{-# OPTIONS --cubical-compatible --safe #-} module non-regular where open import Data.Nat open import Data.Empty open import Data.List open import Data.Maybe hiding ( map ) open import Relation.Binary.PropositionalEquality hiding ( [_] ) open import logic open import automaton open import automaton-ex open import finiteSetUtil open import finiteSet open import Relation.Nullary open import regular-language open import nat open import pumping open FiniteSet list-eq : List In2 → List In2 → Bool list-eq [] [] = true list-eq [] (x ∷ s1) = false list-eq (x ∷ s) [] = false list-eq (i0 ∷ s) (i0 ∷ s1) = false list-eq (i0 ∷ s) (i1 ∷ s1) = false list-eq (i1 ∷ s) (i0 ∷ s1) = false list-eq (i1 ∷ s) (i1 ∷ s1) = list-eq s s1 input-addi0 : ( n : ℕ ) → List In2 → List In2 input-addi0 zero x = x input-addi0 (suc i) x = i0 ∷ input-addi0 i x input-addi1 : ( n : ℕ ) → List In2 input-addi1 zero = [] input-addi1 (suc i) = i1 ∷ input-addi1 i inputnn0 : ( n : ℕ ) → List In2 inputnn0 n = input-addi0 n (input-addi1 n) inputnn1-i1 : (i : ℕ) → List In2 → Bool inputnn1-i1 zero [] = true inputnn1-i1 (suc _) [] = false inputnn1-i1 zero (i1 ∷ x) = false inputnn1-i1 (suc i) (i1 ∷ x) = inputnn1-i1 i x inputnn1-i1 zero (i0 ∷ x) = false inputnn1-i1 (suc _) (i0 ∷ x) = false inputnn1-i0 : (i : ℕ) → List In2 → ℕ ∧ List In2 inputnn1-i0 i [] = ⟪ i , [] ⟫ inputnn1-i0 i (i1 ∷ x) = ⟪ i , (i1 ∷ x) ⟫ inputnn1-i0 i (i0 ∷ x) = inputnn1-i0 (suc i) x open _∧_ inputnn1 : List In2 → Bool inputnn1 x = inputnn1-i1 (proj1 (inputnn1-i0 0 x)) (proj2 (inputnn1-i0 0 x)) t1 = inputnn1 ( i0 ∷ i1 ∷ [] ) t2 = inputnn1 ( i0 ∷ i0 ∷ i1 ∷ i1 ∷ [] ) t3 = inputnn1 ( i0 ∷ i0 ∷ i0 ∷ i1 ∷ i1 ∷ [] ) t4 : inputnn1 ( inputnn0 5 ) ≡ true t4 = refl t5 : ( n : ℕ ) → Set t5 n = inputnn1 ( inputnn0 n ) ≡ true import Level cons-inject : {n : Level.Level } (A : Set n) { a b : A } {x1 x2 : List A} → a ∷ x1 ≡ b ∷ x2 → x1 ≡ x2 cons-inject _ refl = refl append-[] : {A : Set} {x1 : List A } → x1 ++ [] ≡ x1 append-[] {A} {[]} = refl append-[] {A} {x ∷ x1} = cong (λ k → x ∷ k) (append-[] {A} {x1} ) open import Data.Nat.Properties open import Relation.Binary.Definitions open import Relation.Binary.PropositionalEquality nn30 : (y : List In2) → (j : ℕ) → proj2 (inputnn1-i0 (suc j) y) ≡ proj2 (inputnn1-i0 j y ) nn30 [] _ = refl nn30 (i0 ∷ y) j = nn30 y (suc j) nn30 (i1 ∷ y) _ = refl nn31 : (y : List In2) → (j : ℕ) → proj1 (inputnn1-i0 (suc j) y) ≡ suc (proj1 (inputnn1-i0 j y )) nn31 [] _ = refl nn31 (i0 ∷ y) j = nn31 y (suc j) nn31 (i1 ∷ y) _ = refl nn01 : (i : ℕ) → inputnn1 ( inputnn0 i ) ≡ true nn01 i = subst₂ (λ j k → inputnn1-i1 j k ≡ true) (sym (nn07 i 0 refl)) (sym (nn09 i)) (nn04 i) where nn07 : (j x : ℕ) → x + j ≡ i → proj1 ( inputnn1-i0 x (input-addi0 