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You are given the following context-free grammar that produces a language *L*:

`S -> | SS | aSaSb | aSbSa | bSaSa`

It is easy to see that this grammar produces exactly the language *M* of all words over *a*, *b* that contains twice as many *a*'s as *b*'s. But is it easy to prove?

We claim that *L* ⊆ *M* is while proving *M* ⊆ *L* is not, so we give you some guidance: Assume *w* ∈ *M*. We prove *w* ∈ *L* by induction on the length of *w*.

For the induction step, consider *w* ∈ *M* with positive length and assume that the claim holds for all *w*′ of shorter length (IH).

Consider the function *h*(*w*):=2 * |*w*|_{b} − |*w*|_{a}. We have *h*(*w*)=0 iff *w* ∈ *M*. If *h* is zero for any prefix of *w*, then it is easy to establish *w* ∈ *L* with (IH). So assume the contrary for the remainder of the proof.

Now consider three cases:

*w*=*b**u*. Thus*h*(*u*)= − 2. As the value of*h*for*w*does not have a zero, starts with*b*, and can only decrease by 1, we have*u*=*v**a**a*for some*v*. This means*h*(*v*)=0. Use the last production to show that the word is in the grammar.*w*=*u**b*. Analogous.*w*=*a**u**a*. As*w*does not have a zero, we know that*h*(*v*) never takes the value 1 for any prefix of*u*. With*h*(*u*)=2, we can conclude that*u*=*v*_{1}*b**v*_{2}for*v*_{1},*v*_{2}with*h*(*v*_{1})=*h*(*v*_{2})=0. Now use the fourth production to show*w*∈*L*.

You are given four tasks:

- Show
*L*⊆*M* - Prove a theorem that helps you to establish the essential step in (a), (b).
- Prove a theorem that helps you to establish the essential step in (c).
- Show
*M*⊆*L*. You may want to exploit the symmetry of (a) and (b).

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### Definitions File

### Template File

### Check File

theory Defs imports Main begin section ‹Grammars and Languages› text ‹‹#w⇘c⇙› denotes the number of times the character ‹c› occurs in word ‹w›› abbreviation count :: "'a list ⇒ 'a ⇒ nat" ("#_⇘_⇙") where "count xs x ≡ count_list xs x" abbreviation "a ≡ CHR ''a''" abbreviation "b ≡ CHR ''b''" definition prefix :: "'a list ⇒ 'a list ⇒ bool" (infix "≺" 50) where "prefix xs ys ≡ ∃ zs. ys = xs @ zs ∧ zs ≠ []" section ‹Problem statement› text ‹Consider the following grammar:› inductive_set L :: "string set" where "[] ∈ L" | "x @ y ∈ L" if "x ∈ L" "y ∈ L" | "a # x @ a # y @ [b] ∈ L" if "x ∈ L" "y ∈ L" | "a # x @ b # y @ [a] ∈ L" if "x ∈ L" "y ∈ L" | "b # x @ a # y @ [a] ∈ L" if "x ∈ L" "y ∈ L" text ‹Q: Can we delete any of these rules?› definition "M = {w. set w ⊆ {a, b} ∧ #w⇘a⇙ = 2 * #w⇘b⇙}" text ‹ We want to prove that the grammar exactly produces the language of all words that contain twice as many a's than b's:› lemma "L = M" oops text ‹The height function› definition "h w = 2 * int (#w⇘b⇙) - int (#w⇘a⇙)" end

theory Submission imports Defs begin text ‹First task› theorem L_sub: "L ⊆ M" sorry text ‹Second task› theorem h_first_zero: assumes "h (xs @ ys) < 0" "h xs ≥ 0" shows "∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" sorry text ‹Third task› theorem h_2_split: assumes "h w = 2" "∀v. v ≺ w ⟶ h v ≠ 1" obtains x y where "h x = 0" "h y = 0" "w = x @ b # y" sorry text ‹Fourth task› theorem L_sup: assumes h_first_zero: "⋀xs ys. h (xs @ ys) < 0 ⟹ 0 ≤ h xs ⟹ ∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" assumes h_2_split: "⋀t w. h w = 2 ⟹ ∀v. v ≺ w ⟶ h v ≠ 1 ⟹ (⋀x y. h x = 0 ⟹ h y = 0 ⟹ w = x @ b # y ⟹ t) ⟹ t" assumes "h w = 0" "set w ⊆ {a, b}" shows "w ∈ L" sorry text ‹The final theorem› corollary "L = M" using L_sub L_sup[OF h_first_zero h_2_split] unfolding h_def M_def by auto end

