Quelle  Listn.thy

(*  Title:      HOL/MicroJava/BV/Listn.thy
    Author:     Tobias Nipkow
    Copyright   2000 TUM

Lists of a fixed length.
*)

section ‹Fixed Length Lists›

theory Listn
imports Err
begin

definition list :: "nat ==> 'a set ==> 'a list set"
where
  "list n A = {xs. size xs = n ∧ set xs ⊆ A}"

definition le :: "'a ord ==> ('a list)ord"
where
  "le r = list_all2 (λx y. x ⊑_r y)"

abbreviation
  lesublist :: "'a list ==> 'a ord ==> 'a list ==> bool"  (‹(_ /[⊑] _)› [50, 0, 51] 50) where
  "x [⊑] y == x <=_(Listn.le r) y"

abbreviation
  lesssublist :: "'a list ==> 'a ord ==> 'a list ==> bool"  (‹(_ /[⊏] _)› [50, 0, 51] 50) where
  "x [⊏] y == x <_(Listn.le r) y"

(*<*)
notation (ASCII)
  lesublist  (‹(_ /[<=_] _)› [50, 0, 51] 50) and
  lesssublist  (‹(_ /[<_] _)› [50, 0, 51] 50)

abbreviation (input)
  lesublist2 :: "'a list ==> 'a ord ==> 'a list ==> bool"  (‹(_ /[⊑_{_}] _)› [50, 0, 51] 50) where
  "x [⊑_r] y == x [⊑] y"

abbreviation (input)
  lesssublist2 :: "'a list ==> 'a ord ==> 'a list ==> bool"  (‹(_ /[⊏_{_}] _)› [50, 0, 51] 50) where
  "x [⊏_r] y == x [⊏] y"
(*>*)

abbreviation
  plussublist :: "'a list ==> ('a ==> 'b ==> 'c) ==> 'b list ==> 'c list"
    (‹(_ /[⊔] _)› [65, 0, 66] 65) where
  "x [⊔] y == x ⊔ _f y"

(*<*)
notation
  plussublist  (‹(_ /[+_] _)› [65, 0, 66] 65)

abbreviation (input)
  plussublist2 :: "'a list ==> ('a ==> 'b ==> 'c) ==> 'b list ==> 'c list"
    (‹(_ /[⊔_{_}] _)› [65, 0, 66] 65) where
  "x [⊔_f] y == x [⊔] y"
(*>*)


primrec coalesce :: "'a err list ==> 'a list err"
where
  "coalesce [] = OK[]"
| "coalesce (ex#exs) = Err.sup (#) ex (coalesce exs)"

definition sl :: "nat ==> 'a sl ==> 'a list sl"
where
  "sl n = (λ(A,r,f). (list n A, le r, map2 f))"

definition sup :: "('a ==> 'b ==> 'c err) ==> 'a list ==> 'b list ==> 'c list err"
where
  "sup f = (λxs ys. if size xs = size ys then coalesce(xs [⊔] ys) else Err)"

definition upto_esl :: "nat ==> 'a esl ==> 'a list esl"
where
  "upto_esl m = (λ(A,r,f). (Union{list n A |n. n ≤ m}, le r, sup f))"


lemmas [simp] = set_update_subsetI

lemma unfold_lesub_list: "xs [⊑] ys = Listn.le r xs ys"
(*<*) by (simp add: lesub_def) (*>*)

lemma Nil_le_conv [iff]: "([] [⊑] ys) = (ys = [])"
(*<*)
apply (unfold lesub_def Listn.le_def)
apply simp
done
(*>*)

lemma Cons_notle_Nil [iff]: "¬ x#xs [⊑] []"
(*<*)
apply (unfold lesub_def Listn.le_def)
apply simp
done
(*>*)

lemma Cons_le_Cons [iff]: "x#xs [⊑] y#ys = (x ⊑_r y ∧ xs [⊑] ys)"
(*<*)
by (simp add: lesub_def Listn.le_def)
(*>*)