j (input-addi1 i))) ≡ x + j nn07 zero x eq with input-addi1 i in eq1 ... | [] = +-comm 0 _ ... | i0 ∷ t = ⊥-elim ( nn08 i eq1 ) where nn08 : (i : ℕ) → ¬ (input-addi1 i ≡ i0 ∷ t ) nn08 zero () nn08 (suc i) () ... | i1 ∷ t = +-comm 0 _ nn07 (suc j) x eq = trans (nn07 j (suc x) (trans (cong (λ k → k + j) (+-comm 1 _ )) (trans (+-assoc x _ _) eq)) ) (trans (+-assoc 1 x _) (trans (cong (λ k → k + j) (+-comm 1 _) ) (+-assoc x 1 _) )) nn09 : (x : ℕ) → proj2 ( inputnn1-i0 0 (input-addi0 x (input-addi1 i))) ≡ input-addi1 i nn09 zero with input-addi1 i in eq1 ... | [] = refl ... | i0 ∷ t = ⊥-elim ( nn08 i eq1 ) where nn08 : (i : ℕ) → ¬ (input-addi1 i ≡ i0 ∷ t ) nn08 zero () nn08 (suc i) () ... | i1 ∷ t = refl nn09 (suc j) = trans (nn30 (input-addi0 j (input-addi1 i)) 0) (nn09 j ) nn04 : (i : ℕ) → inputnn1-i1 i (input-addi1 i) ≡ true nn04 zero = refl nn04 (suc i) = nn04 i half : (x : List In2) → ℕ half [] = 0 half (x ∷ []) = 0 half (x ∷ x₁ ∷ x₂) = suc (half x₂) top-is-i0 : (x : List In2) → Bool top-is-i0 [] = true top-is-i0 (i0 ∷ _) = true top-is-i0 (i1 ∷ _) = false -- if this is easy, we may have an easy proof -- nn02 : (x : List In2) → inputnn1 x ≡ true → x ≡ inputnn0 (half x) -- -- if there is an automaton with n states , which accespt inputnn1, it has a trasition function. -- The function is determinted by inputs, -- open RegularLanguage open Automaton open _∧_ open RegularLanguage open import Data.Nat.Properties open import Data.List.Properties open import nat lemmaNN : (r : RegularLanguage In2 ) → ¬ ( (s : List In2) → isRegular inputnn1 s r ) lemmaNN r Rg = tann {TA.x TAnn} (TA.non-nil-y TAnn ) (TA.xyz=is TAnn) (tr-accept→ (automaton r) _ (astart r) (TA.trace-xyz TAnn) ) (tr-accept→ (automaton r) _ (astart r) (TA.trace-xyyz TAnn) ) where n : ℕ n = suc (finite (afin r)) nn = inputnn0 n nn03 : accept (automaton r) (astart r) nn ≡ true nn03 = subst (λ k → k ≡ true ) (Rg nn ) (nn01 n) nn09 : (n m : ℕ) → n ≤ n + m nn09 zero m = z≤n nn09 (suc n) m = s≤s (nn09 n m) nn04 : Trace (automaton r) nn (astart r) nn04 = tr-accept← (automaton r) nn (astart r) nn03 nntrace = tr→qs (automaton r) nn (astart r) nn04 nn07 : (n : ℕ) → length (inputnn0 n ) ≡ n + n nn07 i = nn19 i where nn17 : (i : ℕ) → length (input-addi1 i) ≡ i nn17 zero = refl nn17 (suc i)= cong suc (nn17 i) nn18 : (i j : ℕ) → length (input-addi0 j (input-addi1 i)) ≡ j + length (input-addi1 i ) nn18 i zero = refl nn18 i (suc j)= cong suc (nn18 i j) nn19 : (i : ℕ) → length (input-addi0 i ( input-addi1 i )) ≡ i + i nn19 i = begin length (input-addi0 i ( input-addi1 i )) ≡⟨ nn18 i i ⟩ i + length (input-addi1 i) ≡⟨ cong (λ k → i + k) ( nn17 i) ⟩ i + i ∎ where open ≡-Reasoning nn05 : length nntrace > finite (afin r) nn05 = begin