Mercurial > hg > Members > kono > Proof > galois
view Symmetric.agda @ 30:a9fa68dfbd26
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Wed, 19 Aug 2020 16:08:32 +0900 |
parents | a65f3b17eade |
children | 039e8511da2a |
line wrap: on
line source
module Symmetric where open import Level hiding ( suc ; zero ) open import Algebra open import Algebra.Structures open import Data.Fin hiding ( _<_ ) open import Data.Fin.Properties hiding ( <-cmp ; <-trans ) open import Data.Product open import Data.Fin.Permutation open import Function hiding (id ; flip) open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_) open import Function.LeftInverse using ( _LeftInverseOf_ ) open import Function.Equality using (Π) open import Data.Nat -- using (ℕ; suc; zero; s≤s ; z≤n ) open import Data.Nat.Properties -- using (<-trans) open import Relation.Binary.PropositionalEquality open import Data.List using (List; []; _∷_ ; length) open import nat fid : {p : ℕ } → Fin p → Fin p fid x = x -- Data.Fin.Permutation.id pid : {p : ℕ } → Permutation p p pid = permutation fid fid record { left-inverse-of = λ x → refl ; right-inverse-of = λ x → refl } -- Data.Fin.Permutation.flip pinv : {p : ℕ } → Permutation p p → Permutation p p pinv {p} P = permutation (_⟨$⟩ˡ_ P) (_⟨$⟩ʳ_ P ) record { left-inverse-of = λ x → inverseʳ P ; right-inverse-of = λ x → inverseˡ P } record _=p=_ {p : ℕ } ( x y : Permutation p p ) : Set where field peq : ( q : Fin p ) → x ⟨$⟩ʳ q ≡ y ⟨$⟩ʳ q open _=p=_ prefl : {p : ℕ } { x : Permutation p p } → x =p= x peq (prefl {p} {x}) q = refl psym : {p : ℕ } { x y : Permutation p p } → x =p= y → y =p= x peq (psym {p} {x} {y} eq ) q = sym (peq eq q) ptrans : {p : ℕ } { x y z : Permutation p p } → x =p= y → y =p= z → x =p= z peq (ptrans {p} {x} {y} x=y y=z ) q = trans (peq x=y q) (peq y=z q) Symmetric : ℕ → Group Level.zero Level.zero Symmetric p = record { Carrier = Permutation p p ; _≈_ = _=p=_ ; _∙_ = _∘ₚ_ ; ε = pid ; _⁻¹ = pinv ; isGroup = record { isMonoid = record { isSemigroup = record { isMagma = record { isEquivalence = record {refl = prefl ; trans = ptrans ; sym = psym } ; ∙-cong = presp } ; assoc = passoc } ; identity = ( (λ q → record { peq = λ q → refl } ) , (λ q → record { peq = λ q → refl } )) } ; inverse = ( (λ x → record { peq = λ q → inverseʳ x} ) , (λ x → record { peq = λ q → inverseˡ x} )) ; ⁻¹-cong = λ i=j → record { peq = λ q → p-inv i=j q } } } where presp : {x y u v : Permutation p p } → x =p= y → u =p= v → (x ∘ₚ u) =p= (y ∘ₚ v) presp {x} {y} {u} {v} x=y u=v = record { peq = λ q → lemma4 q } where lemma4 : (q : Fin p) → ((x ∘ₚ u) ⟨$⟩ʳ q) ≡ ((y ∘ₚ v) ⟨$⟩ʳ q) lemma4 q = trans (cong (λ k → Inverse.to u Π.