Mercurial > hg > Members > kono > Proof > galois
view Putil.agda @ 98:3a37a8f8cb39
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 31 Aug 2020 09:51:13 +0900 |
| parents | f4ff8e352aa7 |
| children | cadfbf0c810b |
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{-# OPTIONS --allow-unsolved-metas #-} module Putil 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 ( <-trans ; ≤-trans ; ≤-irrelevant ) renaming ( <-cmp to <-fcmp ) 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 ; _++_ ; head ; tail ) renaming (reverse to rev ) open import nat open import Symmetric open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import Relation.Binary.Definitions open import fin -- An inductive construction of permutation -- Todo -- -- describe property of pins ( move 0 to any position) -- describe property of shrink ( remove one column ) -- prove FL→iso -- prove FL←iso -- we already have refl and trans in the Symmetric Group pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n) pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where p→ : Fin (suc n) → Fin (suc n) p→ zero = zero p→ (suc x) = suc ( perm ⟨$⟩ʳ x) p← : Fin (suc n) → Fin (suc n) p← zero = zero p← (suc x) = suc ( perm ⟨$⟩ˡ x) piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x piso← zero = refl piso← (suc x) = cong (λ k → suc k ) (inverseʳ perm) piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x piso→ zero = refl piso→ (suc x) = cong (λ k → suc k ) (inverseˡ perm) pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n )) pswap {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→ zero = suc zero p→ (suc zero) = zero p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) ) p← : Fin (suc (suc n)) → Fin (suc (suc n)) p← zero = suc zero p← (suc zero) = zero p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) ) piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x piso← zero = refl piso← (suc zero) = refl piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm) piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x piso→ zero = refl piso→ (suc zero) = refl piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm) -- enumeration psawpn : {n : ℕ} → 1 < n → Permutation n n psawpn {suc zero} (s≤s ()) psawpn {suc n} (s≤s (s≤s x)) = pswap pid pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n pfill {n} {m} m≤n perm = pfill1 (n - m) (n-m<n n m ) (subst (λ k → Permutation k k ) (n-n-m=m m≤n ) perm) where pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n pfill1 0 _ perm = perm pfill1 (suc i) i<n perm = pfill1 i (≤to< i<n) (subst (λ k → Permutation k k ) (si-sn=i-n i<n ) ( pprep perm ) ) -- -- psawpim (inseert swap at position m ) -- psawpim : {n m : ℕ} → suc (suc m) ≤ n → Permutation n n psawpim {n} {m} m≤n = pfill m≤n ( psawpn (s≤s (s≤s z≤n)) ) n≤ : (i : ℕ ) → {j : ℕ } → i ≤ i + j n≤ (zero) {j} = z≤n n≤ (suc i) {j} = s≤s ( n≤ i ) lem0 : {n : ℕ } → n ≤ n lem0 {zero} = z≤n lem0 {suc n} = s≤s lem0 lem00 : {n m : ℕ } → n ≡ m → n ≤ m lem00 refl = lem0 plist1 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ plist1 {n} perm zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ< {zero} (s≤s z≤n))) ∷ [] plist1 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ< (s≤s lt))) ∷ plist1 perm i (<-trans lt a<sa) plist : {n : ℕ} → Permutation n n → List ℕ plist {0} perm = [] plist {suc n} perm = rev (plist1 perm n a<sa) -- pconcat : {n m : ℕ } → Permutation m m → Permutation n n → Permutation (m + n) (m + n) -- pconcat {n} {m} p q = pfill {n + m} {m} ? p ∘ₚ ? -- inductivley enmumerate permutations -- from n-1 length create n length inserting new element at position m -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] -- 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] plist ( pins {3} (n≤ 1) ) 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] -- 1 ∷ 2 ∷ 0 ∷ 3 ∷ [] plist ( pins {3} (n≤ 2) ) 2 ∷ 0 ∷ 1 ∷ 3 ∷ [] -- 1 ∷ 2 ∷ 3 ∷ 0 ∷ [] plist ( pins {3} (n≤ 3) ) 3 ∷ 0 ∷ 1 ∷ 2 ∷ [] -- pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) -- pins {_} {zero} _ = pid -- pins {suc _} {suc zero} _ = pswap pid -- pins {suc (suc n)} {suc m} (s≤s m<n) = pins1 (suc m) (suc (suc n)) lem0 where -- pins1 : (i j : ℕ ) → j ≤ suc (suc n) → Permutation (suc (suc (suc n ))) (suc (suc (suc n))) -- pins1 _ zero _ = pid -- pins1 zero _ _ = pid -- pins1 (suc i) (suc j) (s≤s si≤n) = psawpim {suc (suc (suc n))} {j} (s≤s (s≤s si≤n)) ∘ₚ pins1 i j (≤-trans si≤n a≤sa ) open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) open ≡-Reasoning pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) pins {_} {zero} _ = pid pins {suc n} {suc m} (s≤s m≤n) = permutation p← p→ record { left-inverse-of = piso← ; right-inverse-of = piso→ } where next : Fin (suc (suc n)) → Fin (suc (suc n)) next zero = suc zero next (suc x) = fromℕ< (≤-trans (fin<n {_} {x} ) a≤sa ) p→ : Fin (suc (suc n)) → Fin (suc (suc n)) p→ x with <-cmp (toℕ x) (suc m) ... | tri< a ¬b ¬c = fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) )) ... | tri≈ ¬a b ¬c = zero ... | tri> ¬a ¬b c = x p← : Fin (suc (suc n)) → Fin (suc (suc n)) p← zero = fromℕ< (s≤s (s≤s m≤n)) p← (suc x) with <-cmp (toℕ x) (suc m) ... | tri< a ¬b ¬c = fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) ... | tri≈ ¬a b ¬c = suc x ... | tri> ¬a ¬b c = suc x mm : toℕ (fromℕ< {suc m} {suc (suc n)} (s≤s (s≤s m≤n))) ≡ suc m mm = toℕ-fromℕ< (s≤s (s≤s m≤n)) mma : (x : Fin (suc n) ) → suc (toℕ x) ≤ suc m → toℕ ( fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) ) ≤ m mma x (s≤s x<sm) = subst (λ k → k ≤ m) (sym (toℕ-fromℕ< (≤-trans fin<n a≤sa ) )) x<sm p3 : (x : Fin (suc n) ) → toℕ (fromℕ< (≤-trans (fin<n {_} {suc x} ) (s≤s a≤sa))) ≡ suc (toℕ x) p3 x = begin toℕ (fromℕ< (≤-trans (fin<n {_} {suc x} ) (s≤s a≤sa))) ≡⟨ toℕ-fromℕ< ( s≤s ( ≤-trans fin<n a≤sa ) ) ⟩ suc (toℕ x) ∎ piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x piso→ zero with <-cmp (toℕ (fromℕ< (≤-trans (s≤s