Mercurial > hg > Members > kono > Proof > automaton
diff automaton-in-agda/src/finiteSetUtil.agda @ 405:af8f630b7e60
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sun, 24 Sep 2023 18:02:04 +0900 |
parents | dfaf230f7b9a |
children | a60132983557 |
line wrap: on
line diff
--- a/automaton-in-agda/src/finiteSetUtil.agda Sun Sep 24 13:20:31 2023 +0900 +++ b/automaton-in-agda/src/finiteSetUtil.agda Sun Sep 24 18:02:04 2023 +0900 @@ -27,7 +27,7 @@ module _ {Q : Set } (F : FiniteSet Q) where open FiniteSet F - equal?-refl : { x : Q } → equal? x x ≡ true + equal?-refl : { x : Q } → equal? x x ≡ true equal?-refl {x} with F←Q x ≟ F←Q x ... | yes eq = refl ... | no ne = ⊥-elim (ne refl) @@ -47,24 +47,24 @@ ... | yes eq = yes (subst₂ (λ j k → j ≡ k ) (finiso→ x) (finiso→ y) (cong Q←F eq) ) ... | no n = no (λ eq → n (cong F←Q eq)) End : (m : ℕ ) → (p : Q → Bool ) → Set - End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false + End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false first-end : ( p : Q → Bool ) → End finite p first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {finite} i) ) next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p → (m<n : m < finite ) → p (Q←F (fromℕ< m<n )) ≡ false → End m p - next-end {m} p prev m<n np i m<i with NP.<-cmp m (toℕ i) + next-end {m} p prev m<n np i m<i with NP.<-cmp m (toℕ i) next-end p prev m<n np i m<i | tri< a ¬b ¬c = prev i a next-end p prev m<n np i m<i | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c ) next-end {m} p prev m<n np i m<i | tri≈ ¬a b ¬c = subst ( λ k → p (Q←F k) ≡ false) (m<n=i i b m<n ) np where m<n=i : {n : ℕ } (i : Fin n) {m : ℕ } → m ≡ (toℕ i) → (m<n : m < n ) → fromℕ< m<n ≡ i - m<n=i i refl m<n = fromℕ<-toℕ i m<n + m<n=i i refl m<n = fromℕ<-toℕ i m<n found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true found {p} q pt = found1 finite (NP.≤-refl ) ( first-end p ) where found1 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → ((i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false ) → exists1 m m<n p ≡ true found1 0 m<n end = ⊥-elim ( ¬-bool (subst (λ k → k ≡ false ) (cong (λ k → p k) (finiso→ q) ) (end (F←Q q) z≤n )) pt ) found1 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true - found1 (suc m) m<n end | yes eq = subst (λ k → k \/ exists1 m (<to≤ m<n) p ≡ true ) (sym eq) (bool-or-4 {exists1 m (<to≤ m<n) p} ) + found1 (suc m) m<n end | yes eq = subst (λ k → k \/ exists1 m (<to≤ m<n) p ≡ true ) (sym eq) (bool-or-4 {exists1 m (<to≤ m<n) p} ) found1 (suc m) m<n end | no np = begin p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p ≡⟨ bool-or-1 (¬-bool-t np ) ⟩ @@ -78,9 +78,9 @@ not-found2 zero _ = refl not-found2 ( suc m ) m<n with pn (Q←F (fromℕ< {m} {finite} m<n)) not-found2 (suc m) m<n | eq = begin - p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p + p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p ≡⟨ bool-or-1 eq ⟩ - exists1 m (<to≤ m<n) p + exists1 m (<to≤ m<n) p ≡⟨ not-found2 m (<to≤ m<n) ⟩ false ∎ where open ≡-Reasoning @@ -92,9 +92,9 @@ f01 = not-found (λ q → subst ( λ k → p k ≡ false ) (finiso→ _) (end (F←Q q) z≤n )) found2 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true found2 (suc m) m<n end | yes eq = record { found-q = Q←F (fromℕ< m<n) ; found-p = eq } - found2 (suc m) m<n end | no np = - found2 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) - not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false + found2 (suc m) m<n end | no np = + found2 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) + not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false not-found← {p} np q = ¬-bool-t ( contra-position {_} {_} {_} {exists p ≡ true} (found q) (λ ep → ¬-bool np ep ) ) Q←F-inject : {x y : Fin finite} → Q←F x ≡ Q←F y → x ≡ y Q←F-inject eq = subst₂ (λ j k → j ≡ k ) (finiso← _) (finiso← _) (cong F←Q eq) @@ -103,16 +103,16 @@ -iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B +iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B iso-fin {A} {B} fin iso = record { Q←F = λ f → fun→ iso ( FiniteSet.Q←F fin f ) ; F←Q = λ b → FiniteSet.F←Q fin (fun← iso b ) - ; finiso→ = finiso→ - ; finiso← = finiso← + ; finiso→ = finiso→ + ; finiso← = finiso← } where finiso→ : (q : B) → fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡ q finiso→ q = begin - fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) + fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡⟨ cong (λ k → fun→ iso k ) (FiniteSet.finiso→ fin _ ) ⟩ fun→ iso (Bijection.fun← iso q) ≡⟨ fiso→ iso _ ⟩ @@ -120,9 +120,9 @@ ∎ where open ≡-Reasoning finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡ f finiso← f = begin - FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) + FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (Bijection.fiso← iso _) ⟩ - FiniteSet.F←Q fin (FiniteSet.Q←F fin f) + FiniteSet.F←Q fin (FiniteSet.Q←F fin f) ≡⟨ FiniteSet.finiso← fin _ ⟩ f ∎ where @@ -136,7 +136,7 @@ fin00 : (q : One) → one ≡ q fin00 one = refl -fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) +fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) fin-∨1 {B} fb = record { Q←F = Q←F ; F←Q = F←Q @@ -149,7 +149,7 @@ Q←F (suc f) = case2 (FiniteSet.Q←F fb f) F←Q : One ∨ B → Fin (suc b) F←Q (case1 one) = zero - F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) + F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) finiso→ : (q : One ∨ B) → Q←F (F←Q q) ≡ q finiso→ (case1 one) = refl finiso→ (case2 b) = cong (λ k → case2 k ) (FiniteSet.finiso→ fb b) @@ -158,7 +158,7 @@ finiso← (suc f) = cong ( λ k → suc k ) (FiniteSet.finiso← fb f) -fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B → FiniteSet (Fin a ∨ B) +fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B → FiniteSet (Fin a ∨ B) fin-∨2 {B} zero fb = iso-fin fb iso where iso : Bijection B (Fin zero ∨ B) iso = record { @@ -168,7 +168,7 @@ ; fiso← = λ _ → refl } where fun←1 : Fin zero ∨ B → B - fun←1 (case2 x) = x + fun←1 (case2 x) = x fiso→1 : (f : Fin zero ∨ B ) → case2 (fun←1 f) ≡ f fiso→1 (case2 x) = refl fin-∨2 {B} (suc a) fb = iso-fin (fin-∨1 (fin-∨2 a fb) ) iso @@ -193,9 +193,9 @@ fun→ (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f fiso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin fiso→ (FiniteSet→Fin fin) = FiniteSet.finiso→ fin - + -fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) +fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) fin-∨ {A} {B} fa fb = iso-fin (fin-∨2 a fb ) iso2 where a = FiniteSet.finite fa ia = FiniteSet→Fin fa @@ -211,14 +211,14 @@ open import Data.Product hiding ( map ) -fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) +fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) fin-× {A} {B} fa fb with FiniteSet→Fin fa ... | a=f = iso-fin (fin-×-f a ) iso-1 where a = FiniteSet.finite fa b = FiniteSet.finite fb iso-1 : Bijection (Fin a × B) ( A × B ) - fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) - fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) + fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) + fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) fiso← iso-1 x = lemma where lemma : (FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj₁ x)) , proj₂ x) ≡ ( proj₁ x , proj₂ x ) lemma = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso← fa _ ) @@ -234,19 +234,19 @@ fiso→ iso-2 (zero , b ) = refl fiso→ iso-2 (suc a , b ) = refl - fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) × B) + fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) × B) fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 open _∧_ -fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) +fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) fin-∧ {A} {B} fa fb with FiniteSet→Fin fa -- same thing for our tool ... | a=f = iso-fin (fin-×-f a ) iso-1 where a = FiniteSet.finite fa b = FiniteSet.finite fb iso-1 : Bijection (Fin a ∧ B) ( A ∧ B ) - fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} + fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} fun→ iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x) ; proj2 = proj2 x} fiso← iso-1 x = lemma where lemma : record { proj1 = FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj1 x)) ; proj2 = proj2 x} ≡ record {proj1 = proj1 x ; proj2 = proj2 x } @@ -263,7 +263,7 @@ fiso→ iso-2 (record { proj1 = zero ; proj2 = b }) = refl fiso→ iso-2 (record { proj1 = suc a ; proj2 = b }) = refl - fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) ∧ B) + fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) ∧ B) fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 @@ -292,12 +292,12 @@ cast-iso refl (suc f) = cong ( λ k → suc k ) ( cast-iso refl f ) -fin2List : {n : ℕ } → FiniteSet (Vec Bool n) +fin2List : {n : ℕ } → FiniteSet (Vec Bool n) fin2List {zero} = record { Q←F = λ _ → Vec.[] ; F←Q = λ _ → # 0 - ; finiso→ = finiso→ - ; finiso← = finiso← + ; finiso→ = finiso→ + ; finiso← = finiso← } where Q = Vec Bool zero finiso→ : (q : Q) → [] ≡ q @@ -309,7 +309,7 @@ QtoR : Vec Bool (suc n) → Vec Bool n ∨ Vec Bool n QtoR ( true ∷ x ) = case1 x QtoR ( false ∷ x ) = case2 x - RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) + RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) RtoQ ( case1 x ) = true ∷ x RtoQ ( case2 x ) = false ∷ x isoRQ : (x : Vec Bool (suc n) ) → RtoQ ( QtoR x ) ≡ x @@ -329,17 +329,17 @@ qb1 : (q : Q) → toℕ (FiniteSet.F←Q fin q) < n → Bool qb1 q q<n = Q→B q (NP.<-trans q<n a<sa) -List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → Q → Bool +List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → Q → Bool List2Func {Q} {zero} fin (s≤s z≤n) [] q = false List2Func {Q} {suc n} fin (s≤s n<m) (h ∷ t) q with FiniteSet.F←Q fin q ≟ fromℕ< n<m ... | yes _ = h ... | no _ = List2Func {Q} fin (NP.<-trans n<m a<sa ) t q -open import Level renaming ( suc to Suc ; zero to Zero) +open import Level renaming ( suc to Suc ; zero to Zero) L2F : {Q : Set } {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → (q : Q ) → toℕ (FiniteSet.F←Q fin q ) < n → Bool -L2F fin n<m x q q<n = List2Func fin n<m x q +L2F fin n<m x q q<n = List2Func fin n<m x q L2F-iso : { Q : Set } → (fin : FiniteSet Q ) → (f : Q → Bool ) → (q : Q ) → (L2F fin a<sa (F2L fin a<sa (λ q _ → f q) )) q (toℕ<n _) ≡ f q L2F-iso {Q} fin f q = l2f m a<sa (toℕ<n _) where @@ -352,93 +352,25 @@ lemma13 {suc n} {suc nq} n<m nt (s≤s nq≤n) = s≤s (lemma13 {n} {nq} (NP.<-trans a<sa n<m ) (λ eq → nt ( cong ( λ k → suc k ) eq )) nq≤n) lemma3f : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt) lemma3f (s≤s lt) = refl - lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m + lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m lemma12f {zero} {suc m} (s≤s z≤n) zero refl = refl lemma12f {suc n} {suc m} (s≤s n<m) (suc f) refl = subst ( λ k → suc f ≡ k ) (sym (lemma3f n<m) ) ( cong ( λ k → suc k ) ( lemma12f {n} {m} n<m f refl ) ) l2f : (n : ℕ ) → (n<m : n < suc m ) → (q<n : toℕ (FiniteSet.F←Q fin q ) < n ) → (L2F fin n<m (F2L fin n<m (λ q _ → f q))) q q<n ≡ f q l2f zero (s≤s z≤n) () - l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m - l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin - f (FiniteSet.Q←F fin (fromℕ< n<m)) + l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m + l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin + f (FiniteSet.Q←F fin (fromℕ< n<m)) ≡⟨ cong ( λ k → f (FiniteSet.Q←F fin k )) (sym p) ⟩ f (FiniteSet.Q←F fin ( FiniteSet.F←Q fin q )) ≡⟨ cong ( λ k → f k ) (FiniteSet.finiso→ fin _ ) ⟩ - f q + f q ∎ where open ≡-Reasoning l2f (suc n) (s≤s n<m) (s≤s n<q) | no ¬p = l2f n (NP.