# Theory AuxLemmas

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theory AuxLemmas
imports JBasis
begin

    (*  Title:      HOL/MicroJava/Comp/AuxLemmas.thy
        ID:         $Id: AuxLemmas.thy,v 1.5 2005/04/27 14:39:44 paulson Exp $
        Author:     Martin Strecker
    *)

    (* Auxiliary Lemmas *)

    theory AuxLemmas
    imports "../J/JBasis"
    begin

    (**********************************************************************)
    (* List.thy *)
    (**********************************************************************)

    lemma app_nth_greater_len [rule_format (no_asm), simp]:
     "∀ ind. length pre ≤ ind --> (pre @ a # post) ! (Suc ind) = (pre @ post) ! ind"
    apply (induct "pre")
    apply auto
    apply (case_tac ind)
    apply auto
    done

    lemma length_takeWhile: "v ∈ set xs ==> length (takeWhile (%z. z~=v) xs) < length xs"
    by (induct xs, auto)

    lemma nth_length_takeWhile [simp]:
      "v ∈ set xs ==> xs ! (length (takeWhile (%z. z~=v) xs)) = v"
    by (induct xs, auto)

    lemma map_list_update [simp]:
      "[| x ∈ set xs; distinct xs|] ==>
      (map f xs) [length (takeWhile (λz. z ≠ 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) o (λ (a,b). ((fa a), (fb b))) =
      (λ (a,b). (f (fa a) (fb b)))"
    by (simp add: split_def o_def)

    lemma split_iter: "(λ (a,b,c). ((g1 a), (g2 b), (g3 c))) =
            (λ (a,p). ((g1 a), (λ (b, c). ((g2 b), (g3 c))) p))"
    by (simp add: split_def o_def)

    (**********************************************************************)
    (* Set.thy *)
    (**********************************************************************)

    lemma singleton_in_set: "A = {a} ==> a ∈ A" by simp

    (**********************************************************************)
    (* Map.thy *)
    (**********************************************************************)

    lemma the_map_upd: "(the o f(x\<mapsto>v)) = (the o f)(x:=v)"
    by (simp add: expand_fun_eq)

    lemma map_of_in_set:
      "(map_of xs x = None) = (x ∉ set (map fst xs))"
    by (induct xs, auto)

    lemma map_map_upd [simp]:
      "y ∉ set xs ==> map (the o f(y\<mapsto>v)) xs = map (the o f) xs"
    by (simp add: the_map_upd)

    lemma map_map_upds [rule_format (no_asm), simp]:
    "∀ f vs. (∀y∈set ys. y ∉ set xs) --> map (the o f(ys[\<mapsto>]vs)) xs = map (the o f) xs"
    apply (induct xs)
     apply simp
    apply fastsimp
    done

    lemma map_upds_distinct [rule_format (no_asm), simp]:
      "∀ f vs. length ys = length vs --> distinct ys --> map (the o f(ys[\<mapsto>]vs)) ys = vs"
    apply (induct ys)
    apply simp
    apply (intro strip)
    apply (case_tac vs)
    apply simp
    apply (simp only: map_upds_Cons distinct.simps hd.simps tl.simps map.simps)
    apply clarify
    apply (simp del: o_apply)
    apply simp
    done

    lemma map_of_map_as_map_upd [rule_format (no_asm)]: "distinct (map f zs) -->
    map_of (map (λ 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)]:
      "∀ m ys. (m(xs[\<mapsto>]ys)) k = Some y --> k ∈ (set xs) ∨ (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
      ==> k ∈ (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) -->
      (∀ x ∈ set xs. (P1 x)) -->
      (∀ x. (P1 x) --> (P2 (f x))) -->
      (P2(k, v))"
    by (induct xs,auto)

    lemma map_of_map2: "∀ x ∈ set xs. (fst (f x)) = (fst x) ==>
      map_of (map f xs) a = option_map (λ 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

lemma app_nth_greater_len:

      length pre ≤ ind ==> (pre @ a # post) ! Suc ind = (pre @ post) ! ind

lemma length_takeWhile:

      v ∈ set xs ==> length (takeWhile (%z. z ≠ v) xs) < length xs

lemma nth_length_takeWhile:

      v ∈ set xs ==> xs ! length (takeWhile (%z. z ≠ v) xs) = v

lemma map_list_update:

      [| x ∈ set xs; distinct xs |]
      ==> map f xs[length (takeWhile (%z. z ≠ x) xs) := v] = map (f(x := v)) xs

lemma split_compose:

      (%(x, y). f x y) o (%(a, b). (fa a, fb b)) = (%(a, b). f (fa a) (fb b))

lemma split_iter:

      (%(a, b, c). (g1.0 a, g2.0 b, g3.0 c)) =
      (%(a, p). (g1.0 a, (%(b, c). (g2.0 b, g3.0 c)) p))

lemma singleton_in_set:

      A = {a} ==> a ∈ A

lemma the_map_upd:

      the o f(x |-> v) = (the o f)(x := v)

lemma map_of_in_set:

      (map_of xs x = None) = (x ∉ set (map fst xs))

lemma map_map_upd:

      y ∉ set xs ==> map (the o f(y |-> v)) xs = map (the o f) xs

lemma map_map_upds:

      ∀y∈set ys. y ∉ set xs ==> map (the o f(ys [|->] vs)) xs = map (the o f) xs

lemma map_upds_distinct:

      [| length ys = length vs; distinct ys |] ==> map (the o f(ys [|->] vs)) ys = vs

lemma map_of_map_as_map_upd:

      distinct (map f zs)
      ==> map_of (map (%p. (f p, g p)) zs) = [map f zs [|->] map g zs]

lemma map_upds_SomeD:

      (m(xs [|->] ys)) k = Some y ==> k ∈ set xs ∨ m k = Some y

lemma map_of_upds_SomeD:

      (map_of m(xs [|->] ys)) k = Some y ==> k ∈ set (xs @ map fst m)

lemma map_of_map_prop:

      [| map_of (map f xs) k = Some v; Ball (set xs) P1.0;
         ∀x. P1.0 x --> P2.0 (f x) |]
      ==> P2.0 (k, v)

lemma map_of_map2:

      ∀x∈set xs. fst (f x) = fst x
      ==> map_of (map f xs) a = option_map (%b. snd (f (a, b))) (map_of xs a)

lemma option_map_of:

      (option_map f (map_of xs k) = None) = (map_of xs k = None)