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src/HOL/Library/List_Prefix.thy

author | wenzelm |

Thu Jan 11 19:37:46 2001 +0100 (2001-01-11) | |

changeset 10870 | 9444e3cf37e1 |

parent 10512 | d34192966cd8 |

child 11780 | d17ee2241257 |

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

added strict_prefixI', strict_prefixE';

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?]: "ys = xs @ z # zs ==> xs < ys"

31 by (unfold strict_prefix_def prefix_def) blast

33 lemma strict_prefixE' [elim?]:

34 "xs < ys ==> (!!z zs. ys = xs @ z # zs ==> C) ==> C"

35 proof -

36 assume r: "!!z zs. ys = xs @ z # zs ==> C"

37 assume "xs < ys"

38 then obtain us where "ys = xs @ us" and "xs \<noteq> ys"

39 by (unfold strict_prefix_def prefix_def) blast

40 with r show ?thesis by (auto simp add: neq_Nil_conv)

41 qed

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

44 by (unfold strict_prefix_def) blast

46 lemma strict_prefixE [elim?]:

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

48 by (unfold strict_prefix_def) blast

51 subsection {* Basic properties of prefixes *}

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

54 by (simp add: prefix_def)

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

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

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

60 proof

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

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

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

64 proof (cases zs rule: rev_cases)

65 assume "zs = []"

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

67 thus ?thesis ..

68 next

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

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

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

72 thus ?thesis ..

73 qed

74 next

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

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

77 proof

78 assume "xs = ys @ [y]"

79 thus ?thesis by simp

80 next

81 assume "xs \<le> ys"

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

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

84 thus ?thesis ..

85 qed

86 qed

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

89 by (auto simp add: prefix_def)

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

92 by (induct xs) simp_all

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

95 proof -

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

97 thus ?thesis by simp

98 qed

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

101 proof -

102 assume "xs \<le> ys"

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

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

105 thus ?thesis ..

106 qed

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

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

111 theorem prefix_append:

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

113 apply (induct zs rule: rev_induct)

114 apply force

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

116 apply simp

117 apply blast

118 done

120 lemma append_one_prefix:

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

122 apply (unfold prefix_def)

123 apply (auto simp add: nth_append)

124 apply (case_tac zs)

125 apply auto

126 done

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

129 by (auto simp add: prefix_def)

132 subsection {* Parallel lists *}

134 constdefs

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

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

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

139 by (unfold parallel_def) blast

141 lemma parallelE [elim]:

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

143 by (unfold parallel_def) blast

145 theorem prefix_cases:

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

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

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

149 by (unfold parallel_def strict_prefix_def) blast

151 theorem parallel_decomp:

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

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

154 proof (induct xs rule: rev_induct)

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

156 thus "?E []" ..

157 next

158 fix x xs

159 assume hyp: "PROP ?P xs"

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

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

162 proof (rule prefix_cases)

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

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

165 show ?thesis

166 proof (cases ys')

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

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

169 hence False by blast

170 thus ?thesis ..

171 next

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

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

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

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

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

177 ultimately show ?thesis by blast

178 qed

179 next

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

181 with asm have False by blast

182 thus ?thesis ..

183 next

184 assume "xs \<parallel> ys"

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

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

187 by blast

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

189 with neq ys show ?thesis by blast

190 qed

191 qed

193 end