diff partfunc.agda @ 387:8b0715e28b33

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 25 Jul 2020 09:09:00 +0900
parents filter.agda@24a66a246316
children 19687f3304c9
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/partfunc.agda	Sat Jul 25 09:09:00 2020 +0900
@@ -0,0 +1,233 @@
+{-# OPTIONS --allow-unsolved-metas #-}
+open import Level
+open import Relation.Nullary 
+open import Relation.Binary.PropositionalEquality
+open import Ordinals
+module partfunc {n : Level } (O : Ordinals {n})  where
+
+open import logic
+open import Relation.Binary 
+open import Data.Empty 
+open import Data.List 
+open import Data.Maybe  
+open import Relation.Binary
+open import Relation.Binary.Core
+open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import filter O
+
+open _∧_
+open _∨_
+open Bool
+
+
+record PFunc  (Dom : Set n) (Cod : Set n) : Set (suc n) where
+   field
+      dom : Dom → Set n
+      pmap : (x : Dom ) → dom x → Cod
+      meq : {x : Dom } → { p q : dom x } → pmap x p ≡ pmap x q
+
+
+data Findp : {Cod : Set n} → List (Maybe Cod) → (x : Nat) → Set where
+   v0 : {Cod : Set n} → {f : List (Maybe Cod)} → ( v : Cod ) → Findp ( just v  ∷ f ) Zero
+   vn : {Cod : Set n} → {f : List (Maybe Cod)} {d : Maybe Cod} → {x : Nat} → Findp f x → Findp (d ∷ f) (Suc x) 
+
+open PFunc
+
+find : {Cod : Set n} → (f : List (Maybe Cod) ) → (x : Nat) → Findp f x → Cod
+find (just v ∷ _) 0 (v0 v) = v
+find (_ ∷ n) (Suc i) (vn p) = find n i p
+
+findpeq : {Cod : Set n} → (f : List (Maybe Cod)) → {x : Nat} {p q : Findp f x } → find f x p ≡ find f x q
+findpeq n {0} {v0 _} {v0 _} = refl
+findpeq [] {Suc x} {()}
+findpeq (just x₁ ∷ n) {Suc x} {vn p} {vn q} = findpeq n {x} {p} {q}
+findpeq (nothing ∷ n) {Suc x} {vn p} {vn q} = findpeq n {x} {p} {q}
+
+List→PFunc : {Cod : Set n} → List (Maybe Cod) → PFunc (Lift n Nat) Cod
+List→PFunc fp = record { dom = λ x → Lift n (Findp fp (lower x))
+                       ; pmap = λ x y → find fp (lower x) (lower y)
+                       ; meq = λ {x} {p} {q} → findpeq fp {lower x} {lower p} {lower q} 
+                       }
+
+_3⊆b_ : (f g : List (Maybe Two)) → Bool
+[] 3⊆b [] = true
+[] 3⊆b (nothing ∷ g) = [] 3⊆b g
+[] 3⊆b (_ ∷ g) = true
+(nothing ∷ f) 3⊆b [] = f 3⊆b []
+(nothing ∷ f) 3⊆b (_ ∷ g)  = f 3⊆b g
+(just i0 ∷ f) 3⊆b (just i0 ∷ g) = f 3⊆b g
+(just i1 ∷ f) 3⊆b (just i1 ∷ g) = f 3⊆b g
+_ 3⊆b _ = false 
+
+_3⊆_ : (f g : List (Maybe Two)) → Set
+f 3⊆ g  = (f 3⊆b g) ≡ true
+
+_3∩_ : (f g : List (Maybe Two)) → List (Maybe Two)
+[] 3∩ (nothing ∷ g) = nothing ∷ ([] 3∩ g)
+[] 3∩ g  = []
+(nothing ∷ f) 3∩ [] = nothing ∷ f 3∩ []
+f 3∩ [] = []
+(just i0 ∷ f) 3∩ (just i0 ∷ g) = just i0 ∷ (  f 3∩ g )
