Theory Coinductive_Languages.Coinductive_Language

(*<*)
(* Author: Dmitriy Traytel *)

section ‹A Codatatype of Formal Languages›

theory Coinductive_Language
imports Main
begin

hide_const (open) Inter
(*>*)

section ‹Introduction›

text ‹
We define formal languages as a codataype of infinite trees branching over the
alphabet @{typ 'a}. Each node in such a tree indicates whether the path to this
node constitutes a word inside or outside of the language.

›

codatatype 'a language = Lang (𝔬: bool) (𝔡: "'a ⇒ 'a language")

text ‹
This codatatype is isormorphic to the set of lists representation of languages,
but caters for definitions by corecursion and proofs by coinduction.

Regular operations on languages are then defined by primitive corecursion.
A difficulty arises here, since the standard definitions of concatenation and
iteration from the coalgebraic literature are not primitively corecursive---they
require guardedness up-to union/concatenation. Without support for up-to corecursion,
these operation must be defined as a composition of primitive ones (and proved being
equal to the standard definitions). As an exercise in coinduction we also prove the
axioms of Kleene algebra for the defined regular operations.

Furthermore, a language for context-free grammars given by productions in Greibach
normal form and an initial nonterminal is constructed by primitive corecursion,
yielding an executable decision procedure for the word problem without further ado.
›
(*<*)
(* custom coinduction theorem (getting rid of rel_fun) *)
declare language.coinduct_strong[unfolded rel_fun_def, simplified, case_names Lang, coinduct pred]
(*>*)

section ‹Regular Languages›

primcorec Zero :: "'a language" where
  "𝔬 Zero = False"
| "𝔡 Zero = (λ_. Zero)"

primcorec One :: "'a language" where
  "𝔬 One = True"
| "𝔡 One = (λ_. Zero)"

primcorec Atom :: "'a ⇒ 'a language" where
  "𝔬 (Atom a) = False"
| "𝔡 (Atom a) = (λb. if a = b then One else Zero)"

primcorec Plus :: "'a language ⇒ 'a language ⇒ 'a language" where
  "𝔬 (Plus r s) = (𝔬 r ∨ 𝔬 s)"
| "𝔡 (Plus r s) = (λa. Plus (𝔡 r a) (𝔡 s a))"

theorem Plus_ZeroL[simp]: "Plus Zero r = r"
  by (coinduction arbitrary: r) simp

theorem Plus_ZeroR[simp]: "Plus r Zero = r"
  by (coinduction arbitrary: r) simp

theorem Plus_assoc: "Plus (Plus r s) t = Plus r (Plus s t)"
  by (coinduction arbitrary: r s t) auto

theorem Plus_comm: "Plus r s = Plus s r"
  by (coinduction arbitrary: r s) auto

lemma Plus_rotate: "Plus r (Plus s t) = Plus s (Plus r t)"
  using Plus_assoc Plus_comm by metis

theorem Plus_idem: "Plus r r = r"
  by (coinduction arbitrary: r) auto

lemma Plus_idem_assoc: "Plus r (Plus r s) = Plus r s"
  by (metis Plus_assoc Plus_idem)

lemmas Plus_ACI[simp] = Plus_rotate Plus_comm Plus_assoc Plus_idem_assoc Plus_idem

lemma Plus_OneL[simp]: "𝔬 r ⟹ Plus One r = r"
  by (coinduction arbitrary: r) auto

lemma Plus_OneR[simp]: "𝔬 r ⟹ Plus r One = r"
  by (coinduction arbitrary: r) auto

text ‹
  Concatenation is not primitively corecursive---the corecursive call of its derivative is
  guarded by @{term Plus}. However, it can be defined as a composition of two primitively
  corecursive functions.
›

primcorec TimesLR :: "'a language ⇒ 'a language ⇒ ('a × bool) language" where
  "𝔬 (TimesLR r s) = (𝔬 r ∧ 𝔬 s)"
| "𝔡 (TimesLR r s) = (λ(a, b).
   if b then TimesLR (𝔡 r a) s else if 𝔬 r then TimesLR (𝔡 s a) One else Zero)"

primcorec Times_Plus :: "('a × bool) language ⇒ 'a language" where
  "𝔬 (Times_Plus r) = 𝔬 r"
| "𝔡 (Times_Plus r) = (λa. Times_Plus (Plus (𝔡 r (a, True)) (𝔡 r (a, False))))"

lemma TimesLR_ZeroL[simp]: "TimesLR Zero r = Zero"
  by (coinduction arbitrary: r) auto