z≤n) (s≤s (s≤s m≤n) )))) (suc m) ... | tri< a ¬b ¬c = refl piso→ (suc x) with <-cmp (toℕ (suc x)) (suc m) ... | tri≈ ¬a refl ¬c = p13 where p13 : fromℕ< (s≤s (s≤s m≤n)) ≡ suc x p13 = cong (λ k → suc k ) (fromℕ<-toℕ _ (s≤s m≤n) ) ... | tri> ¬a ¬b c = p16 (suc x) refl where p16 : (y : Fin (suc (suc n))) → y ≡ suc x → p← y ≡ suc x p16 zero eq = ⊥-elim ( nat-≡< (cong (λ k → suc (toℕ k) ) eq) (s≤s (s≤s (z≤n)))) p16 (suc y) eq with <-cmp (toℕ y) (suc m) -- suc (suc m) < toℕ (suc x) ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< refl ( ≤-trans c (subst (λ k → k < suc m) p17 a )) ) where -- x = suc m case, c : suc (suc m) ≤ suc (toℕ x), a : suc (toℕ y) ≤ suc m, suc y ≡ suc x p17 : toℕ y ≡ toℕ x p17 with <-cmp (toℕ y) (toℕ x) | cong toℕ eq ... | tri< a ¬b ¬c | seq = ⊥-elim ( nat-≡< seq (s≤s a) ) ... | tri≈ ¬a b ¬c | seq = b ... | tri> ¬a ¬b c | seq = ⊥-elim ( nat-≡< (sym seq) (s≤s c)) ... | tri≈ ¬a b ¬c = eq ... | tri> ¬a ¬b c₁ = eq ... | tri< a ¬b ¬c = p10 (fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) ))) refl where p10 : (y : Fin (suc (suc n)) ) → y ≡ fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) )) → p← y ≡ suc x p10 zero () p10 (suc y) eq = p15 where p12 : toℕ y ≡ suc (toℕ x) p12 = begin toℕ y ≡⟨ cong (λ k → Data.Nat.pred (toℕ k)) eq ⟩ toℕ (fromℕ< (≤-trans a (s≤s m≤n))) ≡⟨ toℕ-fromℕ< {suc (toℕ x)} {suc n} (≤-trans a (s≤s m≤n)) ⟩ suc (toℕ x) ∎ p15 : p← (suc y) ≡ suc x p15 with <-cmp (toℕ y) (suc m) -- eq : suc y ≡ suc (suc (fromℕ< (≤-pred (≤-trans a (s≤s m≤n))))) , a : suc x < suc m ... | tri< a₁ ¬b ¬c = p11 where p11 : fromℕ< (≤-trans (fin<n {_} {y}) a≤sa ) ≡ suc x p11 = begin fromℕ< (≤-trans (fin<n {_} {y}) a≤sa ) ≡⟨ fromℕ<-irrelevant _ _ p12 _ (s≤s (fin<n {suc n})) ⟩ suc (fromℕ< (fin<n {suc n} {x} )) ≡⟨ cong suc (fromℕ<-toℕ x _ ) ⟩ suc x ∎ ... | tri≈ ¬a b ¬c = ⊥-elim ( nat-≡< b (subst (λ k → k < suc m) (sym p12) a )) -- suc x < suc m -> y = suc x → toℕ y < suc m ... | tri> ¬a ¬b c = ⊥-elim ( nat-<> c (subst (λ k → k < suc m) (sym p12) a )) piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x piso← zero with <-cmp (toℕ (fromℕ< (s≤s (s≤s m≤n)))) (suc m) | mm ... | tri< a ¬b ¬c | t = ⊥-elim ( ¬b t ) ... | tri≈ ¬a b ¬c | t = refl ... | tri> ¬a ¬b c | t = ⊥-elim ( ¬b t ) piso← (suc x) with <-cmp (toℕ x) (suc m) ... | tri> ¬a ¬b c with <-cmp (toℕ (suc x)) (suc m) ... | tri< a ¬b₁ ¬c = ⊥-elim ( nat-<> a (<-trans c a<sa ) ) ... | tri≈ ¬a₁ b ¬c = ⊥-elim ( nat-≡< (sym b) (<-trans c a<sa )) ... | tri> ¬a₁ ¬b₁ c₁ = refl piso← (suc x) | tri≈ ¬a b ¬c with <-cmp (toℕ (suc x)) (suc m) ... | tri< a ¬b ¬c₁ = ⊥-elim ( nat-≡< b (<-trans a<sa a) ) ... | tri≈ ¬a₁ refl ¬c₁ = ⊥-elim ( nat-≡< b a<sa ) ... | tri> ¬a₁ ¬b c = refl piso← (suc x) | tri< a ¬b ¬c with <-cmp (toℕ ( fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) )) (suc m) ... | tri≈ ¬a b ¬c₁ = ⊥-elim ( ¬a (s≤s (mma x a))) ... | tri> ¬a ¬b₁ c = ⊥-elim ( ¬a (s≤s (mma x a))) ... | tri< a₁ ¬b₁ ¬c₁ = p0 where p2 : suc (suc (toℕ x)) ≤ suc (suc n) p2 = s≤s (fin<n {suc n} {x}) p6 : suc (toℕ (fromℕ< (≤-trans (fin<n {_} {suc x}) (s≤s a≤sa)))) ≤ suc (suc n) p6 = s≤s (≤-trans a₁ (s≤s m≤n)) p0 : fromℕ< (≤-trans (s≤s a₁) (s≤s (s≤s m≤n) )) ≡ suc x p0 = begin fromℕ< (≤-trans (s≤s a₁) (s≤s (s≤s m≤n) )) ≡⟨⟩ fromℕ< (s≤s (≤-trans a₁ (s≤s m≤n))) ≡⟨ lemma10 (p3 x) {p6} {p2} ⟩ fromℕ< ( s≤s (fin<n {suc n} {x}) ) ≡⟨⟩ suc (fromℕ< (fin<n {suc n} {x} )) ≡⟨ cong suc (fromℕ<-toℕ x _ ) ⟩ suc x ∎ t7 = plist (pins {3} (n≤ 3)) ∷ plist (flip ( pins {3} (n≤ 3) )) ∷ plist ( pins {3} (n≤ 3) ∘ₚ flip ( pins {3} (n≤ 3))) ∷ [] -- t8 = {!!} open import logic open _∧_ perm1 : {perm : Permutation 1 1 } {q : Fin 1} → (perm ⟨$⟩ʳ q ≡ # 0) ∧ ((perm ⟨$⟩ˡ q ≡ # 0)) perm1 {p} {q} = ⟪ perm01 _ _ , perm00 _ _ ⟫ where perm01 : (x y : Fin 1) → (p ⟨$⟩ʳ x) ≡ y perm01 x y with p ⟨$⟩ʳ x perm01 zero zero | zero = refl perm00 : (x y : Fin 1) → (p ⟨$⟩ˡ x) ≡ y perm00 x y with p ⟨$⟩ˡ x perm00 zero zero | zero = refl p=0 : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) ≡ # 0 p=0 {zero} perm with ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) ... | zero = refl p=0 {suc n} perm with perm ⟨$⟩ʳ (# 0) | inspect (_⟨$⟩ʳ_ perm ) (# 0)| toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0)) | inspect toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0)) ... | zero | record { eq = e} | m<n | _ = p001 where p001 : perm ⟨$⟩ˡ ( pins m<n ⟨$⟩ʳ zero) ≡ zero p001 = subst (λ k → perm ⟨$⟩ˡ k ≡ zero ) e (inverseˡ perm) ... | suc t | record { eq = e } | m<n | record { eq = e1 } = p002 where -- m<n : suc (toℕ t) ≤ suc n p002 : perm ⟨$⟩ˡ ( pins m<n ⟨$⟩ʳ zero) ≡ zero p002 = p005 zero (toℕ t) refl m<n refl where -- suc (toℕ t) ≤ suc n p003 : (s : Fin (suc (suc n))) → s ≡ (perm ⟨$⟩ʳ (# 0)) → perm ⟨$⟩ˡ s ≡ # 0 p003 s eq = subst (λ k → perm ⟨$⟩ˡ k ≡ zero ) (sym eq) (inverseˡ perm) p005 : (x : Fin (suc (suc n))) → (m : ℕ ) → x ≡ zero → (m≤n : suc m ≤ suc n ) → m ≡ toℕ t → perm ⟨$⟩ˡ ( pins m≤n ⟨$⟩ʳ zero) ≡ zero p005 zero m eq (s≤s m≤n) meq = p004 where p004 : perm ⟨$⟩ˡ (fromℕ< (s≤s (s≤s m≤n))) ≡ zero p004 = p003 (fromℕ< (s≤s (s≤s m≤n))) ( begin fromℕ< (s≤s (s≤s m≤n)) ≡⟨ fromℕ<-irrelevant _ _ (cong suc meq) (s≤s (s≤s m≤n)) (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) ⟩ fromℕ< (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) ≡⟨ fromℕ<-toℕ {suc (suc n)} (suc t) (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) ⟩ suc t ≡⟨ sym e ⟩ (perm ⟨$⟩ʳ (# 0)) ∎ ) -- pp : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → Fin (suc n) -- pp perm → (( perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) plist2 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ plist2 {n} perm zero _ = toℕ ( perm ⟨$⟩ʳ (fromℕ< {zero} (s≤s z≤n))) ∷ [] plist2 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ʳ (fromℕ< (s≤s lt))) ∷ plist2 perm i (<-trans lt a<sa) plist0 : {n : ℕ} → Permutation n n → List ℕ plist0 {0} perm = [] plist0 {suc n} perm = plist2 perm n a<sa open _=p=_ -- -- plist cong -- ←pleq : {n : ℕ} → (x y : Permutation n n ) → x =p= y → plist0 x ≡ plist0 y ←pleq {zero} x y eq = refl ←pleq {suc n} x y eq = ←pleq1 n a<sa where ←pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn ←pleq1 zero _ = cong ( λ k → toℕ k ∷ [] ) ( peq eq (fromℕ< {zero} (s≤s z≤n))) ←pleq1 (suc i) (s≤s lt) = cong₂ ( λ j k → toℕ j ∷ k ) ( peq eq (fromℕ< (s≤s lt))) ( ←pleq1 i (<-trans lt a<sa) ) headeq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → x ≡ y headeq refl = refl taileq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → xt ≡ yt taileq refl = refl -- -- plist equalizer -- pleq : {n : ℕ} → (x y : Permutation n n ) → plist0 x ≡ plist0 y → x =p= y pleq {0} x y refl = record { peq = λ q → pleq0 q } where pleq0 : (q : Fin 0 ) → (x ⟨$⟩ʳ q) ≡ (y ⟨$⟩ʳ q) pleq0 () pleq {suc n} x y eq = record { peq = λ q → pleq1 n a<sa eq q fin<n } where pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn → (q : Fin (suc n)) → toℕ q < suc i → x ⟨$⟩ʳ q ≡ y ⟨$⟩ʳ q pleq1 zero i<sn eq q q<i with <-cmp (toℕ q) zero ... | tri< () ¬b ¬c ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) ... | tri≈ ¬a b ¬c = begin x ⟨$⟩ʳ q ≡⟨ cong ( λ k → x ⟨$⟩ʳ k ) (toℕ-injective b )⟩ x ⟨$⟩ʳ zero ≡⟨ toℕ-injective (headeq eq) ⟩ y ⟨$⟩ʳ zero ≡⟨ cong ( λ k → y ⟨$⟩ʳ k ) (sym (toℕ-injective b )) ⟩ y ⟨$⟩ʳ q ∎ pleq1 (suc i) (s≤s i<sn) eq q q<i with <-cmp (toℕ q) (suc i) ... | tri< a ¬b ¬c = pleq1 i (<-trans i<sn a<sa ) (taileq eq) q a ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) ... | tri≈ ¬a b ¬c = begin x ⟨$⟩ʳ q ≡⟨ cong (λ k → x ⟨$⟩ʳ k) (pleq3 b) ⟩ x ⟨$⟩ʳ (suc (fromℕ< i<sn)) ≡⟨ toℕ-injective pleq2 ⟩ y ⟨$⟩ʳ (suc (fromℕ< i<sn)) ≡⟨ cong (λ k → y ⟨$⟩ʳ k) (sym (pleq3 b)) ⟩ y ⟨$⟩ʳ q ∎ where pleq3 : toℕ q ≡ suc i → q ≡ suc (fromℕ< i<sn) pleq3 tq=si = toℕ-injective ( begin toℕ q ≡⟨ b ⟩ suc i ≡⟨ sym (toℕ-fromℕ< (s≤s i<sn)) ⟩ toℕ (fromℕ< (s≤s i<sn)) ≡⟨⟩ toℕ (suc (fromℕ< i<sn)) ∎ ) pleq2 : toℕ ( x ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) ≡ toℕ ( y ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) pleq2 = headeq eq pprep-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pprep x =p= pprep y pprep-cong {n} {x} {y} x=y = record { peq = pprep-cong1 } where pprep-cong1 : (q : Fin (suc n)) → (pprep x ⟨$⟩ʳ q) ≡ (pprep y ⟨$⟩ʳ q) pprep-cong1 zero = refl pprep-cong1 (suc q) = begin pprep x ⟨$⟩ʳ suc q ≡⟨⟩ suc ( x ⟨$⟩ʳ q ) ≡⟨ cong ( λ k → suc k ) ( peq x=y q ) ⟩ suc ( y ⟨$⟩ʳ q ) ≡⟨⟩ pprep y ⟨$⟩ʳ suc q ∎ pprep-dist : {n : ℕ} → {x y : Permutation n n } → pprep (x ∘ₚ y) =p= (pprep x ∘ₚ pprep y) pprep-dist {n} {x} {y} = record { peq = pprep-dist1 } where pprep-dist1 : (q : Fin (suc n)) → (pprep (x ∘ₚ y) ⟨$⟩ʳ q) ≡ ((pprep x ∘ₚ pprep y) ⟨$⟩ʳ q) pprep-dist1 zero = refl pprep-dist1 (suc q) = cong ( λ k → suc k ) refl pswap-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pswap x =p= pswap y pswap-cong {n} {x} {y} x=y = record { peq = pswap-cong1 } where pswap-cong1 : (q : Fin (suc (suc n))) → (pswap x ⟨$⟩ʳ q) ≡ (pswap y ⟨$⟩ʳ q) pswap-cong1 zero = refl pswap-cong1 (suc zero) = refl pswap-cong1 (suc (suc q)) = begin pswap x ⟨$⟩ʳ suc (suc q) ≡⟨⟩ suc (suc (x ⟨$⟩ʳ q)) ≡⟨ cong ( λ k → suc (suc k) ) ( peq x=y q ) ⟩ suc (suc (y ⟨$⟩ʳ q)) ≡⟨⟩ pswap y ⟨$⟩ʳ suc (suc q) ∎ pswap-dist : {n : ℕ} → {x y : Permutation n n } → pprep (pprep (x ∘ₚ y)) =p= (pswap x ∘ₚ pswap y) pswap-dist {n} {x} {y} = record { peq = pswap-dist1 } where pswap-dist1 : (q : Fin (suc (suc n))) → ((pprep (pprep (x ∘ₚ y))) ⟨$⟩ʳ q) ≡ ((pswap x ∘ₚ pswap y) ⟨$⟩ʳ q) pswap-dist1 zero = refl pswap-dist1 (suc zero) = refl pswap-dist1 (suc (suc q)) = cong ( λ k → suc (suc k) ) refl data FL : (n : ℕ )→ Set where f0 : FL 0 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) shlem→ : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → (x : Fin (suc n) ) → perm ⟨$⟩ˡ x ≡ zero → x ≡ zero shlem→ perm p0=0 x px=0 = begin x ≡⟨ sym ( inverseʳ perm ) ⟩ perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ x) ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) px=0 ⟩ perm ⟨$⟩ʳ zero ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) (sym p0=0) ⟩ perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ zero) ≡⟨ inverseʳ perm ⟩ zero ∎ where open ≡-Reasoning shlem← : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → (x : Fin (suc n)) → perm ⟨$⟩ʳ x ≡ zero → x ≡ zero shlem← perm p0=0 x px=0 = begin x ≡⟨ sym (inverseˡ perm ) ⟩ perm ⟨$⟩ˡ ( perm ⟨$⟩ʳ x ) ≡⟨ cong (λ k → perm ⟨$⟩ˡ k ) px=0 ⟩ perm ⟨$⟩ˡ zero ≡⟨ p0=0 ⟩ zero ∎ where open ≡-Reasoning sh2 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ˡ (suc x) ≡ zero sh2 perm p0=0 {x} eq with shlem→ perm p0=0 (suc x) eq sh2 perm p0=0 {x} eq | () sh1 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ʳ (suc x) ≡ zero sh1 perm p0=0 {x} eq with shlem← perm p0=0 (suc x) eq sh1 perm p0=0 {x} eq | () -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] → 0 ∷ 1 ∷ 2 ∷ [] shrink : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → perm ⟨$⟩ˡ (# 0) ≡ # 0 → Permutation n n shrink {n} perm p0=0 = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where p→ : Fin n → Fin n p→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) p→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) p→ x | suc t | _ = t p← : Fin n → Fin n p← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) p← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) p← x | suc t | _ = t piso← : (x : Fin n ) → p→ ( p← x ) ≡ x piso← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) piso← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) piso← x | suc t | _ with perm ⟨$⟩ʳ (suc t) | inspect (_⟨$⟩ʳ_ perm ) (suc t) piso← x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 e ) piso← x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin t1 ≡⟨ plem0 plem1 ⟩ x ∎ where open ≡-Reasoning plem0 : suc t1 ≡ suc x → t1 ≡ x plem0 refl = refl plem1 : suc t1 ≡ suc x plem1 = begin suc t1 ≡⟨ sym e1 ⟩ Inverse.to perm Π.⟨$⟩ suc t ≡⟨ cong (λ k → Inverse.to perm Π.