+(just i1 ∷ f) 3∩ (just i1 ∷ g) = just i1 ∷ (  f 3∩ g )
+(_ ∷ f) 3∩ (_ ∷ g)   = nothing  ∷ ( f 3∩ g )
+
+3∩⊆f : { f g : List (Maybe Two) } → (f 3∩ g ) 3⊆ f
+3∩⊆f {[]} {[]} = refl
+3∩⊆f {[]} {just _ ∷ g} = refl
+3∩⊆f {[]} {nothing ∷ g} = 3∩⊆f {[]} {g} 
+3∩⊆f {just _ ∷ f} {[]} = refl
+3∩⊆f {nothing ∷ f} {[]} = 3∩⊆f {f} {[]}
+3∩⊆f {just i0 ∷ f} {just i0 ∷ g} = 3∩⊆f {f} {g}
+3∩⊆f {just i1 ∷ f} {just i1 ∷ g} =  3∩⊆f {f} {g}
+3∩⊆f {just i0 ∷ f} {just i1 ∷ g} =   3∩⊆f {f} {g}
+3∩⊆f {just i1 ∷ f} {just i0 ∷ g} =    3∩⊆f {f} {g}
+3∩⊆f {nothing ∷ f} {just _ ∷ g} = 3∩⊆f {f} {g}
+3∩⊆f {just i0  ∷ f} {nothing ∷ g} = 3∩⊆f {f} {g} 
+3∩⊆f {just i1 ∷ f} {nothing ∷ g} =  3∩⊆f {f} {g} 
+3∩⊆f {nothing ∷ f} {nothing ∷ g} = 3∩⊆f {f} {g}
+
+3∩sym : { f g : List (Maybe Two) } → (f 3∩ g ) ≡ (g 3∩ f )
+3∩sym {[]} {[]} = refl
+3∩sym {[]} {just _ ∷ g} = refl
+3∩sym {[]} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {[]} {g})
+3∩sym {just _ ∷ f} {[]} = refl
+3∩sym {nothing ∷ f} {[]} = cong (λ k → nothing ∷ k) (3∩sym {f} {[]})
+3∩sym {just i0 ∷ f} {just i0 ∷ g} = cong (λ k → just i0 ∷ k) (3∩sym {f} {g})
+3∩sym {just i0 ∷ f} {just i1 ∷ g} =  cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {just i1 ∷ f} {just i0 ∷ g} =  cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {just i1 ∷ f} {just i1 ∷ g} =  cong (λ k → just i1 ∷ k) (3∩sym {f} {g})
+3∩sym {just i0 ∷ f} {nothing ∷ g} =  cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {just i1 ∷ f} {nothing ∷ g} =  cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {nothing ∷ f} {just i0 ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {nothing ∷ f} {just i1 ∷ g} =  cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+3∩sym {nothing ∷ f} {nothing ∷ g} = cong (λ k → nothing ∷ k) (3∩sym {f} {g})
+
+3⊆-[] : { h : List (Maybe Two) } → [] 3⊆ h
+3⊆-[] {[]} = refl
+3⊆-[] {just _ ∷ h} = refl
+3⊆-[] {nothing ∷ h} = 3⊆-[] {h}
+
+3⊆trans : { f g h : List (Maybe Two) } → f 3⊆ g → g 3⊆ h → f 3⊆ h
+3⊆trans {[]} {[]} {[]} f<g g<h = refl
+3⊆trans {[]} {[]} {just _ ∷ h} f<g g<h = refl
+3⊆trans {[]} {[]} {nothing ∷ h} f<g g<h = 3⊆trans {[]} {[]} {h} refl g<h
+3⊆trans {[]} {nothing ∷ g} {[]} f<g g<h = refl
+3⊆trans {[]} {just _ ∷ g} {just _ ∷ h} f<g g<h = refl
+3⊆trans {[]} {nothing ∷ g} {just _ ∷ h} f<g g<h = refl
+3⊆trans {[]} {nothing ∷ g} {nothing ∷ h} f<g g<h = 3⊆trans {[]} {g} {h} f<g g<h 
+3⊆trans {nothing ∷ f} {[]} {[]} f<g g<h = f<g
+3⊆trans {nothing ∷ f} {[]} {just _ ∷ h} f<g g<h = 3⊆trans {f} {[]} {h} f<g (3⊆-[] {h})
+3⊆trans {nothing ∷ f} {[]} {nothing ∷ h} f<g g<h = 3⊆trans {f} {[]} {h} f<g g<h 
+3⊆trans {nothing ∷ f} {nothing ∷ g} {[]} f<g g<h = 3⊆trans {f} {g} {[]} f<g g<h 