lemma TimesLR_ZeroR[simp]: "TimesLR r Zero = Zero"
  by (coinduction arbitrary: r) (auto intro: exI[of _ Zero])

lemma TimesLR_PlusL[simp]: "TimesLR (Plus r s) t = Plus (TimesLR r t) (TimesLR s t)"
  by (coinduction arbitrary: r s t) auto

lemma TimesLR_PlusR[simp]: "TimesLR r (Plus s t) = Plus (TimesLR r s) (TimesLR r t)"
  by (coinduction arbitrary: r s t) auto

lemma Times_Plus_Zero[simp]: "Times_Plus Zero = Zero"
  by coinduction simp

lemma Times_Plus_Plus[simp]: "Times_Plus (Plus r s) = Plus (Times_Plus r) (Times_Plus s)"
proof (coinduction arbitrary: r s)
  case (Lang r s)
  then show ?case unfolding Times_Plus.sel Plus.sel
    by (intro conjI[OF refl]) (metis Plus_comm Plus_rotate)
qed

lemma Times_Plus_TimesLR_One[simp]: "Times_Plus (TimesLR r One) = r"
  by (coinduction arbitrary: r) simp

lemma Times_Plus_TimesLR_PlusL[simp]:
  "Times_Plus (TimesLR (Plus r s) t) = Plus (Times_Plus (TimesLR r t)) (Times_Plus (TimesLR s t))"
  by (coinduction arbitrary: r s t) auto

lemma Times_Plus_TimesLR_PlusR[simp]:
  "Times_Plus (TimesLR r (Plus s t)) = Plus (Times_Plus (TimesLR r s)) (Times_Plus (TimesLR r t))"
  by (coinduction arbitrary: r s t) auto

definition Times :: "'a language ⇒ 'a language ⇒ 'a language" where
  "Times r s = Times_Plus (TimesLR r s)"

lemma 𝔬_Times[simp]:
  "𝔬 (Times r s) = (𝔬 r ∧ 𝔬 s)"
  unfolding Times_def by simp

lemma 𝔡_Times[simp]:
  "𝔡 (Times r s) = (λa. if 𝔬 r then Plus (Times (𝔡 r a) s) (𝔡 s a) else Times (𝔡 r a) s)"
  unfolding Times_def by auto

theorem Times_ZeroL[simp]: "Times Zero r = Zero"
  by coinduction simp

theorem Times_ZeroR[simp]: "Times r Zero = Zero"
  by (coinduction arbitrary: r) auto

theorem Times_OneL[simp]: "Times One r = r"
  by (coinduction arbitrary: r rule: language.coinduct_strong) (simp add: rel_fun_def)

theorem Times_OneR[simp]: "Times r One = r"
  by (coinduction arbitrary: r) simp



text ‹
  Coinduction up-to @{term Plus}--congruence relaxes the coinduction hypothesis by requiring
  membership in the congruence closure of the bisimulation rather than in the bisimulation itself.
›

inductive Plus_cong for R where
  Refl[intro]: "x = y ⟹ Plus_cong R x y"
| Base[intro]: "R x y ⟹ Plus_cong R x y"
| Sym: "Plus_cong R x y ⟹ Plus_cong R y x"
| Trans[intro]: "Plus_cong R x y ⟹ Plus_cong R y z ⟹ Plus_cong R x z"
| Plus[intro]: "⟦Plus_cong R x y; Plus_cong R x' y'⟧ ⟹ Plus_cong R (Plus x x') (Plus y y')"

lemma language_coinduct_upto_Plus[unfolded rel_fun_def, simplified, case_names Lang, consumes 1]:
  assumes R: "R L K" and hyp:
    "(⋀L K. R L K ⟹ 𝔬 L = 𝔬 K ∧ rel_fun (=) (Plus_cong R) (𝔡 L) (𝔡 K))"
  shows "L = K"
proof (coinduct rule: language.coinduct[of "Plus_cong R"])
  fix L K assume "Plus_cong R L K"
  then show "𝔬 L = 𝔬 K ∧ rel_fun (=) (Plus_cong R) (𝔡 L) (𝔡 K)"
    by (induct rule: Plus_cong.induct) (auto simp: rel_fun_def intro: Sym dest: hyp)
qed (intro Base R)

theorem Times_PlusL[simp]: "Times (Plus r s) t = Plus (Times r t) (Times s t)"
  by (coinduction arbitrary: r s rule: language_coinduct_upto_Plus) auto

theorem Times_PlusR[simp]: "Times r (Plus s t) = Plus (Times r s) (Times r t)"
  by (coinduction arbitrary: r s rule: language_coinduct_upto_Plus) fastforce