theory Check imports Submission begin text ‹First task› theorem L_sub: "L ⊆ M" by (rule Submission.L_sub) text ‹Second task› theorem h_first_zero: assumes "h (xs @ ys) < 0" "h xs ≥ 0" shows "∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" using assms by (rule Submission.h_first_zero) text ‹Third task› theorem h_2_split: assumes "h w = 2" "∀v. v ≺ w ⟶ h v ≠ 1" obtains x y where "h x = 0" "h y = 0" "w = x @ b # y" using assms by (rule Submission.h_2_split) text ‹Fourth task› theorem L_sup: assumes h_first_zero: "⋀xs ys. h (xs @ ys) < 0 ⟹ 0 ≤ h xs ⟹ ∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" assumes h_2_split: "⋀t w. h w = 2 ⟹ ∀v. v ≺ w ⟶ h v ≠ 1 ⟹ (⋀x y. h x = 0 ⟹ h y = 0 ⟹ w = x @ b # y ⟹ t) ⟹ t" assumes "h w = 0" "set w ⊆ {a, b}" shows "w ∈ L" using assms by (rule Submission.L_sup) text ‹The final theorem› corollary "L = M" using L_sub L_sup[OF h_first_zero h_2_split] unfolding h_def M_def by auto end

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### Definitions File

### Template File

Require Export Ascii List. Require Export Arith ZArith Lia. Require Import ssreflect. Export ListNotations. Open Scope char_scope. (** In this challenge, we assume classical logic. *) Require Export Classical. (** A simple formalization of (possibly infinite) sets as predicates. *) Notation set A := (A -> Prop) (only parsing). Definition Incl {U} (A B: set U) : Prop := forall x, A x -> B x. Definition Union {U} (A B: set U): set U := fun x => A x \/ B x. Definition Single {U} a : set U := fun x => x = a. Definition Pair {U} a b : set U := fun x => x = a \/ x = b. Definition Empty {U} := fun (x: U) => False. Notation "A ⊆ B" := (Incl A B) (at level 70). Notation "A ∪ B" := (Union A B) (at level 50). Notation "'{ x }" := (Single x) (at level 1). Notation "∅" := Empty. Axiom set_ext : forall U (A B: set U), (forall x, A x <-> B x) -> A = B. (* Simple goals involving sets can be solved by the [firstorder] tactic. This should be enough for our needs here. *) Goal forall U (A B: set U) a b, '{a} ∪ ∅ ∪ B ∪ (A ∪ '{b} ∪ B) ⊆ Pair a b ∪ A ∪ B. Proof. firstorder. (* boom. *) Qed. (** Definitions on words (as lists of characters). *) Notation word := (list ascii). (* [#{c} w] denotes the number of times the character [c] occurs in the word [w]. *) Notation "'#' '{' c '}'" := (fun w => count_occ ascii_dec w c) (at level 1). Eval compute in #{"a"} ["a"; "b"; "c"]. (* there is one "a" in "abc" *) Definition a := "a". Definition b := "b". (* The strict prefix relation *) Definition prefix {A} (xs ys: list A) := exists zs, ys = xs ++ zs /\ zs <> []. Notation "xs ≺ ys" := (prefix xs ys) (at level 50). (* The set of letters in a word *) Definition letters (w: word): set ascii := fun c => List.In c w. Lemma letters_nil : letters [] = ∅. Proof. reflexivity. Qed. Lemma letters_cons x xs : letters (x :: xs) = '{x} ∪ letters xs. Proof. rewrite /letters /Union /Single //=. apply set_ext. firstorder. Qed. Lemma letters_app xs ys : letters (xs ++ ys) = letters xs ∪ letters ys. Proof. rewrite /letters. apply set_ext. intro. rewrite in_app_iff //=. Qed. (* Can be used by ssreflect's rewrite to rewrite with any of the lemmas in the tuple: use "rewrite lettersE" *) Definition lettersE := (letters_nil, letters_cons, letters_app). (** Problem statement *) (* Consider the following grammar, which inductively defines a set of words: *) Inductive L : set word := | L_nil : L [] | L_app : forall x y, L x -> L y -> L (x ++ y) | L_aab : forall x y, L x -> L y -> L (a :: x ++ a :: y ++ [b]) | L_aba : forall x y, L x -> L y -> L (a :: x ++ b :: y ++ [a]) | L_baa : forall x y, L x -> L y -> L (b :: x ++ a :: y ++ [a]). (* We want to prove that the grammar exactly produces the language of all words that contain twice as many a's than b's: *) Definition M : set word := fun w => letters w ⊆ Pair a b /\ #{a} w = 2 * #{b} w. Goal L = M. Abort. (* The height function. w is in M iff h(w) = 0. *) Definition h (w: word): Z := 2 * Z.of_nat (#{b} w) - Z.of_nat (#{a} w).