+3⊆trans {nothing ∷ f} {nothing ∷ g} {just _ ∷ h} f<g g<h =  3⊆trans {f} {g} {h} f<g g<h 
+3⊆trans {nothing ∷ f} {nothing ∷ g} {nothing ∷ h} f<g g<h =  3⊆trans {f} {g} {h} f<g g<h 
+3⊆trans {[]} {just i0 ∷ g} {[]} f<g ()
+3⊆trans {[]} {just i1 ∷ g} {[]} f<g ()
+3⊆trans {[]} {just x ∷ g} {nothing ∷ h} f<g g<h = 3⊆-[] {h}
+3⊆trans {just i0 ∷ f} {[]} {h} () g<h
+3⊆trans {just i1 ∷ f} {[]} {h} () g<h
+3⊆trans {just x ∷ f} {just i0 ∷ g} {[]} f<g ()
+3⊆trans {just x ∷ f} {just i1 ∷ g} {[]} f<g ()
+3⊆trans {just i0 ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
+3⊆trans {just i1 ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g g<h = 3⊆trans {f} {g} {h} f<g g<h
+3⊆trans {just x ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g ()
+3⊆trans {just x ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g ()
+3⊆trans {just i0 ∷ f} {nothing ∷ g} {_} () g<h
+3⊆trans {just i1 ∷ f} {nothing ∷ g} {_} () g<h
+3⊆trans {nothing ∷ f} {just i0 ∷ g} {[]} f<g ()
+3⊆trans {nothing ∷ f} {just i1 ∷ g} {[]} f<g ()
+3⊆trans {nothing ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g g<h =  3⊆trans {f} {g} {h} f<g g<h
+3⊆trans {nothing ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g g<h =  3⊆trans {f} {g} {h} f<g g<h
+3⊆trans {nothing ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g ()
+3⊆trans {nothing ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g ()
+
+3⊆∩f : { f g h : List (Maybe Two) }  → f 3⊆ g → f 3⊆ h → f 3⊆  (g 3∩ h ) 
+3⊆∩f {[]} {[]} {[]} f<g f<h = refl
+3⊆∩f {[]} {[]} {x ∷ h} f<g f<h = 3⊆-[] {[] 3∩ (x ∷ h)}
+3⊆∩f {[]} {x ∷ g} {h} f<g f<h = 3⊆-[] {(x ∷ g) 3∩ h}
+3⊆∩f {nothing ∷ f} {[]} {[]} f<g f<h = 3⊆∩f {f} {[]} {[]} f<g f<h 
+3⊆∩f {nothing ∷ f} {[]} {just _ ∷ h} f<g f<h = f<g
+3⊆∩f {nothing ∷ f} {[]} {nothing ∷ h} f<g f<h = 3⊆∩f {f} {[]} {h} f<g f<h 
+3⊆∩f {just i0 ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {just i1 ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just _ ∷ g} {[]} f<g f<h = f<h
+3⊆∩f {nothing ∷ f} {just i0 ∷ g} {just i0 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just i0 ∷ g} {just i1 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just i1 ∷ g} {just i0 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just i1 ∷ g} {just i1 ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just i0 ∷ g} {nothing ∷ h} f<g f<h =   3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {just i1 ∷ g} {nothing ∷ h} f<g f<h =   3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {nothing ∷ g} {[]} f<g f<h = 3⊆∩f {f} {g} {[]} f<g f<h  
+3⊆∩f {nothing ∷ f} {nothing ∷ g} {just _ ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+3⊆∩f {nothing ∷ f} {nothing ∷ g} {nothing ∷ h} f<g f<h =  3⊆∩f {f} {g} {h} f<g f<h 