theorem Times_assoc[simp]: "Times (Times r s) t = Times r (Times s t)"
  by (coinduction arbitrary: r s t rule: language_coinduct_upto_Plus) fastforce

text ‹
  Similarly to @{term Times}, iteration is not primitively corecursive (guardedness by
  @{term Times} is required). We apply a similar trick to obtain its definition.
›

primcorec StarLR :: "'a language ⇒ 'a language ⇒ 'a language" where
  "𝔬 (StarLR r s) = 𝔬 r"
| "𝔡 (StarLR r s) = (λa. StarLR (𝔡 (Times r (Plus One s)) a) s)"

lemma StarLR_Zero[simp]: "StarLR Zero r = Zero"
  by coinduction auto

lemma StarLR_Plus[simp]: "StarLR (Plus r s) t = Plus (StarLR r t) (StarLR s t)"
  by (coinduction arbitrary: r s) (auto simp del: Plus_ACI Times_PlusR)

lemma StarLR_Times_Plus_One[simp]: "StarLR (Times r (Plus One s)) s = StarLR r s"
proof (coinduction arbitrary: r s)
  case Lang
  { fix a
    define L and R where "L = Plus (𝔡 r a) (Plus (Times (𝔡 r a) s) (𝔡 s a))"
    and "R = Times (Plus (𝔡 r a) (Plus (Times (𝔡 r a) s) (𝔡 s a))) s"
    have "Plus L (Plus R (𝔡 s a)) = Plus (Plus L (𝔡 s a)) R" by (metis Plus_assoc Plus_comm)
    also have "Plus L (𝔡 s a) = L" unfolding L_def by simp
    finally have "Plus L (Plus R (𝔡 s a)) = Plus L R" .
  }
  then show ?case by (auto simp del: StarLR_Plus Plus_assoc Times_PlusL)
qed

lemma StarLR_Times: "StarLR (Times r s) t = Times r (StarLR s t)"
  by (coinduction arbitrary: r s t rule: language_coinduct_upto_Plus)
    (fastforce simp del: Plus_ACI Times_PlusR)

definition Star :: "'a language ⇒ 'a language" where
  "Star r = StarLR One r"

lemma 𝔬_Star[simp]: "𝔬 (Star r)"
  unfolding Star_def by simp

lemma 𝔡_Star[simp]: "𝔡 (Star r) = (λa. Times (𝔡 r a) (Star r))"
  unfolding Star_def by (auto simp add: Star_def StarLR_Times[symmetric])

lemma Star_Zero[simp]: "Star Zero = One"
  by coinduction auto

lemma Star_One[simp]: "Star One = One"
  by coinduction auto

lemma Star_unfoldL: "Star r = Plus One (Times r (Star r))"
  by coinduction auto

primcorec Inter :: "'a language ⇒ 'a language ⇒ 'a language" where
  "𝔬 (Inter r s) = (𝔬 r ∧ 𝔬 s)"
| "𝔡 (Inter r s) = (λa. Inter (𝔡 r a) (𝔡 s a))"

primcorec Not :: "'a language ⇒ 'a language" where
  "𝔬 (Not r) = (¬ 𝔬 r)"
| "𝔡 (Not r) = (λa. Not (𝔡 r a))"

primcorec Full :: "'a language" (‹Σ⇧*›) where
  "𝔬 Full = True"
| "𝔡 Full = (λ_. Full)"

text ‹
  Shuffle product is not primitively corecursive---the corecursive call of its derivative is
  guarded by @{term Plus}. However, it can be defined as a composition of two primitively
  corecursive functions.
›

primcorec ShuffleLR :: "'a language ⇒ 'a language ⇒ ('a × bool) language" where
  "𝔬 (ShuffleLR r s) = (𝔬 r ∧ 𝔬 s)"
| "𝔡 (ShuffleLR r s) = (λ(a, b). if b then ShuffleLR (𝔡 r a) s else ShuffleLR r (𝔡 s a))"

lemma ShuffleLR_ZeroL[simp]: "ShuffleLR Zero r = Zero"
  by (coinduction arbitrary: r) auto

lemma ShuffleLR_ZeroR[simp]: "ShuffleLR r Zero = Zero"
  by (coinduction arbitrary: r) (auto intro: exI[of _ Zero])

lemma ShuffleLR_PlusL[simp]: "ShuffleLR (Plus r s) t = Plus (ShuffleLR r t) (ShuffleLR s t)"
  by (coinduction arbitrary: r s t) auto