lemma Cons_less_Conss [simp]:
  "order r ==> x#xs [⊏_r] y#ys = (x ⊏_r y ∧ xs [⊑] ys ∨ x = y ∧ xs [⊏_r] ys)"
(*<*)
apply (unfold lesssub_def)
apply blast
done
(*>*)

lemma list_update_le_cong:
  "[ i<size xs; xs [⊑] ys; x ⊑_r y ] ==> xs[i:=x] [⊑] ys[i:=y]"
(*<*)
apply (unfold unfold_lesub_list)
apply (unfold Listn.le_def)
apply (simp add: list_all2_update_cong)
done
(*>*)


lemma le_listD: "[ xs [⊑] ys; p < size xs ] ==> xs!p ⊑_r ys!p"
(*<*)
by (simp add: Listn.le_def lesub_def list_all2_nthD)
(*>*)

lemma le_list_refl: "∀x. x ⊑_r x ==> xs [⊑] xs"
(*<*)
apply (simp add: unfold_lesub_list lesub_def Listn.le_def list_all2_refl)
done
(*>*)

lemma le_list_trans: "[ order r; xs [⊑] ys; ys [⊑] zs ] ==> xs [⊑] zs"
(*<*)
apply (unfold unfold_lesub_list)
apply (unfold Listn.le_def)
apply (rule list_all2_trans)
apply (erule order_trans)
apply assumption+
done
(*>*)

lemma le_list_antisym: "[ order r; xs [⊑] ys; ys [⊑] xs ] ==> xs = ys"
(*<*)
apply (unfold unfold_lesub_list)
apply (unfold Listn.le_def)
apply (rule list_all2_antisym)
apply (rule order_antisym)
apply assumption+
done
(*>*)

lemma order_listI [simp, intro!]: "order r ==> order(Listn.le r)"
(*<*)
apply (subst order_def)
apply (blast intro: le_list_refl le_list_trans le_list_antisym
             dest: order_refl)
done
(*>*)

lemma lesub_list_impl_same_size [simp]: "xs [⊑] ys ==> size ys = size xs"
(*<*)
apply (unfold Listn.le_def lesub_def)
apply (simp add: list_all2_lengthD)
done
(*>*)

lemma lesssub_lengthD: "xs [⊏_r] ys ==> size ys = size xs"
(*<*)
apply (unfold lesssub_def)
apply auto
done
(*>*)

lemma le_list_appendI: "a [⊑] b ==> c [⊑] d ==> a@c [⊑] b@d"
(*<*)
apply (unfold Listn.le_def lesub_def)
apply (rule list_all2_appendI, assumption+)
done
(*>*)

lemma le_listI:
  assumes "length a = length b"
  assumes "∧n. n < length a ==> a!n ⊑_r b!n"
  shows "a [⊑] b"
(*<*)
proof -
  from assms have "list_all2 r a b"
    by (simp add: list_all2_all_nthI lesub_def)
  then show ?thesis by (simp add: Listn.le_def lesub_def)
qed
(*>*)

lemma listI: "[ size xs = n; set xs ⊆ A ] ==> xs ∈ list n A"
(*<*)
apply (unfold list_def)
apply blast
done
(*>*)

(* FIXME: remove simp *)
lemma listE_length [simp]: "xs ∈ list n A ==> size xs = n"
(*<*)
apply (unfold list_def)
apply blast
done
(*>*)

lemma less_lengthI: "[ xs ∈ list n A; p < n ] ==> p < size xs"
(*<*) by simp (*>*)

lemma listE_set [simp]: "xs ∈ list n A ==> set xs ⊆ A"
(*<*)
apply (unfold list_def)
apply blast
done
(*>*)

lemma list_0 [simp]: "list 0 A = {[]}"
(*<*)
apply (unfold list_def)
apply auto
done
(*>*)

