(*  Title:      HOLCF/Library/List_Cpo.thy

    Author:     Brian Huffman

*)

header {* Lists as a complete partial order *}

theory List_Cpo

imports HOLCF

begin

subsection {* Lists are a partial order *}

instantiation list :: (po) po

begin

definition

  "xs \<sqsubseteq> ys \<longleftrightarrow> list_all2 (op \<sqsubseteq>) xs ys"

instance proof

  fix xs :: "'a list"

  from below_refl show "xs \<sqsubseteq> xs"

    unfolding below_list_def

    by (rule list_all2_refl)

next

  fix xs ys zs :: "'a list"

  assume "xs \<sqsubseteq> ys" and "ys \<sqsubseteq> zs"

  with below_trans show "xs \<sqsubseteq> zs"

    unfolding below_list_def

    by (rule list_all2_trans)

next

  fix xs ys zs :: "'a list"

  assume "xs \<sqsubseteq> ys" and "ys \<sqsubseteq> xs"

  with below_antisym show "xs = ys"

    unfolding below_list_def

    by (rule list_all2_antisym)

qed

end

lemma below_list_simps [simp]:

  "[] \<sqsubseteq> []"

  "x # xs \<sqsubseteq> y # ys \<longleftrightarrow> x \<sqsubseteq> y \<and> xs \<sqsubseteq> ys"

  "\<not> [] \<sqsubseteq> y # ys"

  "\<not> x # xs \<sqsubseteq> []"

by (simp_all add: below_list_def)

lemma Nil_below_iff [simp]: "[] \<sqsubseteq> xs \<longleftrightarrow> xs = []"

by (cases xs, simp_all)

lemma below_Nil_iff [simp]: "xs \<sqsubseteq> [] \<longleftrightarrow> xs = []"

by (cases xs, simp_all)

text "Thanks to Joachim Breitner"

lemma list_Cons_below:

  assumes "a # as \<sqsubseteq> xs"

  obtains b and bs where "a \<sqsubseteq> b" and "as \<sqsubseteq> bs" and "xs = b # bs"

  using assms by (cases xs, auto)

lemma list_below_Cons:

  assumes "xs \<sqsubseteq> b # bs"

  obtains a and as where "a \<sqsubseteq> b" and "as \<sqsubseteq> bs" and "xs = a # as"

  using assms by (cases xs, auto)

lemma hd_mono: "xs \<sqsubseteq> ys \<Longrightarrow> hd xs \<sqsubseteq> hd ys"

by (cases xs, simp, cases ys, simp, simp)

lemma tl_mono: "xs \<sqsubseteq> ys \<Longrightarrow> tl xs \<sqsubseteq> tl ys"

by (cases xs, simp, cases ys, simp, simp)

lemma ch2ch_hd [simp]: "chain (\<lambda>i. S i) \<Longrightarrow> chain (\<lambda>i. hd (S i))"

by (rule chainI, rule hd_mono, erule chainE)

lemma ch2ch_tl [simp]: "chain (\<lambda>i. S i) \<Longrightarrow> chain (\<lambda>i. tl (S i))"

by (rule chainI, rule tl_mono, erule chainE)

lemma below_same_length: "xs \<sqsubseteq> ys \<Longrightarrow> length xs = length ys"

unfolding below_list_def by (rule list_all2_lengthD)

lemma list_chain_cases:

  assumes S: "chain S"

  obtains "\<forall>i. S i = []" |

    A B where "chain A" and "chain B" and "\<forall>i. S i = A i # B i"

proof (cases "S 0")

  case Nil

  have "\<forall>i. S 0 \<sqsubseteq> S i" by (simp add: chain_mono [OF S])

  with Nil have "\<forall>i. S i = []" by simp

  thus ?thesis ..

next

  case (Cons x xs)

  have "\<forall>i. S 0 \<sqsubseteq> S i" by (simp add: chain_mono [OF S])

  hence *: "\<forall>i. S i \<noteq> []" by (rule all_forward) (auto simp add: Cons)

  let ?A = "\<lambda>i. hd (S i)"

  let ?B = "\<lambda>i. tl (S i)"

  from S have "chain ?A" and "chain ?B" by simp_all

  moreover have "\<forall>i. S i = ?A i # ?B i" by (simp add: *)

  ultimately show ?thesis ..

