author | nipkow |

Sun May 13 14:32:48 2018 +0200 (12 months ago) | |

changeset 68161 | 2053ff42214b |

parent 68160 | efce008331f6 |

child 68162 | 61878d2aa6c7 |

tuned

1.1 --- a/src/HOL/Data_Structures/Sorting.thy Sun May 13 13:43:34 2018 +0200 1.2 +++ b/src/HOL/Data_Structures/Sorting.thy Sun May 13 14:32:48 2018 +0200 1.3 @@ -292,23 +292,23 @@ 1.4 1.5 fun c_merge_adj :: "('a::linorder) list list \<Rightarrow> nat" where 1.6 "c_merge_adj [] = 0" | 1.7 -"c_merge_adj [x] = 0" | 1.8 -"c_merge_adj (x # y # zs) = c_merge x y + c_merge_adj zs" 1.9 +"c_merge_adj [xs] = 0" | 1.10 +"c_merge_adj (xs # ys # zss) = c_merge xs ys + c_merge_adj zss" 1.11 1.12 fun c_merge_all :: "('a::linorder) list list \<Rightarrow> nat" where 1.13 "c_merge_all [] = undefined" | 1.14 -"c_merge_all [x] = 0" | 1.15 -"c_merge_all xs = c_merge_adj xs + c_merge_all (merge_adj xs)" 1.16 +"c_merge_all [xs] = 0" | 1.17 +"c_merge_all xss = c_merge_adj xss + c_merge_all (merge_adj xss)" 1.18 1.19 definition c_msort_bu :: "('a::linorder) list \<Rightarrow> nat" where 1.20 "c_msort_bu xs = (if xs = [] then 0 else c_merge_all (map (\<lambda>x. [x]) xs))" 1.21 1.22 lemma length_merge_adj: 1.23 - "\<lbrakk> even(length xs); \<forall>x \<in> set xs. length x = m \<rbrakk> \<Longrightarrow> \<forall>x \<in> set (merge_adj xs). length x = 2*m" 1.24 -by(induction xs rule: merge_adj.induct) (auto simp: length_merge) 1.25 + "\<lbrakk> even(length xss); \<forall>x \<in> set xss. length x = m \<rbrakk> \<Longrightarrow> \<forall>xs \<in> set (merge_adj xss). length xs = 2*m" 1.26 +by(induction xss rule: merge_adj.induct) (auto simp: length_merge) 1.27 1.28 -lemma c_merge_adj: "\<forall>x \<in> set xs. length x = m \<Longrightarrow> c_merge_adj xs \<le> m * length xs" 1.29 -proof(induction xs rule: c_merge_adj.induct) 1.30 +lemma c_merge_adj: "\<forall>xs \<in> set xss. length xs = m \<Longrightarrow> c_merge_adj xss \<le> m * length xss" 1.31 +proof(induction xss rule: c_merge_adj.induct) 1.32 case 1 thus ?case by simp 1.33 next 1.34 case 2 thus ?case by simp 1.35 @@ -316,9 +316,9 @@ 1.36 case (3 x y) thus ?case using c_merge_ub[of x y] by (simp add: algebra_simps) 1.37 qed 1.38 1.39 -lemma c_merge_all: "\<lbrakk> \<forall>x \<in> set xs. length x = m; length xs = 2^k \<rbrakk> 1.40 - \<Longrightarrow> c_merge_all xs \<le> m * k * 2^k" 1.41 -proof (induction xs arbitrary: k m rule: c_merge_all.induct) 1.42 +lemma c_merge_all: "\<lbrakk> \<forall>xs \<in> set xss. length xs = m; length xss = 2^k \<rbrakk> 1.43 + \<Longrightarrow> c_merge_all xss \<le> m * k * 2^k" 1.44 +proof (induction xss arbitrary: k m rule: c_merge_all.induct) 1.45 case 1 thus ?case by simp 1.46 next 1.47 case 2 thus ?case by simp