# Theory List_Ins_Del

theory List_Ins_Del
imports Sorted_Less
```(* Author: Tobias Nipkow *)

section ‹List Insertion and Deletion›

theory List_Ins_Del
imports Sorted_Less
begin

subsection ‹Elements in a list›

lemma sorted_Cons_iff:
"sorted(x # xs) = ((∀y ∈ set xs. x < y) ∧ sorted xs)"

lemma sorted_snoc_iff:
"sorted(xs @ [x]) = (sorted xs ∧ (∀y ∈ set xs. y < x))"
(*
text‹The above two rules introduce quantifiers. It turns out
that in practice this is not a problem because of the simplicity of
the "isin" functions that implement @{const set}. Nevertheless
it is possible to avoid the quantifiers with the help of some rewrite rules:›

lemma sorted_ConsD: "sorted (y # xs) ⟹ x ≤ y ⟹ x ∉ set xs"
by (auto simp: sorted_Cons_iff)

lemma sorted_snocD: "sorted (xs @ [y]) ⟹ y ≤ x ⟹ x ∉ set xs"
by (auto simp: sorted_snoc_iff)

lemmas isin_simps2 = sorted_lems sorted_ConsD sorted_snocD
*)

lemmas isin_simps = sorted_lems sorted_Cons_iff sorted_snoc_iff

subsection ‹Inserting into an ordered list without duplicates:›

fun ins_list :: "'a::linorder ⇒ 'a list ⇒ 'a list" where
"ins_list x [] = [x]" |
"ins_list x (a#xs) =
(if x < a then x#a#xs else if x=a then a#xs else a # ins_list x xs)"

lemma set_ins_list: "set (ins_list x xs) = insert x (set xs)"
by(induction xs) auto

lemma distinct_if_sorted: "sorted xs ⟹ distinct xs"
apply(induction xs rule: induct_list012)
apply auto
by (metis in_set_conv_decomp_first less_imp_not_less sorted_mid_iff2)

lemma sorted_ins_list: "sorted xs ⟹ sorted(ins_list x xs)"
by(induction xs rule: induct_list012) auto

lemma ins_list_sorted: "sorted (xs @ [a]) ⟹
ins_list x (xs @ a # ys) =
(if x < a then ins_list x xs @ (a#ys) else xs @ ins_list x (a#ys))"
by(induction xs) (auto simp: sorted_lems)

text‹In principle, @{thm ins_list_sorted} suffices, but the following two
corollaries speed up proofs.›

corollary ins_list_sorted1: "sorted (xs @ [a]) ⟹ a ≤ x ⟹
ins_list x (xs @ a # ys) = xs @ ins_list x (a#ys)"

corollary ins_list_sorted2: "sorted (xs @ [a]) ⟹ x < a ⟹
ins_list x (xs @ a # ys) = ins_list x xs @ (a#ys)"
by(auto simp: ins_list_sorted)

lemmas ins_list_simps = sorted_lems ins_list_sorted1 ins_list_sorted2

text‹Splay trees need two additional @{const ins_list} lemmas:›

lemma ins_list_Cons: "sorted (x # xs) ⟹ ins_list x xs = x # xs"
by (induction xs) auto

lemma ins_list_snoc: "sorted (xs @ [x]) ⟹ ins_list x xs = xs @ [x]"
by(induction xs) (auto simp add: sorted_mid_iff2)

subsection ‹Delete one occurrence of an element from a list:›

fun del_list :: "'a ⇒ 'a list ⇒ 'a list" where
"del_list x [] = []" |
"del_list x (a#xs) = (if x=a then xs else a # del_list x xs)"

lemma del_list_idem: "x ∉ set xs ⟹ del_list x xs = xs"
by (induct xs) simp_all

lemma set_del_list_eq:
"distinct xs ⟹ set (del_list x xs) = set xs - {x}"
by(induct xs) auto

lemma sorted_del_list: "sorted xs ⟹ sorted(del_list x xs)"
apply(induction xs rule: induct_list012)
apply auto
by (meson order.strict_trans sorted_Cons_iff)

lemma del_list_sorted: "sorted (xs @ a # ys) ⟹
del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
by(induction xs)
(fastforce simp: sorted_lems sorted_Cons_iff intro!: del_list_idem)+

text‹In principle, @{thm del_list_sorted} suffices, but the following
corollaries speed up proofs.›

corollary del_list_sorted1: "sorted (xs @ a # ys) ⟹ a ≤ x ⟹
del_list x (xs @ a # ys) = xs @ del_list x (a # ys)"
by (auto simp: del_list_sorted)

corollary del_list_sorted2: "sorted (xs @ a # ys) ⟹ x < a ⟹
del_list x (xs @ a # ys) = del_list x xs @ a # ys"
by (auto simp: del_list_sorted)

corollary del_list_sorted3:
"sorted (xs @ a # ys @ b # zs) ⟹ x < b ⟹
del_list x (xs @ a # ys @ b # zs) = del_list x (xs @ a # ys) @ b # zs"
by (auto simp: del_list_sorted sorted_lems)

corollary del_list_sorted4:
"sorted (xs @ a # ys @ b # zs @ c # us) ⟹ x < c ⟹
del_list x (xs @ a # ys @ b # zs @ c # us) = del_list x (xs @ a # ys @ b # zs) @ c # us"
by (auto simp: del_list_sorted sorted_lems)

corollary del_list_sorted5:
"sorted (xs @ a # ys @ b # zs @ c # us @ d # vs) ⟹ x < d ⟹
del_list x (xs @ a # ys @ b # zs @ c # us @ d # vs) =
del_list x (xs @ a # ys @ b # zs @ c # us) @ d # vs"
by (auto simp: del_list_sorted sorted_lems)

lemmas del_list_simps = sorted_lems
del_list_sorted1
del_list_sorted2
del_list_sorted3
del_list_sorted4
del_list_sorted5

text‹Splay trees need two additional @{const del_list} lemmas:›

lemma del_list_notin_Cons: "sorted (x # xs) ⟹ del_list x xs = xs"
by(induction xs)(fastforce simp: sorted_Cons_iff)+

lemma del_list_sorted_app:
"sorted(xs @ [x]) ⟹ del_list x (xs @ ys) = xs @ del_list x ys"
by (induction xs) (auto simp: sorted_mid_iff2)

end
```