lemma in_list_Suc_iff:
  "(xs ∈ list (Suc n) A) = (∃y∈A. ∃ys ∈ list n A. xs = y#ys)"
(*<*)
apply (unfold list_def)
apply (case_tac "xs")
apply auto
done
(*>*)

lemma Cons_in_list_Suc [iff]:
  "(x#xs ∈ list (Suc n) A) = (x∈A ∧ xs ∈ list n A)"
(*<*)
apply (simp add: in_list_Suc_iff)
done
(*>*)

lemma list_not_empty:
  "∃a. a∈A ==> ∃xs. xs ∈ list n A"
(*<*)
apply (induct "n")
 apply simp
apply (simp add: in_list_Suc_iff)
apply blast
done
(*>*)


lemma nth_in [rule_format, simp]:
  "∀i n. size xs = n ⟶ set xs ⊆ A ⟶ i < n ⟶ (xs!i) ∈ A"
(*<*)
apply (induct "xs")
 apply simp
apply (simp add: nth_Cons split: nat.split)
done
(*>*)

lemma listE_nth_in: "[ xs ∈ list n A; i < n ] ==> xs!i ∈ A"
(*<*) by auto (*>*)

lemma listn_Cons_Suc [elim!]:
  "l#xs ∈ list n A ==> (∧n'. n = Suc n' ==> l ∈ A ==> xs ∈ list n' A ==> P) ==> P"
(*<*) by (cases n) auto (*>*)

lemma listn_appendE [elim!]:
  "a@b ∈ list n A ==> (∧n1 n2. n=n1+n2 ==> a ∈ list n1 A ==> b ∈ list n2 A ==> P) ==> P"
(*<*)
proof -
  have "∧n. a@b ∈ list n A ==> ∃n1 n2. n=n1+n2 ∧ a ∈ list n1 A ∧ b ∈ list n2 A"
    (is "∧n. ?list a n ==> ∃n1 n2. ?P a n n1 n2")
  proof (induct a)
    fix n assume "?list [] n"
    hence "?P [] n 0 n" by simp
    thus "∃n1 n2. ?P [] n n1 n2" by fast
  next
    fix n l ls
    assume "?list (l#ls) n"
    then obtain n' where n: "n = Suc n'" "l ∈ A" and n': "ls@b ∈ list n' A" by fastforce
    assume "∧n. ls @ b ∈ list n A ==> ∃n1 n2. n = n1 + n2 ∧ ls ∈ list n1 A ∧ b ∈ list n2 A"
    from this and n' have "∃n1 n2. n' = n1 + n2 ∧ ls ∈ list n1 A ∧ b ∈ list n2 A" .
    then obtain n1 n2 where "n' = n1 + n2" "ls ∈ list n1 A" "b ∈ list n2 A" by fast
    with n have "?P (l#ls) n (n1+1) n2" by simp
    thus "∃n1 n2. ?P (l#ls) n n1 n2" by fastforce
  qed
  moreover
  assume "a@b ∈ list n A" "∧n1 n2. n=n1+n2 ==> a ∈ list n1 A ==> b ∈ list n2 A ==> P"
  ultimately
  show ?thesis by blast
qed
(*>*)


lemma listt_update_in_list [simp, intro!]:
  "[ xs ∈ list n A; x∈A ] ==> xs[i := x] ∈ list n A"
(*<*)
apply (unfold list_def)
apply simp
done
(*>*)

lemma list_appendI [intro?]:
  "[ a ∈ list n A; b ∈ list m A ] ==> a @ b ∈ list (n+m) A"
(*<*) by (unfold list_def) auto (*>*)

lemma list_map [simp]: "(map f xs ∈ list (size xs) A) = (f ` set xs ⊆ A)"
(*<*) by (unfold list_def) simp (*>*)

lemma list_replicateI [intro]: "x ∈ A ==> replicate n x ∈ list n A"
(*<*) by (induct n) auto (*>*)

lemma plus_list_Nil [simp]: "[] [⊔] xs = []"
(*<*)
apply (unfold plussub_def)
apply simp
done
(*>*)