<-trans n<m a<sa) (lemma11f n<m ¬p n<q) -Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) +Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) Fin2Finite n = record { F←Q = λ x → x ; Q←F = λ x → x ; finiso← = λ q → refl ; finiso→ = λ q → refl } -data fin-less { n : ℕ } { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) : Set where - elm1 : (elm : A ) → toℕ (FiniteSet.F←Q fa elm ) < n → fin-less fa n<m - -get-elm : { n : ℕ } { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa } → fin-less fa n<m → A -get-elm (elm1 a _ ) = a - -get-< : { n : ℕ } { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa }→ (f : fin-less fa n<m ) → toℕ (FiniteSet.F←Q fa (get-elm f )) < n -get-< (elm1 _ b ) = b - -fin-< : {A : Set} → { n : ℕ } → (fa : FiniteSet A ) → (n<m : n < FiniteSet.finite fa ) → FiniteSet (fin-less fa n<m ) -fin-< {A} {n} fa n<m = iso-fin (Fin2Finite n) iso where - m = FiniteSet.finite fa - iso : Bijection (Fin n) (fin-less fa n<m ) - lemma11f : {n : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) ≡ toℕ x - lemma11f {n} {x} n<m = begin - toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) - ≡⟨ toℕ-fromℕ< _ ⟩ - toℕ x - ∎ where - open ≡-Reasoning - fun← iso (elm1 elm x) = fromℕ< x - fun→ iso x = elm1 (FiniteSet.Q←F fa (fromℕ< (NP.<-trans x<n n<m ))) to<n where - x<n : toℕ x < n - x<n = toℕ<n x - to<n : toℕ (FiniteSet.F←Q fa (FiniteSet.Q←F fa (fromℕ< (NP.<-trans x<n n<m)))) < n - to<n = subst (λ k → toℕ k < n ) (sym (FiniteSet.finiso← fa _ )) (subst (λ k → k < n ) (sym ( toℕ-fromℕ< (NP.<-trans x<n n<m) )) x<n ) - fiso← iso x = lemma2 where - lemma2 : fromℕ< (subst (λ k → toℕ k < n) (sym - (FiniteSet.finiso← fa (fromℕ< (NP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) - (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) ≡ x - lemma2 = begin - fromℕ< (subst (λ k → toℕ k < n) (sym - (FiniteSet.finiso← fa (fromℕ< (NP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) - (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) - ≡⟨⟩ - fromℕ< ( subst (λ k → toℕ ( k ) < n ) (sym (FiniteSet.finiso← fa _ )) lemma6 ) - ≡⟨ lemma10 (cong (λ k → toℕ k) (FiniteSet.finiso← fa _ ) ) ⟩ - fromℕ< lemma6 - ≡⟨ lemma10 (lemma11 n<m ) ⟩ - fromℕ< ( toℕ<n x ) - ≡⟨ fromℕ<-toℕ _ _ ⟩ - x - ∎ where - open ≡-Reasoning - lemma6 : toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) < n - lemma6 = subst ( λ k → k < n ) (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x ) - fiso→ iso (elm1 elm x) = ? where -- fin-less-cong fa n<m _ _ lemma (lemma8 (cong (λ k → toℕ (FiniteSet.F←Q fa k) ) lemma ) ) where - lemma13 : toℕ (fromℕ< x) ≡ toℕ (FiniteSet.F←Q fa elm) - lemma13 = begin - toℕ (fromℕ< x) - ≡⟨ toℕ-fromℕ< _ ⟩ - toℕ (FiniteSet.F←Q fa elm) - ∎ where open ≡-Reasoning - lemma : FiniteSet.Q←F fa (fromℕ< (NP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) ≡ elm - lemma = begin - FiniteSet.Q←F fa (fromℕ< (NP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) - ≡⟨⟩ - FiniteSet.Q←F fa (fromℕ< ( NP.<-trans (toℕ<n ( fromℕ< x ) ) n<m)) - ≡⟨ cong (λ k → FiniteSet.Q←F fa k) (lemma10 lemma13 ) ⟩ - FiniteSet.Q←F fa (fromℕ< ( NP.