src/HOL/Data_Structures/AList_Upd_Del.thy

New subdirectory for functional data structures

(* Author: Tobias Nipkow *)

section {* Association List Update and Deletion *}

theory AList_Upd_Del
imports Sorted_Less
begin

abbreviation "sorted1 ps \<equiv> sorted(map fst ps)"

text{* Define own @{text map_of} function to avoid pulling in an unknown
amount of lemmas implicitly (via the simpset). *}

hide_const (open) map_of

fun map_of :: "('a*'b)list \<Rightarrow> 'a \<Rightarrow> 'b option" where
"map_of [] = (\<lambda>a. None)" |
"map_of ((x,y)#ps) = (\<lambda>a. if x=a then Some y else map_of ps a)"

text \<open>Updating into an association list:\<close>

fun upd_list :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) list \<Rightarrow> ('a*'b) list" where
"upd_list a b [] = [(a,b)]" |
"upd_list a b ((x,y)#ps) =
  (if a < x then (a,b)#(x,y)#ps else
  if a=x then (a,b)#ps else (x,y) # upd_list a b ps)"

fun del_list :: "'a::linorder \<Rightarrow> ('a*'b)list \<Rightarrow> ('a*'b)list" where
"del_list a [] = []" |
"del_list a ((x,y)#ps) = (if a=x then ps else (x,y) # del_list a ps)"


subsection \<open>Lemmas for @{const map_of}\<close>

lemma map_of_ins_list: "map_of (upd_list a b ps) = (map_of ps)(a := Some b)"
by(induction ps) auto

lemma map_of_append: "map_of (ps @ qs) a =
  (case map_of ps a of None \<Rightarrow> map_of qs a | Some b \<Rightarrow> Some b)"
by(induction ps)(auto)

lemma map_of_None: "sorted (a # map fst ps) \<Longrightarrow> map_of ps a = None"
by (induction ps) (auto simp: sorted_lems sorted_Cons_iff)

lemma map_of_None2: "sorted (map fst ps @ [a]) \<Longrightarrow> map_of ps a = None"
by (induction ps) (auto simp: sorted_lems)

lemma map_of_del_list: "sorted1 ps \<Longrightarrow>
  map_of(del_list a ps) = (map_of ps)(a := None)"
by(induction ps) (auto simp: map_of_None sorted_lems fun_eq_iff)

lemma map_of_sorted_Cons: "sorted (a # map fst ps) \<Longrightarrow> x < a \<Longrightarrow>
   map_of ps x = None"
by (meson less_trans map_of_None sorted_Cons_iff)

lemma map_of_sorted_snoc: "sorted (map fst ps @ [a]) \<Longrightarrow> a \<le> x \<Longrightarrow>
  map_of ps x = None"
by (meson le_less_trans map_of_None2 not_less sorted_snoc_iff)

lemmas map_of_sorteds = map_of_sorted_Cons map_of_sorted_snoc


subsection \<open>Lemmas for @{const upd_list}\<close>

lemma sorted_upd_list: "sorted1 ps \<Longrightarrow> sorted1 (upd_list a b ps)"
apply(induction ps) 
 apply simp
apply(case_tac ps)
 apply auto
done

lemma upd_list_sorted1: "\<lbrakk> sorted (map fst ps @ [x]); a < x \<rbrakk> \<Longrightarrow>
  upd_list a b (ps @ (x,y) # qs) =  upd_list a b ps @ (x,y) # qs"
by(induction ps) (auto simp: sorted_lems)

lemma upd_list_sorted2: "\<lbrakk> sorted (map fst ps @ [x]); x \<le> a \<rbrakk> \<Longrightarrow>
  upd_list a b (ps @ (x,y) # qs) = ps @ upd_list a b ((x,y)#qs)"
by(induction ps) (auto simp: sorted_lems)

lemmas upd_list_sorteds = upd_list_sorted1 upd_list_sorted2

(*
lemma set_ins_list[simp]: "set (ins_list x xs) = insert x (set xs)"
by(induction xs) auto

lemma distinct_if_sorted: "sorted xs \<Longrightarrow> distinct xs"
apply(induction xs rule: sorted.induct)
apply auto
by (metis in_set_conv_decomp_first less_imp_not_less sorted_mid_iff2)

lemma set_del_list_eq [simp]: "distinct xs ==> set(del_list x xs) = set xs - {x}"
apply(induct xs)
 apply simp
apply simp
apply blast
done
*)


subsection \<open>Lemmas for @{const del_list}\<close>

lemma sorted_del_list: "sorted1 ps \<Longrightarrow> sorted1 (del_list x ps)"
apply(induction ps)
 apply simp
apply(case_tac ps)
apply auto
by (meson order.strict_trans sorted_Cons_iff)

lemma del_list_idem: "x \<notin> set(map fst xs) \<Longrightarrow> del_list x xs = xs"
by (induct xs) auto

lemma del_list_sorted1: "sorted1 (xs @ [(x,y)]) \<Longrightarrow> x \<le> a \<Longrightarrow>
  del_list a (xs @ (x,y) # ys) = xs @ del_list a ((x,y) # ys)"
by (induction xs) (auto simp: sorted_mid_iff2)

lemma del_list_sorted2: "sorted1 (xs @ (x,y) # ys) \<Longrightarrow> a < x \<Longrightarrow>
  del_list a (xs @ (x,y) # ys) = del_list a xs @ (x,y) # ys"
by (induction xs) (fastforce simp: sorted_Cons_iff intro!: del_list_idem)+

lemma del_list_sorted3:
  "sorted1 (xs @ (x,x') # ys @ (y,y') # zs) \<Longrightarrow> a < y \<Longrightarrow>
  del_list a (xs @ (x,x') # ys @ (y,y') # zs) = del_list a (xs @ (x,x') # ys) @ (y,y') # zs"
by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted2 ball_Un)

lemma del_list_sorted4:
  "sorted1 (xs @ (x,x') # ys @ (y,y') # zs @ (z,z') # us) \<Longrightarrow> a < z \<Longrightarrow>
  del_list a (xs @ (x,x') # ys @ (y,y') # zs @ (z,z') # us) = del_list a (xs @ (x,x') # ys @ (y,y') # zs) @ (z,z') # us"
by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted3)

lemma del_list_sorted5:
  "sorted1 (xs @ (x,x') # ys @ (y,y') # zs @ (z,z') # us @ (u,u') # vs) \<Longrightarrow> a < u \<Longrightarrow>
   del_list a (xs @ (x,x') # ys @ (y,y') # zs @ (z,z') # us @ (u,u') # vs) =
   del_list a (xs @ (x,x') # ys @ (y,y') # zs @ (z,z') # us) @ (u,u') # vs" 
by (induction xs) (auto simp: sorted_Cons_iff del_list_sorted4)

lemmas del_list_sorted =
  del_list_sorted1 del_list_sorted2 del_list_sorted3 del_list_sorted4 del_list_sorted5

end