⟨$⟩ k ) (sym e0) ⟩ Inverse.to perm Π.⟨$⟩ ( Inverse.from perm Π.⟨$⟩ suc x ) ≡⟨ inverseʳ perm ⟩ suc x ∎ piso→ : (x : Fin n ) → p← ( p→ x ) ≡ x piso→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) piso→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) piso→ x | suc t | _ with perm ⟨$⟩ˡ (suc t) | inspect (_⟨$⟩ˡ_ perm ) (suc t) piso→ x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 e ) piso→ x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin t1 ≡⟨ plem2 plem3 ⟩ x ∎ where open ≡-Reasoning plem2 : suc t1 ≡ suc x → t1 ≡ x plem2 refl = refl plem3 : suc t1 ≡ suc x plem3 = begin suc t1 ≡⟨ sym e1 ⟩ Inverse.from perm Π.⟨$⟩ suc t ≡⟨ cong (λ k → Inverse.from perm Π.⟨$⟩ k ) (sym e0 ) ⟩ Inverse.from perm Π.⟨$⟩ ( Inverse.to perm Π.⟨$⟩ suc x ) ≡⟨ inverseˡ perm ⟩ suc x ∎ shrink-iso : { n : ℕ } → {perm : Permutation n n} → shrink (pprep perm) refl =p= perm shrink-iso {n} {perm} = record { peq = λ q → refl } shrink-cong : { n : ℕ } → {x y : Permutation (suc n) (suc n)} → x =p= y → (x=0 : x ⟨$⟩ˡ (# 0) ≡ # 0 ) → (y=0 : y ⟨$⟩ˡ (# 0) ≡ # 0 ) → shrink x x=0 =p= shrink y y=0 shrink-cong {n} {x} {y} x=0 y=0 = {!!} FL→perm : {n : ℕ } → FL n → Permutation n n FL→perm f0 = pid FL→perm (x :: fl) = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) t40 = (# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) t4 = FL→perm ((# 2) :: t40 ) -- t1 = plist (shrink (pid {3} ∘ₚ (pins (n≤ 1))) refl) t2 = plist ((pid {5} ) ∘ₚ transpose (# 2) (# 4)) ∷ plist (pid {5} ∘ₚ reverse ) ∷ [] t3 = plist (FL→perm t40) -- ∷ plist (pprep (FL→perm t40)) -- ∷ plist ( pprep (FL→perm t40) ∘ₚ pins ( n≤ 0 {3} )) -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 1 {2} )) -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 2 {1} )) -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 3 {0} )) ∷ plist ( FL→perm ((# 0) :: t40)) -- (0 ∷ 1 ∷ 2 ∷ []) ∷ ∷ plist ( FL→perm ((# 1) :: t40)) -- (0 ∷ 2 ∷ 1 ∷ []) ∷ ∷ plist ( FL→perm ((# 2) :: t40)) -- (1 ∷ 0 ∷ 2 ∷ []) ∷ ∷ plist ( FL→perm ((# 3) :: t40)) -- (2 ∷ 0 ∷ 1 ∷ []) ∷ -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) -- (1 ∷ 2 ∷ 0 ∷ []) ∷ -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) ))) -- (2 ∷ 1 ∷ 0 ∷ []) ∷ -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))))) -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) ∘ₚ (FL→perm ((# 3) :: (((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) )))) )) ∷ [] perm→FL : {n : ℕ } → Permutation n n → FL n perm→FL {zero} perm = f0 perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) -- perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (remove (# 0) perm) pcong-pF : {n : ℕ } → {x y : Permutation n n} → x =p= y → perm→FL x ≡ perm→FL y pcong-pF = {!!