lemma ShuffleLR_PlusR[simp]: "ShuffleLR r (Plus s t) = Plus (ShuffleLR r s) (ShuffleLR r t)"
  by (coinduction arbitrary: r s t) auto

lemma Shuffle_Plus_ShuffleLR_One[simp]: "Times_Plus (ShuffleLR r One) = r"
  by (coinduction arbitrary: r) simp

lemma Shuffle_Plus_ShuffleLR_PlusL[simp]:
  "Times_Plus (ShuffleLR (Plus r s) t) = Plus (Times_Plus (ShuffleLR r t)) (Times_Plus (ShuffleLR s t))"
  by (coinduction arbitrary: r s t) auto

lemma Shuffle_Plus_ShuffleLR_PlusR[simp]:
  "Times_Plus (ShuffleLR r (Plus s t)) = Plus (Times_Plus (ShuffleLR r s)) (Times_Plus (ShuffleLR r t))"
  by (coinduction arbitrary: r s t) auto

definition Shuffle :: "'a language ⇒ 'a language ⇒ 'a language" where
  "Shuffle r s = Times_Plus (ShuffleLR r s)"

lemma 𝔬_Shuffle[simp]:
  "𝔬 (Shuffle r s) = (𝔬 r ∧ 𝔬 s)"
  unfolding Shuffle_def by simp

lemma 𝔡_Shuffle[simp]:
  "𝔡 (Shuffle r s) = (λa. Plus (Shuffle (𝔡 r a) s) (Shuffle r (𝔡 s a)))"
  unfolding Shuffle_def by auto

theorem Shuffle_ZeroL[simp]: "Shuffle Zero r = Zero"
  by (coinduction arbitrary: r rule: language_coinduct_upto_Plus) (auto 0 4)

theorem Shuffle_ZeroR[simp]: "Shuffle r Zero = Zero"
  by (coinduction arbitrary: r rule: language_coinduct_upto_Plus) (auto 0 4)

theorem Shuffle_OneL[simp]: "Shuffle One r = r"
  by (coinduction arbitrary: r) simp

theorem Shuffle_OneR[simp]: "Shuffle r One = r"
  by (coinduction arbitrary: r) simp

theorem Shuffle_PlusL[simp]: "Shuffle (Plus r s) t = Plus (Shuffle r t) (Shuffle s t)"
  by (coinduction arbitrary: r s t rule: language_coinduct_upto_Plus)
    (force intro!: Trans[OF Plus[OF Base Base] Refl])

theorem Shuffle_PlusR[simp]: "Shuffle r (Plus s t) = Plus (Shuffle r s) (Shuffle r t)"
  by (coinduction arbitrary: r s t rule: language_coinduct_upto_Plus)
    (force intro!: Trans[OF Plus[OF Base Base] Refl])

theorem Shuffle_assoc[simp]: "Shuffle (Shuffle r s) t = Shuffle r (Shuffle s t)"
  by (coinduction arbitrary: r s t rule: language_coinduct_upto_Plus) fastforce

theorem Shuffle_comm[simp]: "Shuffle r s = Shuffle s r"
  by (coinduction arbitrary: r s rule: language_coinduct_upto_Plus)
    (auto intro!: Trans[OF Plus[OF Base Base] Refl])

text ‹
  We generalize coinduction up-to @{term Plus} to coinduction up-to all previously defined concepts.
›

inductive regular_cong for R where
  Refl[intro]: "x = y ⟹ regular_cong R x y"
| Sym[intro]: "regular_cong R x y ⟹ regular_cong R y x"
| Trans[intro]: "⟦regular_cong R x y; regular_cong R y z⟧ ⟹ regular_cong R x z"
| Base[intro]: "R x y ⟹ regular_cong R x y"
| Plus[intro, simp]: "⟦regular_cong R x y; regular_cong R x' y'⟧ ⟹
    regular_cong R (Plus x x') (Plus y y')"
| Times[intro, simp]: "⟦regular_cong R x y; regular_cong R x' y'⟧ ⟹
    regular_cong R (Times x x') (Times y y')"
| Star[intro, simp]: "⟦regular_cong R x y⟧ ⟹
    regular_cong R (Star x) (Star y)"
| Inter[intro, simp]: "⟦regular_cong R x y; regular_cong R x' y'⟧ ⟹
    regular_cong R (Inter x x') (Inter y y')"
| Not[intro, simp]: "⟦regular_cong R x y⟧ ⟹
    regular_cong R (Not x) (Not y)"
| Shuffle[intro, simp]: "⟦regular_cong R x y; regular_cong R x' y'⟧ ⟹
    regular_cong R (Shuffle x x') (Shuffle y y')"

