src/HOL/Data_Structures/Reverse.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Reverse.thy	Wed Nov 18 19:38:19 2020 +0100
@@ -0,0 +1,36 @@
+theory Reverse
+imports Main
+begin
+
+fun T_append :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
+"T_append [] ys = 1" |
+"T_append (x#xs) ys = T_append xs ys + 1"
+
+fun T_rev :: "'a list \<Rightarrow> nat" where
+"T_rev [] = 1" |
+"T_rev (x#xs) = T_rev xs + T_append (rev xs) [x] + 1"
+
+lemma T_append: "T_append xs ys = length xs + 1"
+by(induction xs) auto
+
+lemma T_rev: "T_rev xs \<le> (length xs + 1)^2"
+by(induction xs) (auto simp: T_append power2_eq_square)
+
+fun itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+"itrev [] ys = ys" |
+"itrev (x#xs) ys = itrev xs (x # ys)"
+
+lemma itrev: "itrev xs ys = rev xs @ ys"
+by(induction xs arbitrary: ys) auto
+
+lemma itrev_Nil: "itrev xs [] = rev xs"
+by(simp add: itrev)
+
+fun T_itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
+"T_itrev [] ys = 1" |
+"T_itrev (x#xs) ys = T_itrev xs (x # ys) + 1"
+
+lemma T_itrev: "T_itrev xs ys = length xs + 1"
+by(induction xs arbitrary: ys) auto
+
+end
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