⟨$⟩ k) (peq x=y q) ) (peq u=v _ ) passoc : (x y z : Permutation p p) → ((x ∘ₚ y) ∘ₚ z) =p= (x ∘ₚ (y ∘ₚ z)) passoc x y z = record { peq = λ q → refl } p-inv : {i j : Permutation p p} → i =p= j → (q : Fin p) → pinv i ⟨$⟩ʳ q ≡ pinv j ⟨$⟩ʳ q p-inv {i} {j} i=j q = begin i ⟨$⟩ˡ q ≡⟨ cong (λ k → i ⟨$⟩ˡ k) (sym (inverseʳ j) ) ⟩ i ⟨$⟩ˡ ( j ⟨$⟩ʳ ( j ⟨$⟩ˡ q )) ≡⟨ cong (λ k → i ⟨$⟩ˡ k) (sym (peq i=j _ )) ⟩ i ⟨$⟩ˡ ( i ⟨$⟩ʳ ( j ⟨$⟩ˡ q )) ≡⟨ inverseˡ i ⟩ j ⟨$⟩ˡ q ∎ where open ≡-Reasoning perm0 : Permutation zero zero perm0 = permutation fid fid record { left-inverse-of = λ x → refl ; right-inverse-of = λ x → refl } open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import fin fperm : {n m : ℕ} → m < n → Permutation n n → Permutation (suc n) (suc n) fperm {zero} () fperm {suc n} {m} (s≤s m<n) perm = permutation p→ p← record { left-inverse-of = piso← ; right-inverse-of = piso→ } where p→ : Fin (suc (suc n)) → Fin (suc (suc n)) p→ x with <-cmp (toℕ x) m p→ x | tri< a ¬b ¬c = fin+1 (perm ⟨$⟩ʳ (fromℕ≤ x<sn)) where x<sn : toℕ x < suc n x<sn = <-trans a (s≤s m<n) p→ x | tri≈ ¬a b ¬c = fromℕ≤ a<sa p→ x | tri> ¬a ¬b c = fin+1 (perm ⟨$⟩ʳ (fromℕ≤ (pred<n {_} {x} 0<s ))) p← : Fin (suc (suc n)) → Fin (suc (suc n)) p← x = lemma (suc n) refl x where lemma : (i : ℕ ) → i ≡ suc n → (x : Fin (suc (suc n))) → Fin (suc (suc n)) lemma i refl x with <-cmp (toℕ x) i lemma i refl x | tri< a ¬b ¬c = fin+1 (perm ⟨$⟩ˡ (fromℕ≤ x<sn)) where x<sn : toℕ x < suc n x<sn = a lemma i refl x | tri≈ ¬a b ¬c = fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) lemma i refl x | tri> ¬a ¬b c = ⊥-elim ( nat-≤> c fin<n ) lem8 : {x : Fin (suc (suc n)) } → toℕ ( fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )) ≡ m lem8 {x} = toℕ-fromℕ≤ (<-trans (s≤s m<n ) a<sa ) piso← : (x : Fin (suc (suc n))) → p← ( p→ x ) ≡ x piso← x with <-cmp (toℕ x) m piso← x | tri< a ¬b ¬c = {!!} piso← x | tri> ¬a ¬b c = {!!} piso← x | tri≈ ¬a refl ¬c = begin p← ( fromℕ≤ a<sa ) ≡⟨ lem4 refl ⟩ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) ≡⟨ {!!} ⟩ x ∎ where open ≡-Reasoning lem4 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ {suc n} a<sa → p← x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) lem4 {x} refl with <-cmp (toℕ x) (suc n) lem4 refl | tri< a ¬b ¬c = ⊥-elim ( ¬b {!!} ) lem4 refl | tri≈ ¬a b ¬c = refl lem4 refl | tri> ¬a ¬b c = ⊥-elim ( ¬b {!!} ) piso→ : (x : Fin (suc (suc n))) → p→ ( p← x ) ≡ x piso→ x = lemma2 (suc n) refl x where lemma2 : (i : ℕ ) → i ≡ suc n → (x : Fin (suc (suc n))) → p→ ( p← x ) ≡ x lemma2 i refl x with <-cmp (toℕ x) i lemma2 i refl x | tri< a ¬b ¬c = {!!} lemma2 i refl x | tri> ¬a ¬b c = {!!} lemma2 i refl x | tri≈ ¬a b ¬c = begin p→ (fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )) ≡⟨ lem5 (fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa )) {!!} ⟩ fromℕ≤ a<sa ≡⟨ {!!} ⟩ x ∎ where open ≡-Reasoning lem7 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ (s≤s (s≤s m<n)) → toℕ x ≡ m lem7 refl = trans (toℕ-fromℕ≤ _) {!!} lem6 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ (s≤s (s≤s m<n)) → toℕ x ≡ (suc m) lem6 refl = toℕ-fromℕ≤ _ -- lem5 : {x : Fin (suc (suc n)) } → x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) → p→ x ≡ fromℕ≤ a<sa lem5 : (x : Fin (suc (suc n)) ) → x ≡ fromℕ≤ {m} {suc (suc n)} (<-trans (s≤s m<n ) a<sa ) → p→ x ≡ fromℕ≤ a<sa lem5 x eq with <-cmp (toℕ x) m lem5 x eq | tri< a ¬b ¬c = {!!} lem5 x eq | tri≈ ¬a refl ¬c = refl lem5 x eq | tri> ¬a ¬b c = {!!}