<-trans x n<m)) - ≡⟨ cong (λ k → FiniteSet.Q←F fa k) (lemma10 refl ) ⟩ - FiniteSet.Q←F fa (fromℕ< ( toℕ<n (FiniteSet.F←Q fa elm))) - ≡⟨ cong (λ k → FiniteSet.Q←F fa k ) ( fromℕ<-toℕ _ _ ) ⟩ - FiniteSet.Q←F fa (FiniteSet.F←Q fa elm ) - ≡⟨ FiniteSet.finiso→ fa _ ⟩ - elm - ∎ where open ≡-Reasoning - open import Data.List open FiniteSet @@ -473,28 +405,28 @@ ... | false = phase1 finq q qs dup-in-list : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool -dup-in-list {Q} finq q qs = phase1 finq q qs +dup-in-list {Q} finq q qs = phase1 finq q qs -- -- if length of the list is longer than kinds of a finite set, there is a duplicate -- prove this based on the theorem on Data.Fin -- -dup-in-list+fin : { Q : Set } (finq : FiniteSet Q) +dup-in-list+fin : { Q : Set } (finq : FiniteSet Q) → (q : Q) (qs : List Q ) → fin-dup-in-list (F←Q finq q) (map (F←Q finq) qs) ≡ true → dup-in-list finq q qs ≡ true dup-in-list+fin {Q} finq q qs p = i-phase1 qs p where i-phase2 : (qs : List Q) → fin-phase2 (F←Q finq q) (map (F←Q finq) qs) ≡ true - → phase2 finq q qs ≡ true + → phase2 finq q qs ≡ true i-phase2 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) ... | true | t = refl ... | false | tri< a ¬b ¬c = i-phase2 qs p ... | false | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq (subst₂ (λ j k → equal? finq j k ≡ true) (finiso→ finq q) (subst (λ k → Q←F finq k ≡ x) (sym b) (finiso→ finq x)) ( equal?-refl finq ))) ... | false | tri> ¬a ¬b c = i-phase2 qs p - i-phase1 : (qs : List Q) → fin-phase1 (F←Q finq q) (map (F←Q finq) qs) ≡ true - → phase1 finq q qs ≡ true + i-phase1 : (qs : List Q) → fin-phase1 (F←Q finq q) (map (F←Q finq) qs) ≡ true + → phase1 finq q qs ≡ true i-phase1 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) ... | true | tri< a ¬b ¬c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) ( equal→refl finq eq )) a ) ... | true | tri≈ ¬a b ¬c = i-phase2 qs p @@ -523,13 +455,13 @@ open import bijection using ( InjectiveF ; Is ) --- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) +-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -inject-fin : {A B : Set} (fa : FiniteSet A ) - → (fi : InjectiveF B A) +inject-fin : {A B : Set} (fa : FiniteSet A ) + → (fi : InjectiveF B A) → (is-B : (a : A ) → Dec (Is B A (InjectiveF.f fi) a) ) → FiniteSet B -inject-fin {A} {B} fa fi is-B with finite fa in eq1 +inject-fin {A} {B} fa fi is-B with finite fa in eq1 ... | zero = record { finite = 0 ; Q←F = λ () @@ -551,7 +483,7 @@ pfa<fa : pfa < finite fa pfa<fa = subst (λ k → pfa < k ) (sym eq1) a<sa 0<fa : 0 < finite fa - 0<fa = <-transˡ (s≤s z≤n) pfa<fa + 0<fa = <-transˡ (s≤s z≤n) pfa<fa count-B : ℕ → ℕ count-B zero with is-B (Q←F fa ( fromℕ< {0} 0<fa )) @@ -582,46 +514,93 @@ lem00 : (i j : ℕ) → i < j → count-B i ≤ count-B j lem00 i (suc j) (s≤s i<j) = ≤-trans (count-B-mono i<j) (lem01 j) where lem01 : (j : ℕ) → count-B j ≤ count-B (suc j) - lem01 zero with <-cmp (finite fa) 1 + lem01 zero with <-cmp (finite fa) 1 lem01 zero | tri< a ¬b ¬c = ≤-refl lem01 zero | tri≈ ¬a b ¬c = ≤-refl - lem01 zero | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa ( fromℕ< {0} 0<fa )) + lem01 zero | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb1 | yes isb0 = s≤s z≤n ... | yes isb1 | no nisb0 = z≤n ... | no nisb1 | yes isb0 = refl-≤≡ (sym lem14 ) where - lem14 : count-B 0 ≡ 1 + lem14 : count-B 0 ≡ 1 -- in-equality does not work we have to repeat the proof lem14 with is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb = refl ... | no ne = ⊥-elim (ne isb0) ... | no nisb1 | no nisb0 = z≤n - lem01 (suc i) with <-cmp (finite fa) (suc i) | <-cmp (finite fa) (suc (suc i)) + lem01 (suc i) with <-cmp (finite fa) (suc i) | <-cmp (finite fa) (suc (suc i)) ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ count-B i - lem14 with <-cmp (finite