lemma plus_list_Cons [simp]:
  "(x#xs) [⊔] ys = (case ys of [] ==> [] | y#ys ==> (x ⊔_f y)#(xs [⊔] ys))"
(*<*) by (simp add: plussub_def split: list.split) (*>*)

lemma length_plus_list [rule_format, simp]:
  "∀ys. size(xs [⊔] ys) = min(size xs) (size ys)"
(*<*)
apply (induct xs)
 apply simp
apply clarify
apply (simp (no_asm_simp) split: list.split)
done
(*>*)

lemma nth_plus_list [rule_format, simp]:
  "∀xs ys i. size xs = n ⟶ size ys = n ⟶ i<n ⟶ (xs [⊔] ys)!i = (xs!i) ⊔_f (ys!i)"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (case_tac xs)
 apply simp
apply (force simp add: nth_Cons split: list.split nat.split)
done
(*>*)


lemma (in Semilat) plus_list_ub1 [rule_format]:
 "[ set xs ⊆ A; set ys ⊆ A; size xs = size ys ]
  ==> xs [⊑] xs [⊔] ys"
(*<*)
apply (unfold unfold_lesub_list)
apply (simp add: Listn.le_def list_all2_conv_all_nth)
done
(*>*)

lemma (in Semilat) plus_list_ub2:
 "[set xs ⊆ A; set ys ⊆ A; size xs = size ys ] ==> ys [⊑] xs [⊔] ys"
(*<*)
apply (unfold unfold_lesub_list)
apply (simp add: Listn.le_def list_all2_conv_all_nth)
done
(*>*)

lemma (in Semilat) plus_list_lub [rule_format]:
shows "∀xs ys zs. set xs ⊆ A ⟶ set ys ⊆ A ⟶ set zs ⊆ A
  ⟶ size xs = n ∧ size ys = n ⟶
  xs [⊑] zs ∧ ys [⊑] zs ⟶ xs [⊔] ys [⊑] zs"
(*<*)
apply (unfold unfold_lesub_list)
apply (simp add: Listn.le_def list_all2_conv_all_nth)
done
(*>*)

lemma (in Semilat) list_update_incr [rule_format]:
 "x∈A ==> set xs ⊆ A ⟶
  (∀i. i<size xs ⟶ xs [⊑] xs[i := x ⊔_f xs!i])"
(*<*)
apply (unfold unfold_lesub_list)
apply (simp add: Listn.le_def list_all2_conv_all_nth)
apply (induct xs)
 apply simp
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp add: nth_Cons split: nat.split)
done
(*>*)

lemma acc_le_listI' [intro!]:
  "[ order r; acc A r ] ==> acc (∪n. list n A) (Listn.le r)"
(*<*)
apply (unfold acc_def)
apply (subgoal_tac
 "wf(UN n. {(ys,xs). xs ∈ list n A ∧ ys ∈ list n A ∧ xs <_(Listn.le r) ys})")
 apply (erule wf_subset)
 apply clarify
 apply(rule UN_I)
  prefer 2
  apply clarify
  apply(frule lesssub_lengthD)
  apply fastforce
 apply simp
apply (rule wf_UN)
 prefer 2
 apply (rename_tac m n)
 apply (case_tac "m=n")
  apply simp
 apply (clarsimp intro!: equals0I)
 apply (drule lesssub_lengthD)+
 apply simp
apply (induct_tac n)
 apply (simp add: lesssub_def cong: conj_cong)
apply (rename_tac k)
apply (simp add: wf_eq_minimal)
apply (simp (no_asm) add: in_list_Suc_iff cong: conj_cong)
apply clarify
apply (rename_tac M m)
apply (case_tac "∃x∈A. ∃xs∈list k A. x#xs ∈ M")
 prefer 2
 apply (erule thin_rl)
 apply (erule thin_rl)
 apply blast
apply (erule_tac x = "{a. a ∈ A ∧ (∃xs∈list k A. a#xs∈M)}" in allE)
apply (erule impE)
 apply blast
apply (thin_tac "∃x∈A. ∃xs∈list k A. P x xs" for P)
apply clarify
apply (rename_tac maxA xs)
apply (erule_tac x = "{ys. ys ∈ list k A ∧ maxA#ys ∈ M}" in allE)
apply (erule impE)
 apply blast
apply clarify
apply (thin_tac "m ∈ M")
apply (thin_tac "maxA#xs ∈ M")
apply (rule bexI)
 prefer 2
 apply assumption
apply clarify
apply simp
apply (erule disjE)
 prefer 2
 apply blast
by fastforce