suc (finite (afin r)) ≤⟨ nn09 _ _ ⟩ n + n ≡⟨ sym (nn07 n) ⟩ length (inputnn0 n ) ≡⟨ tr→qs=is (automaton r) (inputnn0 n ) (astart r) nn04 ⟩ length nntrace ∎ where open ≤-Reasoning nn06 : Dup-in-list ( afin r) (tr→qs (automaton r) nn (astart r) nn04) nn06 = dup-in-list>n (afin r) nntrace nn05 TAnn : TA (automaton r) (astart r) nn TAnn = pumping-lemma (automaton r) (afin r) (astart r) (Dup-in-list.dup nn06) nn nn04 (Dup-in-list.is-dup nn06) open import Tactic.MonoidSolver using (solve; solve-macro) -- there is a counter example -- tann : {x y z : List In2} → ¬ y ≡ [] → x ++ y ++ z ≡ nn → accept (automaton r) (astart r) (x ++ y ++ z) ≡ true → ¬ (accept (automaton r) (astart r) (x ++ y ++ y ++ z) ≡ true ) tann {x} {y} {z} ny eq axyz axyyz = ¬-bool (nn10 x y z ny (trans (Rg (x ++ y ++ z)) axyz ) ) (trans (Rg (x ++ y ++ y ++ z)) axyyz ) where count0 : (x : List In2) → ℕ count0 [] = 0 count0 (i0 ∷ x) = suc (count0 x) count0 (i1 ∷ x) = count0 x count1 : (x : List In2) → ℕ count1 [] = 0 count1 (i1 ∷ x) = suc (count1 x) count1 (i0 ∷ x) = count1 x -- -- prove some obvious fact -- c0+1=n : (x : List In2 ) → count0 x + count1 x ≡ length x c0+1=n [] = refl c0+1=n (i0 ∷ t) = cong suc ( c0+1=n t ) c0+1=n (i1 ∷ t) = begin count0 t + suc (count1 t) ≡⟨ sym (+-assoc (count0 t) _ _) ⟩ (count0 t + 1 ) + count1 t ≡⟨ cong (λ k → k + count1 t) (+-comm _ 1 ) ⟩ suc (count0 t + count1 t) ≡⟨ cong suc ( c0+1=n t ) ⟩ suc (length t) ∎ where open ≡-Reasoning -- nn15 : (x : List In2 ) → inputnn1 x ≡ true → count0 x ≡ count1 x nn15 x eq = nn18 where nn17 : (x : List In2 ) → (count0 x ≡ proj1 (inputnn1-i0 0 x) + count0 (proj2 (inputnn1-i0 0 x))) ∧ (count1 x ≡ 0 + count1 (proj2 (inputnn1-i0 0 x))) nn17 [] = ⟪ refl , refl ⟫ nn17 (i0 ∷ t ) with nn17 t ... | ⟪ eq1 , eq2 ⟫ = ⟪ begin suc (count0 t ) ≡⟨ cong suc eq1 ⟩ suc (proj1 (inputnn1-i0 0 t) + count0 (proj2 (inputnn1-i0 0 t))) ≡⟨ cong₂ _+_ (sym (nn31 t 0)) (cong count0 (sym (nn30 t 0))) ⟩ proj1 (inputnn1-i0 1 t) + count0 (proj2 (inputnn1-i0 1 t)) ∎ , trans eq2 (cong count1 (sym (nn30 t 0))) ⟫ where open ≡-Reasoning nn20 : proj2 (inputnn1-i0 1 t) ≡ proj2 (inputnn1-i0 0 t) nn20 = nn30 t 0 nn17 (i1 ∷ x₁) = ⟪ refl , refl ⟫ nn19 : (n : ℕ) → (y : List In2 ) → inputnn1-i1 n y ≡ true → (count0 y ≡ 0) ∧ (count1 y ≡ n) nn19 zero [] eq = ⟪ refl , refl ⟫ nn19 zero (i0 ∷ y) () nn19 zero (i1 ∷ y) () nn19 (suc i) (i1 ∷ y) eq with nn19 i y eq ... | t = ⟪ proj1 t , cong suc (proj2 t ) ⟫ nn18 : count0 x ≡ count1 x nn18 = begin count0 x ≡⟨ proj1 (nn17 x) ⟩ proj1 (inputnn1-i0 0 x) + count0 (proj2 (inputnn1-i0 0 x)) ≡⟨ cong (λ k → proj1 (inputnn1-i0 0 x) + k) (proj1 (nn19 (proj1 (inputnn1-i0 0 x)) (proj2 (inputnn1-i0 0 x)) eq)) ⟩ proj1 (inputnn1-i0 0 x) + 0 ≡⟨ +-comm _ 0 ⟩ 