Require Import Defs. (* It is not required, but it is a good idea to import ssreflect to benefit from the better rewrite tactic. (by uncommenting the two lines below) ssreflect's rewrite cheatsheet: a rewrite invocation is of the form [rewrite foo bar baz], where each of "foo", "bar", "baz" can be: - "foo" where foo is a lemma: rewrites with the lemma ("-foo" rewrites in the other direction) - "!foo" where foo is a lemma: rewrites repeatedly with foo - "/foo" where foo is a definition: unfolds the definition ("-/foo" folds the definition) - "//": try to prove the goal or side-condition if it is trivial - "//=": like "//" but also simplify the goal (using "simp") - "(_: foo = bar)": ask the user to prove "foo = bar" as a subgoal, and rewrite with it *) (* Require Import ssreflect. Local Ltac done ::= first [ solve [ trivial | eauto | lia ] ]. *) (* Prevent simpl/cbn from unfolding the multiplication, which is never a good idea. *) Arguments Nat.mul : simpl never. Arguments Z.mul : simpl never. (** First task *) Theorem L_sub : L ⊆ M. Proof. (* todo *) Admitted. (** Second task *) Theorem h_first_zero xs ys : (h (xs ++ ys) < 0)%Z -> (0 <= h xs)%Z -> exists zs ws, ys = zs ++ ws /\ h (xs ++ zs) = 0%Z. Proof. (* todo *) Admitted. (** Third task *) (* Hint: this is useful to reason by well-founded induction on words. *) Definition word_len_lt := ltof _ (@length ascii). Lemma word_len_wf : well_founded word_len_lt. Proof. apply well_founded_ltof. Qed. Definition smallest_word_st (P: word -> Prop) (w: word) := P w /\ forall v, P v -> length w <= length v. (* If [P] holds on some word [w], then we can consider "the" smallest word that satisfies [P]... *) Lemma ex_smallest_word_st : forall (P: set word) (w: word), P w -> exists v, smallest_word_st P v. Proof. (* todo *) Admitted. Theorem h_2_split w : h w = 2%Z -> (forall v, v ≺ w -> h v <> 1%Z) -> exists x y, h x = 0%Z /\ h y = 0%Z /\ w = x ++ [b] ++ y. Proof. (* todo *) Admitted. (** Fourth task *) Section L_sup. Hypothesis h_first_zero : forall xs ys, (h (xs ++ ys) < 0)%Z -> (0 <= h xs)%Z -> exists zs ws, ys = zs ++ ws /\ h (xs ++ zs) = 0%Z. Hypothesis h_2_split : forall w, h w = 2%Z -> (forall v, v ≺ w -> h v <> 1%Z) -> exists x y, h x = 0%Z /\ h y = 0%Z /\ w = x ++ [b] ++ y. Theorem L_sup w : h w = 0%Z -> letters w ⊆ Pair a b -> L w. Proof. (* todo *) Admitted. End L_sup. (** The final theorem *) Theorem final : L = M. Proof. (* todo *) Admitted.

