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src/HOL/Library/List_Prefix.thy

author | wenzelm |

Wed Nov 22 21:47:04 2000 +0100 (2000-11-22) | |

changeset 10512 | d34192966cd8 |

parent 10408 | d8b3613158b1 |

child 10870 | 9444e3cf37e1 |

permissions | -rw-r--r-- |

tuned;

1 (* Title: HOL/Library/List_Prefix.thy

2 ID: $Id$

3 Author: Tobias Nipkow and Markus Wenzel, TU Muenchen

4 *)

6 header {*

7 \title{List prefixes}

8 \author{Tobias Nipkow and Markus Wenzel}

9 *}

11 theory List_Prefix = Main:

13 subsection {* Prefix order on lists *}

15 instance list :: ("term") ord ..

17 defs (overloaded)

18 prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs"

19 strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)"

21 instance list :: ("term") order

22 by intro_classes (auto simp add: prefix_def strict_prefix_def)

24 lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"

25 by (unfold prefix_def) blast

27 lemma prefixE [elim?]: "xs \<le> ys ==> (!!zs. ys = xs @ zs ==> C) ==> C"

28 by (unfold prefix_def) blast

30 lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"

31 by (unfold strict_prefix_def) blast

33 lemma strict_prefixE [elim?]:

34 "xs < ys ==> (xs \<le> ys ==> xs \<noteq> (ys::'a list) ==> C) ==> C"

35 by (unfold strict_prefix_def) blast

38 subsection {* Basic properties of prefixes *}

40 theorem Nil_prefix [iff]: "[] \<le> xs"

41 by (simp add: prefix_def)

43 theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"

44 by (induct xs) (simp_all add: prefix_def)

46 lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"

47 proof

48 assume "xs \<le> ys @ [y]"

49 then obtain zs where zs: "ys @ [y] = xs @ zs" ..

50 show "xs = ys @ [y] \<or> xs \<le> ys"

51 proof (cases zs rule: rev_cases)

52 assume "zs = []"

53 with zs have "xs = ys @ [y]" by simp

54 thus ?thesis ..

55 next

56 fix z zs' assume "zs = zs' @ [z]"

57 with zs have "ys = xs @ zs'" by simp

58 hence "xs \<le> ys" ..

59 thus ?thesis ..

60 qed

61 next

62 assume "xs = ys @ [y] \<or> xs \<le> ys"

63 thus "xs \<le> ys @ [y]"

64 proof

65 assume "xs = ys @ [y]"

66 thus ?thesis by simp

67 next

68 assume "xs \<le> ys"

69 then obtain zs where "ys = xs @ zs" ..

70 hence "ys @ [y] = xs @ (zs @ [y])" by simp

71 thus ?thesis ..

72 qed

73 qed

75 lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"

76 by (auto simp add: prefix_def)

78 lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"

79 by (induct xs) simp_all

81 lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"

82 proof -

83 have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)

84 thus ?thesis by simp

85 qed

87 lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"

88 proof -

89 assume "xs \<le> ys"

90 then obtain us where "ys = xs @ us" ..

91 hence "ys @ zs = xs @ (us @ zs)" by simp

92 thus ?thesis ..

93 qed

95 theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"

96 by (cases xs) (auto simp add: prefix_def)

98 theorem prefix_append:

99 "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"

100 apply (induct zs rule: rev_induct)

101 apply force

102 apply (simp del: append_assoc add: append_assoc [symmetric])

103 apply simp

104 apply blast

105 done

107 lemma append_one_prefix:

108 "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"

109 apply (unfold prefix_def)

110 apply (auto simp add: nth_append)

111 apply (case_tac zs)

112 apply auto

113 done

115 theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"

116 by (auto simp add: prefix_def)

119 subsection {* Parallel lists *}

121 constdefs

122 parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50)

123 "xs \<parallel> ys == \<not> xs \<le> ys \<and> \<not> ys \<le> xs"

125 lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"

126 by (unfold parallel_def) blast

128 lemma parallelE [elim]:

129 "xs \<parallel> ys ==> (\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> C) ==> C"

130 by (unfold parallel_def) blast

132 theorem prefix_cases:

133 "(xs \<le> ys ==> C) ==>

134 (ys < xs ==> C) ==>

135 (xs \<parallel> ys ==> C) ==> C"

136 by (unfold parallel_def strict_prefix_def) blast

138 theorem parallel_decomp:

139 "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"

140 (is "PROP ?P xs" concl is "?E xs")

141 proof (induct xs rule: rev_induct)

142 assume "[] \<parallel> ys" hence False by auto

143 thus "?E []" ..

144 next

145 fix x xs

146 assume hyp: "PROP ?P xs"

147 assume asm: "xs @ [x] \<parallel> ys"

148 show "?E (xs @ [x])"

149 proof (rule prefix_cases)

150 assume le: "xs \<le> ys"

151 then obtain ys' where ys: "ys = xs @ ys'" ..

152 show ?thesis

153 proof (cases ys')

154 assume "ys' = []" with ys have "xs = ys" by simp

155 with asm have "[x] \<parallel> []" by auto

156 hence False by blast

157 thus ?thesis ..

158 next

159 fix c cs assume ys': "ys' = c # cs"

160 with asm ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)

161 hence "x \<noteq> c" by auto

162 moreover have "xs @ [x] = xs @ x # []" by simp

163 moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)

164 ultimately show ?thesis by blast

165 qed

166 next

167 assume "ys < xs" hence "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)

168 with asm have False by blast

169 thus ?thesis ..

170 next

171 assume "xs \<parallel> ys"

172 with hyp obtain as b bs c cs where neq: "(b::'a) \<noteq> c"

173 and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"

174 by blast

175 from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp

176 with neq ys show ?thesis by blast

177 qed

178 qed

180 end