explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
tuned;
(* Title: HOL/MicroJava/Comp/AuxLemmas.thy
Author: Martin Strecker
*)
(* Auxiliary Lemmas *)
theory AuxLemmas
imports "../J/JBasis"
begin
(**********************************************************************)
(* List.thy *)
(**********************************************************************)
lemma app_nth_greater_len [rule_format (no_asm), simp]:
"\<forall> ind. length pre \<le> ind \<longrightarrow> (pre @ a # post) ! (Suc ind) = (pre @ post) ! ind"
apply (induct "pre")
apply auto
apply (case_tac ind)
apply auto
done
lemma length_takeWhile: "v \<in> set xs \<Longrightarrow> length (takeWhile (%z. z~=v) xs) < length xs"
by (induct xs, auto)
lemma nth_length_takeWhile [simp]:
"v \<in> set xs \<Longrightarrow> xs ! (length (takeWhile (%z. z~=v) xs)) = v"
by (induct xs, auto)
lemma map_list_update [simp]:
"\<lbrakk> x \<in> set xs; distinct xs\<rbrakk> \<Longrightarrow>
(map f xs) [length (takeWhile (\<lambda>z. z \<noteq> x) xs) := v] =
map (f(x:=v)) xs"
apply (induct xs)
apply simp
apply (case_tac "x=a")
apply auto
done
(**********************************************************************)
(* Product_Type.thy *)
(**********************************************************************)
lemma split_compose: "(split f) \<circ> (\<lambda> (a,b). ((fa a), (fb b))) =
(\<lambda> (a,b). (f (fa a) (fb b)))"
by (simp add: split_def o_def)
lemma split_iter: "(\<lambda> (a,b,c). ((g1 a), (g2 b), (g3 c))) =
(\<lambda> (a,p). ((g1 a), (\<lambda> (b, c). ((g2 b), (g3 c))) p))"
by (simp add: split_def o_def)
(**********************************************************************)
(* Set.thy *)
(**********************************************************************)
lemma singleton_in_set: "A = {a} \<Longrightarrow> a \<in> A" by simp
(**********************************************************************)
(* Map.thy *)
(**********************************************************************)
lemma the_map_upd: "(the \<circ> f(x\<mapsto>v)) = (the \<circ> f)(x:=v)"
by (simp add: fun_eq_iff)
lemma map_of_in_set:
"(map_of xs x = None) = (x \<notin> set (map fst xs))"
by (induct xs, auto)
lemma map_map_upd [simp]:
"y \<notin> set xs \<Longrightarrow> map (the \<circ> f(y\<mapsto>v)) xs = map (the \<circ> f) xs"
by (simp add: the_map_upd)
lemma map_map_upds [simp]:
"(\<forall>y\<in>set ys. y \<notin> set xs) \<Longrightarrow> map (the \<circ> f(ys[\<mapsto>]vs)) xs = map (the \<circ> f) xs"
apply (induct xs arbitrary: f vs)
apply simp
apply fastforce
done
lemma map_upds_distinct [simp]:
"distinct ys \<Longrightarrow> length ys = length vs \<Longrightarrow> map (the \<circ> f(ys[\<mapsto>]vs)) ys = vs"
apply (induct ys arbitrary: f vs)
apply simp
apply (case_tac vs)
apply simp_all
done
lemma map_of_map_as_map_upd:
"distinct (map f zs) \<Longrightarrow> map_of (map (\<lambda> p. (f p, g p)) zs) = empty (map f zs [\<mapsto>] map g zs)"
by (induct zs) auto
(* In analogy to Map.map_of_SomeD *)
lemma map_upds_SomeD [rule_format (no_asm)]:
"\<forall> m ys. (m(xs[\<mapsto>]ys)) k = Some y \<longrightarrow> k \<in> (set xs) \<or> (m k = Some y)"
apply (induct xs)
apply simp
apply auto
apply(case_tac ys)
apply auto
done
lemma map_of_upds_SomeD: "(map_of m (xs[\<mapsto>]ys)) k = Some y
\<Longrightarrow> k \<in> (set (xs @ map fst m))"
by (auto dest: map_upds_SomeD map_of_SomeD fst_in_set_lemma)
lemma map_of_map_prop [rule_format (no_asm)]:
"(map_of (map f xs) k = Some v) \<longrightarrow>
(\<forall> x \<in> set xs. (P1 x)) \<longrightarrow>
(\<forall> x. (P1 x) \<longrightarrow> (P2 (f x))) \<longrightarrow>
(P2(k, v))"
by (induct xs,auto)
lemma map_of_map2: "\<forall> x \<in> set xs. (fst (f x)) = (fst x) \<Longrightarrow>
map_of (map f xs) a = Option.map (\<lambda> b. (snd (f (a, b)))) (map_of xs a)"
by (induct xs, auto)
lemma option_map_of [simp]: "(Option.map f (map_of xs k) = None) = ((map_of xs k) = None)"
by (simp add: Option.map_def split: option.split)
end