} -- t5 = plist t4 ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 3 ) )) t5 = plist (t4) ∷ plist (flip t4) ∷ ( toℕ (t4 ⟨$⟩ˡ fromℕ< a<sa) ∷ [] ) ∷ ( toℕ (t4 ⟨$⟩ʳ (# 0)) ∷ [] ) -- ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 1 ) )) ∷ plist (remove (# 0) t4 ) ∷ plist ( FL→perm t40 ) ∷ [] t6 = perm→FL t4 FL→iso : {n : ℕ } → (fl : FL n ) → perm→FL ( FL→perm fl ) ≡ fl FL→iso f0 = refl FL→iso (x :: fl) = cong₂ ( λ j k → j :: k ) f001 f002 where perm = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) f001 : perm ⟨$⟩ʳ (# 0) ≡ x f001 = {!!} x=0 : (perm ∘ₚ flip (pins (toℕ≤pred[n] x))) ⟨$⟩ˡ (# 0) ≡ # 0 x=0 = {!!} x=0' : (pprep (FL→perm fl) ∘ₚ pid) ⟨$⟩ˡ (# 0) ≡ # 0 x=0' = {!!} f002 : perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) ≡ fl f002 = begin perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) ≡⟨ pcong-pF (shrink-cong (presp prefl {!!}) (p=0 perm) x=0) ⟩ perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] x))) x=0 ) ≡⟨⟩ perm→FL (shrink ((pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x )) ∘ₚ flip (pins (toℕ≤pred[n] x))) x=0 ) ≡⟨ pcong-pF (shrink-cong (passoc _ _ _ ) x=0 x=0) ⟩ perm→FL (shrink (pprep (FL→perm fl) ∘ₚ (pins ( toℕ≤pred[n] x ) ∘ₚ flip (pins (toℕ≤pred[n] x)))) x=0 ) ≡⟨ pcong-pF (shrink-cong record { peq = λ q → inverseʳ _ } x=0 x=0') ⟩ perm→FL (shrink (pprep (FL→perm fl) ∘ₚ pid) x=0' ) ≡⟨ pcong-pF (shrink-cong prefl x=0' x=0') ⟩ perm→FL (shrink (pprep (FL→perm fl)) x=0' ) ≡⟨ pcong-pF shrink-iso ⟩ perm→FL ( FL→perm fl ) ≡⟨ FL→iso fl ⟩ fl ∎ open _=p=_ -- FL←iso : {n : ℕ } → (perm : Permutation n n ) → FL→perm ( perm→FL perm ) =p= perm -- FL←iso {0} perm = record { peq = λ () } -- FL←iso {suc n} perm = {!!} lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n lem2 i≤n = ≤-trans i≤n ( a≤sa ) ∀-FL : (n : ℕ ) → List (FL (suc n)) ∀-FL x = fls6 x where fls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → FL n → List (FL (suc n)) → List (FL (suc n)) fls4 zero n i≤n perm x = (zero :: perm ) ∷ x fls4 (suc i) n i≤n perm x = fls4 i n (≤-trans a≤sa i≤n ) perm ((fromℕ< (s≤s i≤n) :: perm ) ∷ x) fls5 : ( n : ℕ ) → List (FL n) → List (FL (suc n)) → List (FL (suc n)) fls5 n [] x = x fls5 n (h ∷ x) y = fls5 n x (fls4 n n lem0 h y) fls6 : ( n : ℕ ) → List (FL (suc n)) fls6 zero = (zero :: f0) ∷ [] fls6 (suc n) = fls5 (suc n) (fls6 n) [] tf1 = ∀-FL 4 tf2 = Data.List.map (λ k → ⟪ plist (FL→perm k ) , k ⟫ ) tf1 all-perm : (n : ℕ ) → List (Permutation (suc n) (suc n) ) all-perm n = pls6 n where lem1 : {i n : ℕ } → i ≤ n → i < suc n lem1 z≤n = s≤s z≤n lem1 (s≤s lt) = s≤s (lem1 lt) pls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation n n → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) pls4 zero n i≤n perm x = (pprep perm ∘ₚ pins i≤n ) ∷ x pls4 (suc i) n i≤n perm x = pls4 i n (≤-trans a≤sa i≤n ) perm (pprep perm ∘ₚ pins {n} {suc i} i≤n ∷ x) pls5 : ( n : ℕ ) → List (Permutation n n) → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) pls5 n [] x = x pls5 n (h ∷ x) y = pls5 n x (pls4 n n lem0 h y) pls6 : ( n : ℕ ) → List (Permutation (suc n) (suc n)) pls6 zero = pid ∷ [] pls6 (suc n) = pls5 (suc n) (rev (pls6 n) ) [] -- rev to put id first pls : (n : ℕ ) → List (List ℕ ) pls n = Data.List.map plist (all-perm n)