|
72642
|
1 |
theory Reverse
|
|
|
2 |
imports Main
|
|
|
3 |
begin
|
|
|
4 |
|
|
|
5 |
fun T_append :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
|
|
|
6 |
"T_append [] ys = 1" |
|
|
|
7 |
"T_append (x#xs) ys = T_append xs ys + 1"
|
|
|
8 |
|
|
|
9 |
fun T_rev :: "'a list \<Rightarrow> nat" where
|
|
|
10 |
"T_rev [] = 1" |
|
|
|
11 |
"T_rev (x#xs) = T_rev xs + T_append (rev xs) [x] + 1"
|
|
|
12 |
|
|
|
13 |
lemma T_append: "T_append xs ys = length xs + 1"
|
|
|
14 |
by(induction xs) auto
|
|
|
15 |
|
|
|
16 |
lemma T_rev: "T_rev xs \<le> (length xs + 1)^2"
|
|
|
17 |
by(induction xs) (auto simp: T_append power2_eq_square)
|
|
|
18 |
|
|
|
19 |
fun itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
|
|
|
20 |
"itrev [] ys = ys" |
|
|
|
21 |
"itrev (x#xs) ys = itrev xs (x # ys)"
|
|
|
22 |
|
|
|
23 |
lemma itrev: "itrev xs ys = rev xs @ ys"
|
|
|
24 |
by(induction xs arbitrary: ys) auto
|
|
|
25 |
|
|
|
26 |
lemma itrev_Nil: "itrev xs [] = rev xs"
|
|
|
27 |
by(simp add: itrev)
|
|
|
28 |
|
|
|
29 |
fun T_itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
|
|
|
30 |
"T_itrev [] ys = 1" |
|
|
|
31 |
"T_itrev (x#xs) ys = T_itrev xs (x # ys) + 1"
|
|
|
32 |
|
|
|
33 |
lemma T_itrev: "T_itrev xs ys = length xs + 1"
|
|
|
34 |
by(induction xs arbitrary: ys) auto
|
|
|
35 |
|
|
|
36 |
end |