0 + proj1 (inputnn1-i0 0 x) ≡⟨ cong (λ k → 0 + k) (sym (proj2 (nn19 _ _ eq))) ⟩ 0 + count1 (proj2 (inputnn1-i0 0 x)) ≡⟨ sym (proj2 (nn17 x)) ⟩ count1 x ∎ where open ≡-Reasoning distr0 : (x y : List In2 ) → count0 (x ++ y ) ≡ count0 x + count0 y distr0 [] y = refl distr0 (i0 ∷ x) y = cong suc (distr0 x y) distr0 (i1 ∷ x) y = distr0 x y distr1 : (x y : List In2 ) → count1 (x ++ y ) ≡ count1 x + count1 y distr1 [] y = refl distr1 (i1 ∷ x) y = cong suc (distr1 x y) distr1 (i0 ∷ x) y = distr1 x y -- -- i0 .. i0 ∷ i1 .. i1 sequece does not contains i1 → i0 transition -- record i1i0 (z : List In2) : Set where field a b : List In2 i10 : z ≡ a ++ (i1 ∷ i0 ∷ b ) nn12 : (x : List In2 ) → inputnn1 x ≡ true → ¬ i1i0 x nn12 x eq = nn17 x 0 eq where nn17 : (x : List In2 ) → (i : ℕ) → inputnn1-i1 (proj1 (inputnn1-i0 i x)) (proj2 (inputnn1-i0 i x)) ≡ true → ¬ i1i0 x nn17 [] i eq li with i1i0.a li | i1i0.i10 li ... | [] | () ... | x ∷ s | () nn17 (i0 ∷ x₁) i eq li = nn17 x₁ (suc i) eq record { a = nn18 (i1i0.a li) (i1i0.i10 li) ; b = i1i0.b li ; i10 = nn19 (i1i0.a li) (i1i0.i10 li) } where -- first half nn18 : (a : List In2 ) → i0 ∷ x₁ ≡ a ++ ( i1 ∷ i0 ∷ i1i0.b li) → List In2 nn18 (i0 ∷ t) eq = t nn19 : (a : List In2 ) → (eq : i0 ∷ x₁ ≡ a ++ ( i1 ∷ i0 ∷ i1i0.b li) ) → x₁ ≡ nn18 a eq ++ i1 ∷ i0 ∷ i1i0.b li nn19 (i0 ∷ a) eq = cons-inject In2 eq nn17 (i1 ∷ x₁) i eq li = nn20 (i1 ∷ x₁) i eq li where -- second half nn20 : (x : List In2) → (i : ℕ) → inputnn1-i1 i x ≡ true → i1i0 x → ⊥ nn20 x i eq li = nn21 (i1i0.a li) x i eq (i1i0.i10 li) where nn21 : (a x : List In2) → (i : ℕ) → inputnn1-i1 i x ≡ true → x ≡ a ++ i1 ∷ i0 ∷ i1i0.b li → ⊥ nn21 [] [] zero eq1 () nn21 (i0 ∷ a) [] zero eq1 () nn21 (i1 ∷ a) [] zero eq1 () nn21 a (i0 ∷ x₁) zero () eq0 nn21 [] (i0 ∷ x₁) (suc i) () eq0 nn21 (x ∷ a) (i0 ∷ x₁) (suc i) () eq0 nn21 [] (i1 ∷ i0 ∷ x₁) (suc zero) () eq0 nn21 [] (i1 ∷ i0 ∷ x₁) (suc (suc i)) () eq0 nn21 (i1 ∷ a) (i1 ∷ x₁) (suc i) eq1 eq0 = nn21 a x₁ i eq1 (cons-inject In2 eq0) nn11 : (x y z : List In2 ) → ¬ y ≡ [] → inputnn1 (x ++ y ++ z) ≡ true → ¬ ( inputnn1 (x ++ y ++ y ++ z) ≡ true ) nn11 x y z ny xyz xyyz = ⊥-elim ( nn12 (x ++ y ++ y ++ z ) xyyz record { a = x ++ i1i0.a (bb23 bb22 ) ; b = i1i0.b (bb23 bb22) ++ z ; i10 = bb24 } ) where -- -- we need simple calcuraion to obtain count0 y ≡ count1 y -- nn21 : count0 x + count0 y + count0 y + count0 z ≡ count1 x + count1 y + count1 y + count1 z nn21 = begin (count0 x + count0 y + count0 y) + count0 z ≡⟨ solve +-0-monoid ⟩ count0 x + (count0 y + (count0 y + count0 z)) ≡⟨ sym (cong (λ k → count0 x + (count0 y + k)) (distr0 y _ )) ⟩ count0 x + (count0 y + count0 (y ++ z)) ≡⟨ sym (cong (λ k → count0 x + k) (distr0 y _ )) ⟩ count0 x + (count0 (y ++ y ++ z)) ≡⟨ sym (distr0 x _ ) ⟩ count0 (x ++ y ++ y ++ z) ≡⟨ nn15 (x ++ y ++ y ++ z) xyyz ⟩ count1 (x ++ y ++ y ++ z) ≡⟨ distr1 x _ ⟩ count1 x + (count1 (y ++ y ++ z)) ≡⟨ cong (λ k → count1 x + k) (distr1 y _ ) ⟩ count1 x + (count1 y + count1 (y ++ z)) ≡⟨ (cong (λ k → count1 x + (count1 y + k)) (distr1 y _ )) ⟩ count1 x + (count1 y + (count1 y + count1 z)) ≡⟨ solve +-0-monoid ⟩ count1 x + count1 y + count1 y + count1 z ∎ where open ≡-Reasoning nn20 : count0 x + count0 y + count0 z ≡ count1 x + count1 y + count1 z nn20 = begin count0 x + count0 y + count0 z ≡⟨ solve +-0-monoid ⟩ count0 x + (count0 y + count0 z) ≡⟨ cong (λ k → count0 x + k) (sym (distr0 y z)) ⟩ count0 x + (count0 (y ++ z)) ≡⟨ sym (distr0 x _) ⟩ count0 (x ++ (y ++ z)) ≡⟨ nn15 (x ++ y ++ z) xyz ⟩ count1 (x ++ (y ++ z)) ≡⟨ distr1 x _ ⟩ count1 x + count1 (y ++ z) ≡⟨ cong (λ k → count1 x + k) (distr1 y z) ⟩ count1 x + (count1 y + count1 z) ≡⟨ solve +-0-monoid ⟩ count1 x + count1 y + count1 z ∎ where open ≡-Reasoning -- this takes very long time to check and needs 10GB bb22 : count0 y ≡ count1 y bb22 = begin count0 y ≡⟨ ? ⟩ -- count0 y ≡⟨ sym ( +-cancelʳ-≡ (count1 z + count0 x + count0 y + count0 z) (count1 y) (count0 y) (+-cancelˡ-≡ _ (count1 x + count1 y) ( -- begin -- count1 x + count1 y + (count1 y + (count1 z + count0 x + count0 y + count0 z)) ≡⟨ solve +-0-monoid ⟩ -- (count1 x + count1 y + count1 y + count1 z) + (count0 x + count0 y + count0 z) ≡⟨ sym (cong₂ _+_ nn21 (sym nn20)) ⟩ -- (count0 x + count0 y + count0 y + count0 z) + (count1 x + count1 y + count1 z) ≡⟨ +-comm _ (count1 x + count1 y + count1 z) ⟩ -- (count1 x + count1 y + count1 z) + (count0 x + count0 y + count0 y + count0 z) ≡⟨ solve +-0-monoid ⟩ -- count1 x + count1 y + (count1 z + (count0 x + count0 y)) + count0 y + count0 z -- ≡⟨ cong (λ k → count1 x + count1 y + (count1 z + k) + count0 y + count0 z) (+-comm (count0 x) _) ⟩ -- count1 x + count1 y + (count1 z + (count0 y + count0 x)) + count0 y + count0 z ≡⟨ solve +-0-monoid ⟩ -- count1 x + count1 y + ((count1 z + count0 y) + count0 x) + count0 y + count0 z -- ≡⟨ cong (λ k → count1 x + count1 y + (k + count0 x) + count0 y + count0 z ) (+-comm (count1 z) _) ⟩ -- count1 x + count1 y + (count0 y + count1 z + count0 x) + count0 y + count0 z ≡⟨ solve +-0-monoid ⟩ -- count1 x + count1 y + (count0 y + (count1 z + count0 x + count0 y + count0 z)) ∎ ))) ⟩ count1 y ∎ where open ≡-Reasoning -- -- y contains i0 and i1 , so we have i1 → i0 transition in y ++ y -- bb23 : count0 y ≡ count1 y → i1i0 (y ++ y) bb23 eq = bb25 y y bb26 (subst (λ k → 0 < k ) (sym eq) bb26) where bb26 : 0 < count1 y bb26 with <-cmp 0 (count1 y) ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim (nat-≡< (sym bb27 ) (0