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### Definitions File

### Template File

### Check File

theory Defs imports Main begin section ‹Grammars and Languages› text ‹‹#w⇘c⇙› denotes the number of times the character ‹c› occurs in word ‹w›› abbreviation count :: "'a list ⇒ 'a ⇒ nat" ("#_⇘_⇙") where "count xs x ≡ count_list xs x" abbreviation "a ≡ CHR ''a''" abbreviation "b ≡ CHR ''b''" definition prefix :: "'a list ⇒ 'a list ⇒ bool" (infix "≺" 50) where "prefix xs ys ≡ ∃ zs. ys = xs @ zs ∧ zs ≠ []" section ‹Problem statement› text ‹Consider the following grammar:› inductive_set L :: "string set" where "[] ∈ L" | "x @ y ∈ L" if "x ∈ L" "y ∈ L" | "a # x @ a # y @ [b] ∈ L" if "x ∈ L" "y ∈ L" | "a # x @ b # y @ [a] ∈ L" if "x ∈ L" "y ∈ L" | "b # x @ a # y @ [a] ∈ L" if "x ∈ L" "y ∈ L" text ‹Q: Can we delete any of these rules?› definition "M = {w. set w ⊆ {a, b} ∧ #w⇘a⇙ = 2 * #w⇘b⇙}" text ‹ We want to prove that the grammar exactly produces the language of all words that contain twice as many a's than b's:› lemma "L = M" oops text ‹The height function› definition "h w = 2 * int (#w⇘b⇙) - int (#w⇘a⇙)" end

theory Submission imports Defs begin text ‹First task› theorem L_sub: "L ⊆ M" sorry text ‹Second task› theorem h_first_zero: assumes "h (xs @ ys) < 0" "h xs ≥ 0" shows "∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" sorry text ‹Third task› theorem h_2_split: assumes "h w = 2" "∀v. v ≺ w ⟶ h v ≠ 1" obtains x y where "h x = 0" "h y = 0" "w = x @ b # y" sorry text ‹Fourth task› theorem L_sup: assumes h_first_zero: "⋀xs ys. h (xs @ ys) < 0 ⟹ 0 ≤ h xs ⟹ ∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" assumes h_2_split: "⋀t w. h w = 2 ⟹ ∀v. v ≺ w ⟶ h v ≠ 1 ⟹ (⋀x y. h x = 0 ⟹ h y = 0 ⟹ w = x @ b # y ⟹ t) ⟹ t" assumes "h w = 0" "set w ⊆ {a, b}" shows "w ∈ L" sorry text ‹The final theorem› corollary "L = M" using L_sub L_sup[OF h_first_zero h_2_split] unfolding h_def M_def by auto end

theory Check imports Submission begin text ‹First task› theorem L_sub: "L ⊆ M" by (rule Submission.L_sub) text ‹Second task› theorem h_first_zero: assumes "h (xs @ ys) < 0" "h xs ≥ 0" shows "∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" using assms by (rule Submission.h_first_zero) text ‹Third task› theorem h_2_split: assumes "h w = 2" "∀v. v ≺ w ⟶ h v ≠ 1" obtains x y where "h x = 0" "h y = 0" "w = x @ b # y" using assms by (rule Submission.h_2_split) text ‹Fourth task› theorem L_sup: assumes h_first_zero: "⋀xs ys. h (xs @ ys) < 0 ⟹ 0 ≤ h xs ⟹ ∃as bs. ys = as @ bs ∧ h (xs @ as) = 0" assumes h_2_split: "⋀t w. h w = 2 ⟹ ∀v. v ≺ w ⟶ h v ≠ 1 ⟹ (⋀x y. h x = 0 ⟹ h y = 0 ⟹ w = x @ b # y ⟹ t) ⟹ t" assumes "h w = 0" "set w ⊆ {a, b}" shows "w ∈ L" using assms by (rule Submission.L_sup) text ‹The final theorem› corollary "L = M" using L_sub L_sup[OF h_first_zero h_2_split] unfolding h_def M_def by auto end

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