qed

subsection {* Lists are a complete partial order *}

lemma is_lub_Cons:

  assumes A: "range A <<| x"

  assumes B: "range B <<| xs"

  shows "range (\<lambda>i. A i # B i) <<| x # xs"

using assms

unfolding is_lub_def is_ub_def Ball_def [symmetric]

by (clarsimp, case_tac u, simp_all)

lemma list_cpo_lemma:

  fixes S :: "nat \<Rightarrow> 'a::cpo list"

  assumes "chain S" and "\<forall>i. length (S i) = n"

  shows "\<exists>x. range S <<| x"

using assms

 apply (induct n arbitrary: S)

  apply (subgoal_tac "S = (\<lambda>i. [])")

  apply (fast intro: lub_const)

  apply (simp add: ext_iff)

 apply (drule_tac x="\<lambda>i. tl (S i)" in meta_spec, clarsimp)

 apply (rule_tac x="(\<Squnion>i. hd (S i)) # x" in exI)

 apply (subgoal_tac "range (\<lambda>i. hd (S i) # tl (S i)) = range S")

  apply (erule subst)

  apply (rule is_lub_Cons)

   apply (rule thelubE [OF _ refl])

   apply (erule ch2ch_hd)

  apply assumption

 apply (rule_tac f="\<lambda>S. range S" in arg_cong)

 apply (rule ext)

 apply (rule hd_Cons_tl)

 apply (drule_tac x=i in spec, auto)

done

instance list :: (cpo) cpo

proof

  fix S :: "nat \<Rightarrow> 'a list"

  assume "chain S"

  hence "\<forall>i. S 0 \<sqsubseteq> S i" by (simp add: chain_mono)

  hence "\<forall>i. length (S i) = length (S 0)"

    by (fast intro: below_same_length [symmetric])

  with `chain S` show "\<exists>x. range S <<| x"

    by (rule list_cpo_lemma)

qed

subsection {* Continuity of list operations *}

lemma cont2cont_Cons [simp, cont2cont]:

  assumes f: "cont (\<lambda>x. f x)"

  assumes g: "cont (\<lambda>x. g x)"

  shows "cont (\<lambda>x. f x # g x)"

apply (rule contI)

apply (rule is_lub_Cons)

apply (erule contE [OF f])

apply (erule contE [OF g])

done

lemma lub_Cons:

  fixes A :: "nat \<Rightarrow> 'a::cpo"

  assumes A: "chain A" and B: "chain B"

  shows "(\<Squnion>i. A i # B i) = (\<Squnion>i. A i) # (\<Squnion>i. B i)"

by (intro thelubI is_lub_Cons cpo_lubI A B)

lemma cont2cont_list_case:

  assumes f: "cont (\<lambda>x. f x)"

  assumes g: "cont (\<lambda>x. g x)"

  assumes h1: "\<And>y ys. cont (\<lambda>x. h x y ys)"

  assumes h2: "\<And>x ys. cont (\<lambda>y. h x y ys)"

  assumes h3: "\<And>x y. cont (\<lambda>ys. h x y ys)"

  shows "cont (\<lambda>x. case f x of [] \<Rightarrow> g x | y # ys \<Rightarrow> h x y ys)"

apply (rule cont_apply [OF f])

apply (rule contI)

apply (erule list_chain_cases)

apply (simp add: lub_const)

apply (simp add: lub_Cons)

apply (simp add: cont2contlubE [OF h2])

apply (simp add: cont2contlubE [OF h3])

apply (simp add: diag_lub ch2ch_cont [OF h2] ch2ch_cont [OF h3])

apply (rule cpo_lubI, rule chainI, rule below_trans)

apply (erule cont2monofunE [OF h2 chainE])

apply (erule cont2monofunE [OF h3 chainE])

apply (case_tac y, simp_all add: g h1)

done

lemma cont2cont_list_case' [simp, cont2cont]:

  assumes f: "cont (\<lambda>x. f x)"

  assumes g: "cont (\<lambda>x. g x)"

  assumes h: "cont (\<lambda>p. h (fst p) (fst (snd p)) (snd (snd p)))"

  shows "cont (\<lambda>x. case f x of [] \<Rightarrow> g x | y # ys \<Rightarrow> h x y ys)"

proof -

  have "\<And>y ys. cont (\<lambda>x. h x (fst (y, ys)) (snd (y, ys)))"

    by (rule h [THEN cont_fst_snd_D1])

  hence h1: "\<And>y ys. cont (\<lambda>x. h x y ys)" by simp

  note h2 = h [THEN cont_fst_snd_D2, THEN cont_fst_snd_D1]

  note h3 = h [THEN cont_fst_snd_D2, THEN cont_fst_snd_D2]

  from f g h1 h2 h3 show ?thesis by (rule cont2cont_list_case)

qed

text {* The simple version (due to Joachim Breitner) is needed if the

  element type of the list is not a cpo. *}

lemma cont2cont_list_case_simple [simp, cont2cont]:

  assumes "cont (\<lambda>x. f1 x)"

  assumes "\<And>y ys. cont (\<lambda>x. f2 x y ys)"

  shows "cont (\<lambda>x. case l of [] \<Rightarrow> f1 x | y # ys \<Rightarrow> f2 x y ys)"

using assms by (cases l) auto

text {* There are probably lots of other list operations that also

deserve to have continuity lemmas.  I'll add more as they are

needed. *}

end