Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/partfunc.agda @ 923:85f6238a38db
use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 19 Oct 2022 08:41:42 +0900 |
parents | 5f22af3562b7 |
children | 55ab5de1ae02 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Relation.Nullary open import Relation.Binary.PropositionalEquality -- open import Ordinals module partfunc {n : Level } where -- (O : Ordinals {n}) where open import logic open import Relation.Binary open import Data.Empty open import Data.List hiding (filter) 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 ---- -- -- Partial Function without ZF -- 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 ---- -- -- PFunc (Lift n Nat) Cod is equivalent to List (Maybe Cod) -- 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} } ---- -- -- to List (Maybe Two) is a Latice -- _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 -- min = Data.Nat._⊓_ -- m≤m⊔n = Data.Nat._⊔_ -- open import Data.Nat.Properties