lemma language_coinduct_upto_regular[unfolded rel_fun_def, simplified, case_names Lang, consumes 1]:
  assumes R: "R L K" and hyp:
    "(⋀L K. R L K ⟹ 𝔬 L = 𝔬 K ∧ rel_fun (=) (regular_cong R) (𝔡 L) (𝔡 K))"
  shows "L = K"
proof (coinduct rule: language.coinduct[of "regular_cong R"])
  fix L K assume "regular_cong R L K"
  then show "𝔬 L = 𝔬 K ∧ rel_fun (=) (regular_cong R) (𝔡 L) (𝔡 K)"
    by (induct rule: regular_cong.induct) (auto dest: hyp simp: rel_fun_def)
qed (intro Base R)

lemma Star_unfoldR: "Star r = Plus One (Times (Star r) r)"
proof (coinduction arbitrary: r rule: language_coinduct_upto_regular)
  case Lang
  { fix a have "Plus (Times (𝔡 r a) (Times (Star r) r)) (𝔡 r a) =
      Times (𝔡 r a) (Plus One (Times (Star r) r))" by simp
  }
  then show ?case by (auto simp del: Times_PlusR)
qed

lemma Star_Star[simp]: "Star (Star r) = Star r"
  by (subst Star_unfoldL, coinduction arbitrary: r rule: language_coinduct_upto_regular) auto

lemma Times_Star[simp]: "Times (Star r) (Star r) = Star r"
proof (coinduction arbitrary: r rule: language_coinduct_upto_regular)
  case Lang
  have *: "⋀r s. Plus (Times r s) r = Times r (Plus s One)" by simp
  show ?case by (auto simp del: Times_PlusR Plus_ACI simp: Times_PlusR[symmetric] *)
qed

instantiation language :: (type) "{semiring_1, order}"
begin

lemma Zero_One[simp]: "Zero ≠ One"
  by (metis One.simps(1) Zero.simps(1))

definition "zero_language = Zero"
definition "one_language = One"
definition "plus_language = Plus"
definition "times_language = Times"

definition "less_eq_language r s = (Plus r s = s)"
definition "less_language r s = (Plus r s = s ∧ r ≠ s)"

lemmas language_defs = zero_language_def one_language_def plus_language_def times_language_def
  less_eq_language_def less_language_def

instance proof intro_classes
  fix x y z :: "'a language" assume "x ≤ y" "y ≤ z"
  then show "x ≤ z" unfolding language_defs by (metis Plus_assoc)
next
  fix x y z :: "'a language"
  show "x + y + z = x + (y + z)" unfolding language_defs by (rule Plus_assoc)
qed (auto simp: language_defs)

end

lemma 𝔬_mono[dest]: "r ≤ s ⟹ 𝔬 r ⟹ 𝔬 s"
  unfolding less_eq_language_def by (auto dest: arg_cong[of _ _ 𝔬])

lemma 𝔡_mono[dest]: "r ≤ s ⟹ 𝔡 r a ≤ 𝔡 s a"
  unfolding less_eq_language_def by (metis Plus.simps(2))

text ‹
  For reasoning about @{term "(≤)"}, we prove a coinduction principle and generalize it
  to support up-to reasoning.
›

theorem language_simulation_coinduction[consumes 1, case_names Lang, coinduct pred]:
  assumes "R L K"
      and "(⋀L K. R L K ⟹ 𝔬 L ≤ 𝔬 K ∧ (∀x. R (𝔡 L x) (𝔡 K x)))"
  shows "L ≤ K"
  using ‹R L K› unfolding less_eq_language_def
  by (coinduction arbitrary: L K) (auto dest!: assms(2))

lemma le_PlusL[intro!, simp]: "r ≤ Plus r s"
  by (coinduction arbitrary: r s) auto

lemma le_PlusR[intro!, simp]: "s ≤ Plus r s"
  by (coinduction arbitrary: r s) auto

inductive Plus_Times_pre_cong for R where
  pre_Less[intro, simp]: "x ≤ y ⟹ Plus_Times_pre_cong R x y"
| pre_Trans[intro]: "⟦Plus_Times_pre_cong R x y; Plus_Times_pre_cong R y z⟧ ⟹ Plus_Times_pre_cong R x z"
| pre_Base[intro, simp]: "R x y ⟹ Plus_Times_pre_cong R x y"
| pre_Plus[intro!, simp]: "⟦Plus_Times_pre_cong R x y; Plus_Times_pre_cong R x' y'⟧ ⟹
    Plus_Times_pre_cong R (Plus x x') (Plus y y')"
| pre_Times[intro!, simp]: "⟦Plus_Times_pre_cong R x y; Plus_Times_pre_cong R x' y'⟧ ⟹
    Plus_Times_pre_cong R (Times x x') (Times y y')"