lemma acc_le_listI [intro!]:
  "[ order r; acc A r ] ==> acc (list n A) (Listn.le r)"
apply(drule (1) acc_le_listI')
apply(erule thin_rl)
apply(unfold acc_def)
apply(erule wf_subset)
apply blast
done

lemma acc_le_list_uptoI [intro!]:
  "[ order r; acc A r ] ==> acc (∪{list n A|n. n ≤ mxs}) (Listn.le r)"
apply(drule (1) acc_le_listI')
apply(erule thin_rl)
apply(unfold acc_def)
apply(erule wf_subset)
apply blast
done

lemma closed_listI:
  "closed S f ==> closed (list n S) (map2 f)"
(*<*)
apply (unfold closed_def)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply simp
done
(*>*)


lemma Listn_sl_aux:
assumes "Semilat A r f" shows "semilat (Listn.sl n (A,r,f))"
(*<*)
proof -
  interpret Semilat A r f by fact
  show ?thesis
  apply (unfold Listn.sl_def)
  apply (simp (no_asm) only: semilat_Def split_conv)
  apply (rule conjI)
   apply simp
  apply (rule conjI)
   apply (simp only: closedI closed_listI)
  apply (simp (no_asm) only: list_def)
  apply (simp (no_asm_simp) add: plus_list_ub1 plus_list_ub2 plus_list_lub)
  done
qed
(*>*)

lemma Listn_sl: "semilat L ==> semilat (Listn.sl n L)"
(*<*) apply (cases L) apply simp
apply (drule Semilat.intro)
by (simp add: Listn_sl_aux split_tupled_all) (*>*)

lemma coalesce_in_err_list [rule_format]:
  "∀xes. xes ∈ list n (err A) ⟶ coalesce xes ∈ err(list n A)"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp (no_asm) add: plussub_def Err.sup_def lift2_def split: err.split)
apply force
done
(*>*)

lemma lem: "∧x xs. x ⊔_#) xs = x#xs"
(*<*) by (simp add: plussub_def) (*>*)

lemma coalesce_eq_OK1_D [rule_format]:
  "semilat(err A, Err.le r, lift2 f) ==>
  ∀xs. xs ∈ list n A ⟶ (∀ys. ys ∈ list n A ⟶
  (∀zs. coalesce (xs [⊔] ys) = OK zs ⟶ xs [⊑] zs))"
(*<*)
apply (induct n)
  apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
apply (force simp add: semilat_le_err_OK1)
done
(*>*)

lemma coalesce_eq_OK2_D [rule_format]:
  "semilat(err A, Err.le r, lift2 f) ==>
  ∀xs. xs ∈ list n A ⟶ (∀ys. ys ∈ list n A ⟶
  (∀zs. coalesce (xs [⊔] ys) = OK zs ⟶ ys [⊑] zs))"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
apply (force simp add: semilat_le_err_OK2)
done
(*>*)

lemma lift2_le_ub:
  "[ semilat(err A, Err.le r, lift2 f); x∈A; y∈A; x ⊔_f y = OK z;
      u∈A; x ⊑_r u; y ⊑_r u ] ==> z ⊑_r u"
(*<*)
apply (unfold semilat_Def plussub_def err_def')
apply (simp add: lift2_def)
apply clarify
apply (rotate_tac -3)
apply (erule thin_rl)
apply (erule thin_rl)
apply force
done
(*>*)

