
178 lemma Zero (Suc y) i z≤n | i1 = 3⊆-[] {3↑22 f (Suc y) i}
179 lemma (Suc x) (Suc y) i (s≤s x<y) with f i
180 lemma (Suc x) (Suc y) i (s≤s x<y) | i0 = lemma x y (Suc i) x<y
181 lemma (Suc x) (Suc y) i (s≤s x<y) | i1 = lemma x y (Suc i) x<y
182
183 Finite3b : (p : List (Maybe Two) ) → Bool
184 Finite3b [] = true
185 Finite3b (just _ ∷ f) = Finite3b f
186 Finite3b (nothing ∷ f) = false
187
188 finite3cov : (p : List (Maybe Two) ) → List (Maybe Two)
189 finite3cov [] = []
190 finite3cov (just y ∷ x) = just y ∷ finite3cov x
191 finite3cov (nothing ∷ x) = just i0 ∷ finite3cov x
192
193 Dense-3 : F-Dense (List (Maybe Two) ) (λ x → One) _3⊆_ _3∩_
194 Dense-3 = record {
195 dense = λ x → Finite3b x ≡ true
196 ; d⊆P = OneObj
197 ; dense-f = λ x → finite3cov x
198 ; dense-d = λ {p} d → lemma1 p
199 ; dense-p = λ {p} d → lemma2 p
200 } where
201 lemma1 : (p : List (Maybe Two) ) → Finite3b (finite3cov p) ≡ true
202 lemma1 [] = refl
203 lemma1 (just i0 ∷ p) = lemma1 p
204 lemma1 (just i1 ∷ p) = lemma1 p
205 lemma1 (nothing ∷ p) = lemma1 p
206 lemma2 : (p : List (Maybe Two)) → p 3⊆ finite3cov p
207 lemma2 [] = refl
208 lemma2 (just i0 ∷ p) = lemma2 p
209 lemma2 (just i1 ∷ p) = lemma2 p
210 lemma2 (nothing ∷ p) = lemma2 p
211
212 record 3Gf (f : Nat → Two) (p : List (Maybe Two)) : Set where
213 field
214 3gn : Nat
215 3f<n : p 3⊆ (f 3↑ 3gn)
216
217 open 3Gf
218
219 min = Data.Nat._⊓_
220 -- m≤m⊔n = Data.Nat._⊔_
221 open import Data.Nat.Properties
222
223 3GF : {n : Level } → (Nat → Two ) → F-Filter (List (Maybe Two)) (λ x → One) _3⊆_ _3∩_
224 3GF {n} f = record {
225 filter = λ p → 3Gf f p
226 ; f⊆P = OneObj
227 ; filter1 = λ {p} {q} _ fp p⊆q → lemma1 p q fp p⊆q
228 ; filter2 = λ {p} {q} fp fq → record { 3gn = min (3gn fp) (3gn fq) ; 3f<n = lemma2 p q fp fq }
229 } where
230 lemma1 : (p q : List (Maybe Two) ) → (fp : 3Gf f p) → (p⊆q : p 3⊆ q) → 3Gf f q
231 lemma1 p q fp p⊆q = record { 3gn = 3gn fp ; 3f<n = {!!} }
232 lemma2 : (p q : List (Maybe Two) ) → (fp : 3Gf f p) → (fq : 3Gf f q) → (p 3∩ q) 3⊆ (f 3↑ min (3gn fp) (3gn fq))
233 lemma2 p q fp fq = ?