theorem language_simulation_coinduction_upto_Plus_Times[consumes 1, case_names Lang, coinduct pred]:
  assumes R: "R L K"
      and hyp: "(⋀L K. R L K ⟹ 𝔬 L ≤ 𝔬 K ∧ (∀x. Plus_Times_pre_cong R (𝔡 L x) (𝔡 K x)))"
    shows "L ≤ K"
proof (coinduct rule: language_simulation_coinduction[of "Plus_Times_pre_cong R"])
  fix L K assume "Plus_Times_pre_cong R L K"
  then show "𝔬 L ≤ 𝔬 K ∧ (∀x. Plus_Times_pre_cong R (𝔡 L x) (𝔡 K x))"
    by (induct rule: Plus_Times_pre_cong.induct) (auto dest: hyp)
qed (intro pre_Base R)

lemma ge_One[simp]: "One ≤ r ⟷ 𝔬 r"
  unfolding less_eq_language_def by (metis One.sel(1) Plus.sel(1) Plus_OneL)

lemma Plus_mono: "⟦r1 ≤ s1; r2 ≤ s2⟧ ⟹ Plus r1 r2 ≤ Plus s1 s2"
  by (coinduction arbitrary: r1 r2 s1 s2) (auto elim!: 𝔡_mono)

lemma Plus_upper: "⟦r1 ≤ s; r2 ≤ s⟧ ⟹ Plus r1 r2 ≤ s"
  by (coinduction arbitrary: r1 r2 s) auto

lemma Inter_mono: "⟦r1 ≤ s1; r2 ≤ s2⟧ ⟹ Inter r1 r2 ≤ Inter s1 s2"
  by (coinduction arbitrary: r1 r2 s1 s2) (force elim!: 𝔡_mono)

lemma Times_mono: "⟦r1 ≤ s1; r2 ≤ s2⟧ ⟹ Times r1 r2 ≤ Times s1 s2"
  by (coinduction arbitrary: r1 r2 s1 s2) (auto 0 3 elim!: 𝔡_mono)

text ‹
  We prove the missing axioms of Kleene Algebras about @{term Star}, as well as monotonicity
  properties and three standard interesting rules: bisimulation, sliding, and denesting.
›

theorem le_StarL: "Plus One (Times r (Star r)) ≤ Star r"
  by coinduction auto

theorem le_StarR: "Plus One (Times (Star r) r) ≤ Star r"
  by (rule order_eq_refl[OF Star_unfoldR[symmetric]])

lemma le_TimesL[intro, simp]: "𝔬 s ⟹ r ≤ Times r s"
  by (coinduction arbitrary: r) auto

lemma le_TimesR[intro, simp]: "𝔬 r ⟹ s ≤ Times r s"
  by coinduction auto

lemma Plus_le_iff: "Plus r s ≤ t ⟷ r ≤ t ∧ s ≤ t"
  unfolding less_eq_language_def
  by (metis Plus_assoc Plus_idem_assoc Plus_rotate)

lemma Plus_Times_pre_cong_mono:
  "L' ≤ L ⟹ K ≤ K' ⟹ Plus_Times_pre_cong R L K ⟹ Plus_Times_pre_cong R L' K'"
  by (metis pre_Trans pre_Less)

theorem ardenL: "Plus r (Times s x) ≤ x ⟹ Times (Star s) r ≤ x"
proof (coinduction arbitrary: r s x)
  case Lang
  then show ?case
    by (subst Plus_Times_pre_cong_mono[OF order_refl 𝔡_mono[OF Lang]]) auto
qed

theorem ardenR: "Plus r (Times x s) ≤ x ⟹ Times r (Star s) ≤ x"
proof (coinduction arbitrary: r s x)
  case Lang
  then have "Plus_Times_pre_cong (λL K. ∃r s. L = Times r (Star s) ∧ Plus r (Times K s) ≤ K)
    (𝔡 (Times r (Star s)) a) (𝔡 x a)" for a using 𝔡_mono[OF Lang, of a]
    by (auto 0 4 simp del: Times_PlusL simp: Times_PlusL[symmetric] Plus_le_iff split: if_splits)
  with Lang show ?case
    by auto
qed

