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(*
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 \ 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 \ nat) \ 'a list \ 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 = (\x\xs. T_f x) + length xs + 1"
by (induction xs) auto
lemmas [simp del] = T_map.simps
fun T_filter :: "('a \ nat) \ 'a list \ 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 = (\x\xs. T_p x) + length xs + 1"
by (induction xs) auto
lemmas [simp del] = T_filter.simps
fun T_nth :: "'a list \ nat \ nat" where
"T_nth [] n = 1"
| "T_nth (x # xs) n = (case n of 0 \ 1 | Suc n' \ 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 \ 'a list \ nat" where
"T_take n [] = 1"
| "T_take n (x # xs) = (case n of 0 \ 1 | Suc n' \ 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 \ 'a list \ nat" where
"T_drop n [] = 1"
| "T_drop n (x # xs) = (case n of 0 \ 1 | Suc n' \ 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
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