lemma coalesce_eq_OK_ub_D [rule_format]:
  "semilat(err A, Err.le r, lift2 f) ==>
  ∀xs. xs ∈ list n A ⟶ (∀ys. ys ∈ list n A ⟶
  (∀zs us. coalesce (xs [⊔] ys) = OK zs ∧ xs [⊑] us ∧ ys [⊑] us
           ∧ us ∈ list n A ⟶ zs [⊑] us))"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp (no_asm_use) split: err.split_asm add: lem Err.sup_def lift2_def)
apply clarify
apply (rule conjI)
 apply (blast intro: lift2_le_ub)
apply blast
done
(*>*)

lemma lift2_eq_ErrD:
  "[ x ⊔_f y = Err; semilat(err A, Err.le r, lift2 f); x∈A; y∈A ]
  ==> ¬(∃u∈A. x ⊑_r u ∧ y ⊑_r u)"
(*<*) by (simp add: OK_plus_OK_eq_Err_conv [THEN iffD1]) (*>*)


lemma coalesce_eq_Err_D [rule_format]:
  "[ semilat(err A, Err.le r, lift2 f) ]
  ==> ∀xs. xs ∈ list n A ⟶ (∀ys. ys ∈ list n A ⟶
      coalesce (xs [⊔] ys) = Err ⟶
      ¬(∃zs ∈ list n A. xs [⊑] zs ∧ ys [⊑] zs))"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
 apply (blast dest: lift2_eq_ErrD)
done
(*>*)

lemma closed_err_lift2_conv:
  "closed (err A) (lift2 f) = (∀x∈A. ∀y∈A. x ⊔_f y ∈ err A)"
(*<*)
apply (unfold closed_def)
apply (simp add: err_def')
done
(*>*)

lemma closed_map2_list [rule_format]:
  "closed (err A) (lift2 f) ==>
  ∀xs. xs ∈ list n A ⟶ (∀ys. ys ∈ list n A ⟶
  map2 f xs ys ∈ list n (err A))"
(*<*)
apply (induct n)
 apply simp
apply clarify
apply (simp add: in_list_Suc_iff)
apply clarify
apply (simp add: plussub_def closed_err_lift2_conv)
done
(*>*)

lemma closed_lift2_sup:
  "closed (err A) (lift2 f) ==>
  closed (err (list n A)) (lift2 (sup f))"
(*<*) by (fastforce  simp add: closed_def plussub_def sup_def lift2_def
                          coalesce_in_err_list closed_map2_list
                split: err.split) (*>*)

lemma err_semilat_sup:
  "err_semilat (A,r,f) ==>
  err_semilat (list n A, Listn.le r, sup f)"
(*<*)
apply (unfold Err.sl_def)
apply (simp only: split_conv)
apply (simp (no_asm) only: semilat_Def plussub_def)
apply (simp (no_asm_simp) only: Semilat.closedI [OF Semilat.intro] closed_lift2_sup)
apply (rule conjI)
 apply (drule Semilat.orderI [OF Semilat.intro])
 apply simp
apply (simp (no_asm) only: unfold_lesub_err Err.le_def err_def' sup_def lift2_def)
apply (simp (no_asm_simp) add: coalesce_eq_OK1_D coalesce_eq_OK2_D split: err.split)
apply (blast intro: coalesce_eq_OK_ub_D dest: coalesce_eq_Err_D)
done
(*>*)

lemma err_semilat_upto_esl:
  "∧L. err_semilat L ==> err_semilat(upto_esl m L)"
(*<*)
apply (unfold Listn.upto_esl_def)
apply (simp (no_asm_simp) only: split_tupled_all)
apply simp
apply (fastforce intro!: err_semilat_UnionI err_semilat_sup
                dest: lesub_list_impl_same_size
                simp add: plussub_def Listn.sup_def)
done
(*>*)

end