src/HOL/Data_Structures/Time_Funs.thy

+(*
+  File:    Data_Structures/Time_Functions.thy
+  Author:  Manuel Eberl, TU München
+*)
+section \<open>Time functions for various standard library operations\<close>
+theory Time_Funs
+  imports Main
+begin
+
+fun T_length :: "'a list \<Rightarrow> nat" where
+  "T_length [] = 1"
+| "T_length (x # xs) = T_length xs + 1"
+
+lemma T_length_eq: "T_length xs = length xs + 1"
+  by (induction xs) auto
+
+lemmas [simp del] = T_length.simps
+
+
+fun T_map  :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat" where
+  "T_map T_f [] = 1"
+| "T_map T_f (x # xs) = T_f x + T_map T_f xs + 1"
+
+lemma T_map_eq: "T_map T_f xs = (\<Sum>x\<leftarrow>xs. T_f x) + length xs + 1"
+  by (induction xs) auto
+
+lemmas [simp del] = T_map.simps
+
+
+fun T_filter  :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat" where
+  "T_filter T_p [] = 1"
+| "T_filter T_p (x # xs) = T_p x + T_filter T_p xs + 1"
+
+lemma T_filter_eq: "T_filter T_p xs = (\<Sum>x\<leftarrow>xs. T_p x) + length xs + 1"
+  by (induction xs) auto
+
+lemmas [simp del] = T_filter.simps
+
+
+fun T_nth :: "'a list \<Rightarrow> nat \<Rightarrow> nat" where
+  "T_nth [] n = 1"
+| "T_nth (x # xs) n = (case n of 0 \<Rightarrow> 1 | Suc n' \<Rightarrow> T_nth xs n' + 1)"
+
+lemma T_nth_eq: "T_nth xs n = min n (length xs) + 1"
+  by (induction xs n rule: T_nth.induct) (auto split: nat.splits)
+
+lemmas [simp del] = T_nth.simps
+
+
+fun T_take :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where
+  "T_take n [] = 1"
+| "T_take n (x # xs) = (case n of 0 \<Rightarrow> 1 | Suc n' \<Rightarrow> T_take n' xs + 1)"
+
+lemma T_take_eq: "T_take n xs = min n (length xs) + 1"
+  by (induction xs arbitrary: n) (auto split: nat.splits)
+
+fun T_drop :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where
+  "T_drop n [] = 1"
+| "T_drop n (x # xs) = (case n of 0 \<Rightarrow> 1 | Suc n' \<Rightarrow> T_drop n' xs + 1)"
+
+lemma T_drop_eq: "T_drop n xs = min n (length xs) + 1"
+  by (induction xs arbitrary: n) (auto split: nat.splits)
+  
+  
+end