fa) (suc i) + lem14 with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a ) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a ) ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (nat-≡< b (<-trans a a<sa)) ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = ⊥-elim (nat-<> a (<-trans a<sa c) ) - ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = refl-≤≡ (sym ?) + ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = refl-≤≡ (sym lem14 ) where + lem14 : count-B (suc i) ≡ count-B i + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a ¬b ¬c = refl + ... | tri≈ ¬a b ¬c = refl + ... | tri> ¬a ¬b c = ⊥-elim ( ¬c c ) ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ = ⊥-elim (nat-≡< (sym b) (subst (λ k → _ < k ) (sym b₁) a<sa) ) ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim (nat-≡< (sym b) (<-trans a<sa c)) ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = ⊥-elim (nat-≤> a (<-transʳ c a<sa ) ) - ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c with is-B (Q←F fa (fromℕ< c)) - ... | yes isb = refl-≤≡ (sym ?) - ... | no nisb = refl-≤≡ (sym ?) - lem01 (suc i) | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ - with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa (fromℕ< c₁)) - ... | yes isb0 | yes isb1 = ≤-trans (refl-≤≡ (sym ?)) a≤sa - ... | yes isb0 | no nisb1 = refl-≤≡ (sym ?) - ... | no nisb0 | yes isb1 = ≤-trans (refl-≤≡ (sym ?)) a≤sa - ... | no nisb0 | no nisb1 = refl-≤≡ (sym ?) + ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c with is-B (Q←F fa (fromℕ< c)) + ... | yes isb = refl-≤≡ (sym lem14) where + lem14 : count-B (suc i) ≡ suc (count-B i) + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | yes isb = refl + ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | no nisb = refl-≤≡ (sym lem14) where + lem14 : count-B (suc i) ≡ count-B i + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | yes isb = ⊥-elim (nisb record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | no ne = refl + lem01 (suc i) | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ + with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa (fromℕ< c₁)) + ... | yes isb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where + lem14 : count-B (suc i) ≡ suc (count-B i) + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | yes isb = refl + ... | yes isb0 | no nisb1 = refl-≤≡ (sym lem14) where + lem14 : count-B (suc i) ≡ suc (count-B i) + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | yes isb = refl + ... | no nisb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where + lem14 : count-B (suc i) ≡ count-B i + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | no ne = refl + ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | no nisb0 | no nisb1 = refl-≤≡ (sym lem14) where + lem14 : count-B (suc i) ≡ count-B i + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | no ne = refl + ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) lem31 : (b : B) → 0 < count-B (toℕ (F←Q fa (f b))) lem31 b = lem32 (toℕ (F←Q fa (f b))) refl where - lem32 : (i : ℕ) → toℕ (F←Q fa (f b)) ≡ i → 0 < count-B i + lem32 : (i : ℕ) → toℕ (F←Q fa (f b)) ≡ i → 0 < count-B i lem32 zero eq with is-B (Q←F fa ( fromℕ< {0} 0<fa )) - ... | yes isb = s≤s z≤n + ... | yes isb = s≤s z≤n ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where lem33 : f b ≡ Q←F fa ( fromℕ< {0} 0<fa ) lem33 = begin @@ -630,11 +609,11 @@ Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) 0<fa) ⟩ Q←F fa ( fromℕ< {0} 0<fa ) ∎ where open ≡-Reasoning - lem32 (suc i) eq with <-cmp (finite fa) (suc i) + lem32 (suc i) eq with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (<-trans (fin<n _) a) ) ... | tri≈ ¬a eq1 ¬c = ⊥-elim ( nat-≡< eq (subst (λ k → toℕ (F←Q fa (f b)) < k ) eq1 (fin<n _))) - ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) - ... | yes isb = s≤s z≤n + ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) + ... | yes isb = s≤s z≤n ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where lem33 : f b ≡ Q←F fa ( fromℕ< c) lem33 = begin @@ -643,41 +622,45 @@ Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) c ) ⟩ Q←F fa ( fromℕ< c ) ∎ where open ≡-Reasoning - + cb<mb : (b : B) → pred (count-B (toℕ (F←Q fa (f b)))) < maxb cb<mb b = sx≤y→x<y ( begin - suc ( pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ - count-B (toℕ (F←Q fa (f b))) ≤⟨ lem02 ⟩ + suc ( pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ + count-B (toℕ (F←Q fa (f b))) ≤⟨ lem02 ⟩ count-B (finite fa) ∎ ) where open ≤-Reasoning lem02 : count-B (toℕ (F←Q fa (f b))) ≤ count-B (finite fa) - lem02 = count-B-mono (<to≤ (fin<n {_} (F←Q fa (f b)))) + lem02 = count-B-mono (<to≤ (fin<n {_} (F←Q fa (f b)))) cb00 : (n : ℕ) → n < count-B (finite fa) → CountB n cb00 n n<cb = lem09 (finite fa) (count-B (finite fa)) (<-transˡ a<sa n<cb) refl where - lem06 : (i j : ℕ) → (i<fa : i < finite fa) (j<fa : j < finite fa) + lem06 : (i j : ℕ) → (i<fa : i < finite fa) (j<fa : j < finite fa) → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B i ≡ count-B j → i ≡ j lem06 i j i<fa j<fa bi bj eq = lem08 where - lem20 : (i j : ℕ) → i < j → (i<fa : i < finite fa) (j<fa : j < finite fa) + lem20 : (i j : ℕ) → i < j → (i<fa : i < finite fa) (j<fa : j < finite fa) → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B i < count-B j - lem20 zero (suc j) i<j i<fa j<fa bi bj with <-cmp (finite fa) (suc j) + lem20 zero (suc j) i<j i<fa j<fa bi bj with <-cmp (finite fa) (suc j) ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) - ... | tri> ¬a ¬b c with is-B (Q←F fa ( fromℕ< 0<fa )) | is-B (Q←F fa (fromℕ< c)) + ... | tri> ¬a ¬b c with is-B (Q←F fa ( fromℕ< 0<fa )) | is-B (Q←F fa (fromℕ< c)) ... | no nisc | _ = ⊥-elim (nisc record { a = Is.a bi ; fa=c = lem26 } ) where lem26 : f (Is.a bi) ≡ Q←F fa (fromℕ< 0<fa) lem26 = trans (Is.fa=c bi) (cong (Q←F fa) (fromℕ<-cong _ _ refl i<fa 0<fa) ) ... | yes _ | no nisc = ⊥-elim (nisc record { a = Is.a bj ; fa=c = lem26 } ) where lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) - ... | yes _ | yes _ = lem25 where + ... | yes isb1 | yes _ = lem25 where + lem14 : count-B 0 ≡ 1 + lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) + ... | no ne = ⊥-elim (ne record {a = Is.a isb1 ; fa=c = trans (Is.fa=c isb1) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | yes isb = refl lem25 : 2 ≤ suc (count-B j) lem25 = begin - 2 ≡⟨ cong suc (sym ?) ⟩ + 2 ≡⟨ cong suc (sym lem14) ⟩ suc (count-B 0) ≤⟨ s≤s (count-B-mono {0} {j} z≤n) ⟩ suc (count-B j) ∎ where open ≤-Reasoning - lem20 (suc i) zero () i<fa j<fa bi bj + lem20 (suc i) zero () i<fa j<fa bi bj lem20 (suc i) (suc j) (s≤s i<j) i<fa j<fa bi bj = lem21 where -- -- i < suc i ≤ j @@ -687,7 +670,7 @@ lem23 with <-cmp (finite fa) (suc j) ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) - ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) + ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | yes _ = refl ... | no nisa = ⊥-elim ( nisa record { a = Is.a bj ; fa=c = lem26 } ) where lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) @@ -706,11 +689,15 @@ ... | tri> ¬a ¬b c₁ = ⊥-elim (nat-≡< (sym eq) ( lem20 j i c₁ j<fa i<fa bj bi )) lem09 : (i j : ℕ) → suc n ≤ j → j ≡ count-B i → CountB n - lem09 0 (suc j) (s≤s le) eq with is-B (Q←F fa (fromℕ< {0} 0<fa )) + lem09 0 (suc j) (s≤s le) eq with is-B (Q←F fa (fromℕ< {0} 0<fa )) ... | no nisb = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) ... | yes