lemma le_Star[intro!, simp]: "s ≤ Star s"
  by coinduction auto

lemma Star_mono: "r ≤ s ⟹ Star r ≤ Star s"
  by coinduction auto

lemma Not_antimono: "r ≤ s ⟹ Not s ≤ Not r"
  by (coinduction arbitrary: r s) auto

lemma Not_Plus[simp]: "Not (Plus r s) = Inter (Not r) (Not s)"
  by (coinduction arbitrary: r s) auto

lemma Not_Inter[simp]: "Not (Inter r s) = Plus (Not r) (Not s)"
  by (coinduction arbitrary: r s) auto

lemma Inter_assoc[simp]: "Inter (Inter r s) t = Inter r (Inter s t)"
  by (coinduction arbitrary: r s t) auto

lemma Inter_comm: "Inter r s = Inter s r"
  by (coinduction arbitrary: r s) auto

lemma Inter_idem[simp]: "Inter r r = r"
  by (coinduction arbitrary: r) auto

lemma Inter_ZeroL[simp]: "Inter Zero r = Zero"
  by (coinduction arbitrary: r) auto

lemma Inter_ZeroR[simp]: "Inter r Zero = Zero"
  by (coinduction arbitrary: r) auto

lemma Inter_FullL[simp]: "Inter Full r = r"
  by (coinduction arbitrary: r) auto

lemma Inter_FullR[simp]: "Inter r Full = r"
  by (coinduction arbitrary: r) auto

lemma Plus_FullL[simp]: "Plus Full r = Full"
  by (coinduction arbitrary: r) auto

lemma Plus_FullR[simp]: "Plus r Full = Full"
  by (coinduction arbitrary: r) auto

lemma Not_Not[simp]: "Not (Not r) = r"
  by (coinduction arbitrary: r) auto

lemma Not_Zero[simp]: "Not Zero = Full"
  by coinduction simp

lemma Not_Full[simp]: "Not Full = Zero"
  by coinduction simp

lemma bisimulation:
  assumes "Times r s = Times s t"
  shows "Times (Star r) s = Times s (Star t)"
proof (rule antisym[OF ardenL[OF Plus_upper[OF le_TimesL]] ardenR[OF Plus_upper[OF le_TimesR]]])
  show "Times r (Times s (Star t)) ≤ Times s (Star t)"
    by (rule order_trans[OF _
        Times_mono[OF order_refl ord_le_eq_trans[OF le_PlusR Star_unfoldL[symmetric]]]])
      (simp only: assms Times_assoc[symmetric])
next
  show "Times (Times (Star r) s) t ≤ Times (Star r) s"
    by (rule order_trans[OF _
        Times_mono[OF ord_le_eq_trans[OF le_PlusR Star_unfoldR[symmetric]] order_refl]])
      (simp only: assms Times_assoc)
qed simp_all

lemma sliding: "Times (Star (Times r s)) r = Times r (Star (Times s r))"
proof (rule antisym[OF ardenL[OF Plus_upper[OF le_TimesL]] ardenR[OF Plus_upper[OF le_TimesR]]])
  show "Times (Times r s) (Times r (Star (Times s r))) ≤ Times r (Star (Times s r))"
    by (rule order_trans[OF _
        Times_mono[OF order_refl ord_le_eq_trans[OF le_PlusR Star_unfoldL[symmetric]]]]) simp
next
  show "Times (Times (Star (Times r s)) r) (Times s r) ≤ Times (Star (Times r s)) r"
    by (rule order_trans[OF _
        Times_mono[OF ord_le_eq_trans[OF le_PlusR Star_unfoldR[symmetric]] order_refl]]) simp
qed simp_all

lemma denesting: "Star (Plus r s) = Times (Star r) (Star (Times s (Star r)))"
proof (rule antisym[OF ord_eq_le_trans[OF Times_OneR[symmetric] ardenL[OF Plus_upper]]
   ardenR[OF Plus_upper[OF Star_mono[OF le_PlusL]]]])
  show "Times (Plus r s) (Times (Star r) (Star (Times s (Star r))))
    ≤ Times (Star r) (Star (Times s (Star r)))" (is "Times _ ?L ≤ ?R")
    unfolding Times_PlusL
    by (rule Plus_upper,
      metis Star_unfoldL Times_assoc Times_mono le_PlusR order_refl,
      metis Star_unfoldL Times_assoc 𝔬_Star le_PlusR le_TimesR order_trans)
next
  show "Times (Star (Plus r s)) (Times s (Star r)) ≤ Star (Plus r s)"
    by (metis Plus_comm Star_unfoldL Times_PlusR Times_assoc ardenR bisimulation le_PlusR)
qed simp

