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Datei: Subarray.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/Imperative_HOL/ex/Subarray.thy
    Author:     Lukas Bulwahn, TU Muenchen
*)

section \<open>Theorems about sub arrays\<close>

theory Subarray
imports "../Array" List_Sublist
begin

definition subarray :: "nat \ nat \ ('a::heap) array \ heap \ 'a list" where
  "subarray n m a h \ nths' n m (Array.get h a)"

lemma subarray_upd: "i \ m \ subarray n m a (Array.update a i v h) = subarray n m a h"
apply (simp add: subarray_def Array.update_def)
apply (simp add: nths'_update1)
done

lemma subarray_upd2: " i < n \ subarray n m a (Array.update a i v h) = subarray n m a h"
apply (simp add: subarray_def Array.update_def)
apply (subst nths'_update2)
apply fastforce
apply simp
done

lemma subarray_upd3: "\ n \ i; i < m\ \ subarray n m a (Array.update a i v h) = (subarray n m a h)[i - n := v]"
unfolding subarray_def Array.update_def
by (simp add: nths'_update3)

lemma subarray_Nil: "n \ m \ subarray n m a h = []"
by (simp add: subarray_def nths'_Nil')

lemma subarray_single: "\ n < Array.length h a \ \ subarray n (Suc n) a h = [Array.get h a ! n]"
by (simp add: subarray_def length_def nths'_single)

lemma length_subarray: "m \ Array.length h a \ List.length (subarray n m a h) = m - n"
by (simp add: subarray_def length_def length_nths')

lemma length_subarray_0: "m \ Array.length h a \ List.length (subarray 0 m a h) = m"
by (simp add: length_subarray)

lemma subarray_nth_array_Cons: "\ i < Array.length h a; i < j \ \ (Array.get h a ! i) # subarray (Suc i) j a h = subarray i j a h"
unfolding Array.length_def subarray_def
by (simp add: nths'_front)

lemma subarray_nth_array_back: "\ i < j; j \ Array.length h a\ \ subarray i j a h = subarray i (j - 1) a h @ [Array.get h a ! (j - 1)]"
unfolding Array.length_def subarray_def
by (simp add: nths'_back)

lemma subarray_append: "\ i \ j; j \ k \ \ subarray i j a h @ subarray j k a h = subarray i k a h"
unfolding subarray_def
by (simp add: nths'_append)

lemma subarray_all: "subarray 0 (Array.length h a) a h = Array.get h a"
unfolding Array.length_def subarray_def
by (simp add: nths'_all)

lemma nth_subarray: "\ k < j - i; j \ Array.length h a \ \ subarray i j a h ! k = Array.get h a ! (i + k)"
unfolding Array.length_def subarray_def
by (simp add: nth_nths')

lemma subarray_eq_samelength_iff: "Array.length h a = Array.length h' a \ (subarray i j a h = subarray i j a h') = (\i'. i \ i' \ i' < j \ Array.get h a ! i' = Array.get h' a ! i')"
unfolding Array.length_def subarray_def by (rule nths'_eq_samelength_iff)

lemma all_in_set_subarray_conv: "(\j. j \ set (subarray l r a h) \ P j) = (\k. l \ k \ k < r \ k < Array.length h a \ P (Array.get h a ! k))"
unfolding subarray_def Array.length_def by (rule all_in_set_nths'_conv)

lemma ball_in_set_subarray_conv: "(\j \ set (subarray l r a h). P j) = (\k. l \ k \ k < r \ k < Array.length h a \ P (Array.get h a ! k))"
unfolding subarray_def Array.length_def by (rule ball_in_set_nths'_conv)

end

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