+
+3↑22 : (f : Nat → Two) (i j : Nat) → List (Maybe Two)
+3↑22 f Zero j = []
+3↑22 f (Suc i) j = just (f j) ∷ 3↑22 f i (Suc j)
+
+_3↑_ : (Nat → Two) → Nat → List (Maybe Two)
+_3↑_ f i = 3↑22 f i 0 
+
+3↑< : {f : Nat → Two} → { x y : Nat } → x ≤ y → (_3↑_ f x)  3⊆ (_3↑_ f y)
+3↑< {f} {x} {y} x<y = lemma x y 0 x<y where
+     lemma : (x y i : Nat) → x ≤ y → (3↑22 f x i ) 3⊆ (3↑22 f y i )
+     lemma 0 y i z≤n with f i
+     lemma Zero Zero i z≤n | i0 = refl
+     lemma Zero (Suc y) i z≤n | i0 = 3⊆-[]  {3↑22 f (Suc y) i}
+     lemma Zero Zero i z≤n | i1 = refl
+     lemma Zero (Suc y) i z≤n | i1 = 3⊆-[]  {3↑22 f (Suc y) i}
+     lemma (Suc x) (Suc y) i (s≤s x<y) with f i  
+     lemma (Suc x) (Suc y) i (s≤s x<y) | i0 = lemma x y (Suc i) x<y 
+     lemma (Suc x) (Suc y) i (s≤s x<y) | i1 = lemma x y (Suc i) x<y 
+
+Finite3b : (p : List (Maybe Two) ) → Bool 
+Finite3b [] = true
+Finite3b (just _ ∷ f) = Finite3b f
+Finite3b (nothing ∷ f) = false
+
+finite3cov : (p : List (Maybe Two) ) → List (Maybe Two)
+finite3cov [] = []
+finite3cov (just y ∷ x) = just y ∷ finite3cov x
+finite3cov (nothing ∷ x) = just i0 ∷ finite3cov x
+
+Dense-3 : F-Dense (List (Maybe Two) ) (λ x → One) _3⊆_ _3∩_
+Dense-3 = record {
+       dense =  λ x → Finite3b x ≡ true
+    ;  d⊆P = OneObj
+    ;  dense-f = λ x → finite3cov x
+    ;  dense-d = λ {p} d → lemma1 p
+    ;  dense-p = λ {p} d → lemma2 p
+  } where
+      lemma1 : (p : List (Maybe Two) ) → Finite3b (finite3cov p) ≡ true
+      lemma1 [] = refl
+      lemma1 (just i0 ∷ p) = lemma1 p
+      lemma1 (just i1 ∷ p) = lemma1 p
+      lemma1 (nothing ∷ p) = lemma1 p
+      lemma2 : (p : List (Maybe Two)) → p 3⊆ finite3cov p
+      lemma2 [] = refl
+      lemma2 (just i0 ∷ p) = lemma2 p
+      lemma2 (just i1 ∷ p) = lemma2 p
+      lemma2 (nothing ∷ p) = lemma2 p
+
+record 3Gf (f : Nat → Two) (p : List (Maybe Two)) : Set where
+   field
+     3gn  : Nat
+     3f<n : p 3⊆ (f 3↑ 3gn)
+
+open 3Gf
+
+min  = Data.Nat._⊓_
+-- m≤m⊔n  = Data.Nat._⊔_
+open import Data.Nat.Properties
+
+3GF : {n : Level } → (Nat → Two ) → F-Filter (List (Maybe Two)) (λ x → One) _3⊆_ _3∩_
+3GF {n} f = record {  
+       filter = λ p → 3Gf f p
+     ; f⊆P = OneObj
+     ; filter1 = λ {p} {q} _ fp p⊆q → lemma1 p q fp p⊆q
+     ; filter2 = λ {p} {q} fp fq  → record { 3gn = min (3gn fp) (3gn fq) ; 3f<n = lemma2 p q fp fq }
+ }  where
+      lemma1 : (p q : List (Maybe Two) ) → (fp : 3Gf f p) →  (p⊆q : p 3⊆ q) → 3Gf f q
+      lemma1 p q fp p⊆q  = record { 3gn = 3gn fp  ; 3f<n = {!!} }
+      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))
+      lemma2 p q fp fq = ?