text ‹
It is useful to lift binary operators @{term Plus} and @{term Times}
to $n$-ary operators (that take a list as input).
›

definition PLUS :: "'a language list ⇒ 'a language" where
  "PLUS xs ≡ foldr Plus xs Zero"

lemma 𝔬_foldr_Plus: "𝔬 (foldr Plus xs s) = (∃x∈set (s # xs). 𝔬 x)"
  by (induct xs arbitrary: s) auto

lemma 𝔡_foldr_Plus: "𝔡 (foldr Plus xs s) a = foldr Plus (map (λr. 𝔡 r a) xs) (𝔡 s a)"
  by (induct xs arbitrary: s) simp_all

lemma 𝔬_PLUS[simp]: "𝔬 (PLUS xs) = (∃x∈set xs. 𝔬 x)"
  unfolding PLUS_def 𝔬_foldr_Plus by simp

lemma 𝔡_PLUS[simp]: "𝔡 (PLUS xs) a = PLUS (map (λr. 𝔡 r a) xs)"
  unfolding PLUS_def 𝔡_foldr_Plus by simp

definition TIMES :: "'a language list ⇒ 'a language" where
  "TIMES xs ≡ foldr Times xs One"

lemma 𝔬_foldr_Times: "𝔬 (foldr Times xs s) = (∀x∈set (s # xs). 𝔬 x)"
  by (induct xs) (auto simp: PLUS_def)

primrec tails where
  "tails [] = [[]]"
| "tails (x # xs) = (x # xs) # tails xs"

lemma tails_snoc[simp]: "tails (xs @ [x]) = map (λys. ys @ [x]) (tails xs) @ [[]]"
  by (induct xs) auto

lemma length_tails[simp]: "length (tails xs) = Suc (length xs)"
  by (induct xs) auto

lemma 𝔡_foldr_Times: "𝔡 (foldr Times xs s) a =
  (let n = length (takeWhile 𝔬 xs)
  in PLUS (map (λzs. TIMES (𝔡 (hd zs) a # tl zs)) (take (Suc n) (tails (xs @ [s])))))"
  by (induct xs) (auto simp: TIMES_def PLUS_def Let_def foldr_map o_def)

lemma 𝔬_TIMES[simp]: "𝔬 (TIMES xs) = (∀x∈set xs. 𝔬 x)"
  unfolding TIMES_def 𝔬_foldr_Times by simp

lemma TIMES_snoc_One[simp]: "TIMES (xs @ [One]) = TIMES xs"
  by (induct xs) (auto simp: TIMES_def)

lemma 𝔡_TIMES[simp]: "𝔡 (TIMES xs) a = (let n = length (takeWhile 𝔬 xs)
  in PLUS (map (λzs. TIMES (𝔡 (hd zs) a # tl zs)) (take (Suc n) (tails (xs @ [One])))))"
  unfolding TIMES_def 𝔡_foldr_Times by simp



section ‹Word-theoretic Semantics of Languages›

text ‹
We show our @{type language} codatatype being isomorphic to the standard
language representation as a set of lists.
›

primrec in_language :: "'a language ⇒ 'a list ⇒ bool" where
  "in_language L [] = 𝔬 L"
| "in_language L (x # xs) = in_language (𝔡 L x) xs"

primcorec to_language :: "'a list set ⇒ 'a language" where
  "𝔬 (to_language L) = ([] ∈ L)"
| "𝔡 (to_language L) = (λa. to_language {w. a # w ∈ L})"

lemma in_language_to_language[simp]: "Collect (in_language (to_language L)) = L"
proof (rule set_eqI, unfold mem_Collect_eq)
  fix w show "in_language (to_language L) w = (w ∈ L)" by (induct w arbitrary: L) auto
qed

lemma to_language_in_language[simp]: "to_language (Collect (in_language L)) = L"
  by (coinduction arbitrary: L) auto

lemma in_language_bij: "bij (Collect o in_language)"
proof (rule bijI', unfold o_apply, safe)
  fix L R :: "'a language" assume "Collect (in_language L) = Collect (in_language R)"
  then show "L = R" unfolding set_eq_iff mem_Collect_eq
    by (coinduction arbitrary: L R) (metis in_language.simps)
next
  fix L :: "'a list set"
  have "L = Collect (in_language (to_language L))" by simp
  then show "∃K. L = Collect (in_language K)" by blast
qed

lemma to_language_bij: "bij to_language"
  by (rule o_bij[of "Collect o in_language"]) (simp_all add: fun_eq_iff)

(*<*)
hide_const (open) TimesLR Times_Plus StarLR ShuffleLR

end
(*>*)