isb with ≤-∨ (s≤s le) - ... | case1 eq2 = record { b = Is.a isb ; cb = 0 ; b=cn = lem10 ; cb=n = trans ? (sym (trans eq2 eq)) + ... | case1 eq2 = record { b = Is.a isb ; cb = 0 ; b=cn = lem10 ; cb=n = trans lem14 (sym (trans eq2 eq)) ; cb-inject = λ cb1 c1<fa b1 eq → lem06 0 cb1 0<fa c1<fa isb b1 eq } where + lem14 : count-B 0 ≡ 1 + lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) + ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) + ... | yes isb = refl lem10 : 0 ≡ toℕ (F←Q fa (f (Is.a isb))) lem10 = begin 0 ≡⟨ sym ( toℕ-fromℕ< 0<fa ) ⟩ @@ -718,14 +705,21 @@ toℕ (F←Q fa (Q←F fa (fromℕ< {0} 0<fa ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning ... | case2 (s≤s lt) = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-transʳ z≤n lt) )) - lem09 (suc i) (suc j) (s≤s le) eq with <-cmp (finite fa) (suc i) + lem09 (suc i) (suc j) (s≤s le) eq with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = lem09 i (suc j) (s≤s le) eq ... | tri≈ ¬a b ¬c = lem09 i (suc j) (s≤s le) eq ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | no nisb = lem09 i (suc j) (s≤s le) eq ... | yes isb with ≤-∨ (s≤s le) - ... | case1 eq2 = record { b = Is.a isb ; cb = suc i ; b=cn = lem11 ; cb=n = trans ? (sym (trans eq2 eq )) + ... | case1 eq2 = record { b = Is.a isb ; cb = suc i ; b=cn = lem11 ; cb=n = trans lem14 (sym (trans eq2 eq )) ; cb-inject = λ cb1 c1<fa b1 eq → lem06 (suc i) cb1 c c1<fa isb b1 eq } where + lem14 : count-B (suc i) ≡ suc (count-B i) + lem14 with <-cmp (finite fa) (suc i) + ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) + ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) + ... | yes isb = refl + ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) lem11 : suc i ≡ toℕ (F←Q fa (f (Is.a isb))) lem11 = begin suc i ≡⟨ sym ( toℕ-fromℕ< c) ⟩ @@ -734,14 +728,14 @@ toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning ... | case2 (s≤s lt) = lem09 i j lt (cong pred eq) - iso0 : (x : Fin maxb) → fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡ x + iso0 : (x : Fin maxb) → fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡ x iso0 x = begin - fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡⟨ fromℕ<-cong _ _ ( begin + fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡⟨ fromℕ<-cong _ _ ( begin pred (count-B (toℕ (F←Q fa (f (CountB.b (cb00 (toℕ x) (fin<n _))))))) ≡⟨ sym (cong (λ k → pred (count-B k)) (CountB.b=cn CB)) ⟩ pred (count-B (CountB.cb CB)) ≡⟨ cong pred (CountB.cb=n CB) ⟩ pred (suc (toℕ x)) ≡⟨ refl ⟩ - toℕ x ∎ ) (cb<mb (CountB.b CB)) (fin<n _) ⟩ - fromℕ< (fin<n {_} x) ≡⟨ fromℕ<-toℕ _ (fin<n {_} x) ⟩ + toℕ x ∎ ) (cb<mb (CountB.b CB)) (fin<n _) ⟩ + fromℕ< (fin<n {_} x) ≡⟨ fromℕ<-toℕ _ (fin<n {_} x) ⟩ x ∎ where open ≡-Reasoning CB = cb00 (toℕ x) (fin<n _) @@ -757,17 +751,17 @@ CB = cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _) isb : Is B A f (Q←F fa (fromℕ< (fin<n {_} (F←Q fa (f b)) ))) isb = record { a = b ; fa=c = lem33 } where - lem33 : f b ≡ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) + lem33 : f b ≡ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) lem33 = begin f b ≡⟨ sym (finiso→ fa _) ⟩ Q←F fa (F←Q fa (f b)) ≡⟨ cong (Q←F fa) (sym (fromℕ<-toℕ _ (fin<n (F←Q fa (f b))))) ⟩ - Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) ∎ + Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) ∎ lem30 : count-B (CountB.cb CB) ≡ count-B (toℕ (F←Q fa (InjectiveF.f fi b))) lem30 = begin count-B (CountB.cb CB) ≡⟨ CountB.cb=n CB ⟩ suc (toℕ (fromℕ< (cb<mb b))) ≡⟨ cong suc (toℕ-fromℕ< (cb<mb b)) ⟩ suc (pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ - count-B (toℕ (F←Q fa (f b))) ∎ + count-B (toℕ (F←Q fa (f b))) ∎ -- end