--- a/src/HOL/Data_Structures/Array_Braun.thy Thu Nov 05 19:09:11 2020 +0000
+++ b/src/HOL/Data_Structures/Array_Braun.thy Fri Nov 06 12:48:31 2020 +0100
@@ -444,12 +444,12 @@
definition brauns1 :: "'a list \<Rightarrow> 'a tree" where
"brauns1 xs = (if xs = [] then Leaf else brauns 0 xs ! 0)"
-fun t_brauns :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where
-"t_brauns k xs = (if xs = [] then 0 else
+fun T_brauns :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where
+"T_brauns k xs = (if xs = [] then 0 else
let ys = take (2^k) xs;
zs = drop (2^k) xs;
ts = brauns (k+1) zs
- in 4 * min (2^k) (length xs) + t_brauns (k+1) zs)"
+ in 4 * min (2^k) (length xs) + T_brauns (k+1) zs)"
paragraph "Functional correctness"
@@ -498,8 +498,8 @@
paragraph "Running Time Analysis"
-theorem t_brauns:
- "t_brauns k xs = 4 * length xs"
+theorem T_brauns:
+ "T_brauns k xs = 4 * length xs"
proof (induction xs arbitrary: k rule: measure_induct_rule[where f = length])
case (less xs)
show ?case
@@ -509,7 +509,7 @@
next
assume "xs \<noteq> []"
let ?zs = "drop (2^k) xs"
- have "t_brauns k xs = t_brauns (k+1) ?zs + 4 * min (2^k) (length xs)"
+ have "T_brauns k xs = T_brauns (k+1) ?zs + 4 * min (2^k) (length xs)"
using \<open>xs \<noteq> []\<close> by(simp)
also have "\<dots> = 4 * length ?zs + 4 * min (2^k) (length xs)"
using less[of ?zs "k+1"] \<open>xs \<noteq> []\<close>
@@ -555,10 +555,10 @@
definition list_fast :: "'a tree \<Rightarrow> 'a list" where
"list_fast t = list_fast_rec [t]"
-function t_list_fast_rec :: "'a tree list \<Rightarrow> nat" where
-"t_list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> Leaf) ts
+function T_list_fast_rec :: "'a tree list \<Rightarrow> nat" where
+"T_list_fast_rec ts = (let us = filter (\<lambda>t. t \<noteq> Leaf) ts
in length ts + (if us = [] then 0 else
- 5 * length us + t_list_fast_rec (map left us @ map right us)))"
+ 5 * length us + T_list_fast_rec (map left us @ map right us)))"
by (pat_completeness, auto)
termination
@@ -567,7 +567,7 @@
apply (simp add: list_fast_rec_term)
done
-declare t_list_fast_rec.simps[simp del]
+declare T_list_fast_rec.simps[simp del]
paragraph "Functional Correctness"
@@ -637,21 +637,21 @@
(\<Sum>t\<leftarrow>ts. k * size t) = (\<Sum>t \<leftarrow> map left ts @ map right ts. k * size t) + k * length ts"
by(induction ts)(auto simp add: neq_Leaf_iff algebra_simps)
-theorem t_list_fast_rec_ub:
- "t_list_fast_rec ts \<le> sum_list (map (\<lambda>t. 7*size t + 1) ts)"
+theorem T_list_fast_rec_ub:
+ "T_list_fast_rec ts \<le> sum_list (map (\<lambda>t. 7*size t + 1) ts)"
proof (induction ts rule: measure_induct_rule[where f="sum_list o map size"])
case (less ts)
let ?us = "filter (\<lambda>t. t \<noteq> Leaf) ts"
show ?case
proof cases
assume "?us = []"
- thus ?thesis using t_list_fast_rec.simps[of ts]
+ thus ?thesis using T_list_fast_rec.simps[of ts]
by(simp add: sum_list_Suc)
next
assume "?us \<noteq> []"
let ?children = "map left ?us @ map right ?us"
- have "t_list_fast_rec ts = t_list_fast_rec ?children + 5 * length ?us + length ts"
- using \<open>?us \<noteq> []\<close> t_list_fast_rec.simps[of ts] by(simp)
+ have "T_list_fast_rec ts = T_list_fast_rec ?children + 5 * length ?us + length ts"
+ using \<open>?us \<noteq> []\<close> T_list_fast_rec.simps[of ts] by(simp)
also have "\<dots> \<le> (\<Sum>t\<leftarrow>?children. 7 * size t + 1) + 5 * length ?us + length ts"
using less[of "?children"] list_fast_rec_term[of "?us"] \<open>?us \<noteq> []\<close>
by (simp)
@@ -667,4 +667,4 @@
qed
qed
-end
\ No newline at end of file
+end