src/HOL/ex/Bubblesort.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 58644 8171ef293634 child 60515 484559628038 permissions -rw-r--r--
     1 (* Author: Tobias Nipkow *)

     2

     3 section {* Bubblesort *}

     4

     5 theory Bubblesort

     6 imports "~~/src/HOL/Library/Multiset"

     7 begin

     8

     9 text{* This is \emph{a} version of bubblesort. *}

    10

    11 context linorder

    12 begin

    13

    14 fun bubble_min where

    15 "bubble_min [] = []" |

    16 "bubble_min [x] = [x]" |

    17 "bubble_min (x#xs) =

    18   (case bubble_min xs of y#ys \<Rightarrow> if x>y then y#x#ys else x#y#ys)"

    19

    20 lemma size_bubble_min: "size(bubble_min xs) = size xs"

    21 by(induction xs rule: bubble_min.induct) (auto split: list.split)

    22

    23 lemma bubble_min_eq_Nil_iff[simp]: "bubble_min xs = [] \<longleftrightarrow> xs = []"

    24 by (metis length_0_conv size_bubble_min)

    25

    26 lemma bubble_minD_size: "bubble_min (xs) = ys \<Longrightarrow> size xs = size ys"

    27 by(auto simp: size_bubble_min)

    28

    29 function (sequential) bubblesort where

    30 "bubblesort []  = []" |

    31 "bubblesort [x] = [x]" |

    32 "bubblesort xs  = (case bubble_min xs of y#ys \<Rightarrow> y # bubblesort ys)"

    33 by pat_completeness auto

    34

    35 termination

    36 proof

    37   show "wf(measure size)" by simp

    38 next

    39   fix x1 x2 y :: 'a fix xs ys :: "'a list"

    40   show "bubble_min(x1#x2#xs) = y#ys \<Longrightarrow> (ys, x1#x2#xs) \<in> measure size"

    41     by(auto simp: size_bubble_min dest!: bubble_minD_size split: list.splits if_splits)

    42 qed

    43

    44 lemma mset_bubble_min: "multiset_of (bubble_min xs) = multiset_of xs"

    45 apply(induction xs rule: bubble_min.induct)

    46   apply simp

    47  apply simp

    48 apply (auto simp: add_eq_conv_ex split: list.split)

    49 done

    50

    51 lemma bubble_minD_mset:

    52   "bubble_min (xs) = ys \<Longrightarrow> multiset_of xs = multiset_of ys"

    53 by(auto simp: mset_bubble_min)

    54

    55 lemma mset_bubblesort:

    56   "multiset_of (bubblesort xs) = multiset_of xs"

    57 apply(induction xs rule: bubblesort.induct)

    58   apply simp

    59  apply simp

    60 by(auto split: list.splits if_splits dest: bubble_minD_mset)

    61   (metis add_eq_conv_ex mset_bubble_min multiset_of.simps(2))

    62

    63 lemma set_bubblesort: "set (bubblesort xs) = set xs"

    64 by(rule mset_bubblesort[THEN multiset_of_eq_setD])

    65

    66 lemma bubble_min_min: "bubble_min xs = y#ys \<Longrightarrow> z \<in> set ys \<Longrightarrow> y \<le> z"

    67 apply(induction xs arbitrary: y ys z rule: bubble_min.induct)

    68   apply simp

    69  apply simp

    70 apply (fastforce split: list.splits if_splits dest!: sym[of "a#b" for a b])

    71 done

    72

    73 lemma sorted_bubblesort: "sorted(bubblesort xs)"

    74 apply(induction xs rule: bubblesort.induct)

    75   apply simp

    76  apply simp

    77 apply (fastforce simp: set_bubblesort split: list.split if_splits

    78   intro!: sorted.Cons dest: bubble_min_min)

    79 done

    80

    81 end

    82

    83 end