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1 (* Title: HOL/Library/Permutation.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1995 University of Cambridge |
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5 |
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6 TODO: it would be nice to prove (for "multiset", defined on |
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7 HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x) |
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8 *) |
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9 |
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10 header {* |
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11 \title{Permutations} |
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12 \author{Lawrence C Paulson and Thomas M Rasmussen} |
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13 *} |
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14 |
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15 theory Permutation = Main: |
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16 |
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17 consts |
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18 perm :: "('a list * 'a list) set" |
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19 |
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20 syntax |
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21 "_perm" :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50) |
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22 translations |
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23 "x <~~> y" == "(x, y) \<in> perm" |
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24 |
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25 inductive perm |
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26 intros [intro] |
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27 Nil: "[] <~~> []" |
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28 swap: "y # x # l <~~> x # y # l" |
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29 Cons: "xs <~~> ys ==> z # xs <~~> z # ys" |
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30 trans: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" |
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31 |
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32 lemma perm_refl [iff]: "l <~~> l" |
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33 apply (induct l) |
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34 apply auto |
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35 done |
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36 |
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37 |
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38 subsection {* Some examples of rule induction on permutations *} |
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39 |
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40 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []" |
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41 -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *} |
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42 apply (erule perm.induct) |
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43 apply (simp_all (no_asm_simp)) |
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44 done |
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45 |
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46 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" |
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47 apply (insert xperm_empty_imp_aux) |
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48 apply blast |
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49 done |
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50 |
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51 |
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52 text {* |
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53 \medskip This more general theorem is easier to understand! |
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54 *} |
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55 |
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56 lemma perm_length: "xs <~~> ys ==> length xs = length ys" |
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57 apply (erule perm.induct) |
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58 apply simp_all |
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59 done |
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60 |
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61 lemma perm_empty_imp: "[] <~~> xs ==> xs = []" |
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62 apply (drule perm_length) |
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63 apply auto |
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64 done |
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65 |
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66 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" |
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67 apply (erule perm.induct) |
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68 apply auto |
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69 done |
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70 |
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71 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys" |
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72 apply (erule perm.induct) |
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73 apply auto |
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74 done |
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75 |
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76 |
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77 subsection {* Ways of making new permutations *} |
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78 |
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79 text {* |
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80 We can insert the head anywhere in the list. |
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81 *} |
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82 |
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83 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" |
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84 apply (induct xs) |
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85 apply auto |
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86 done |
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87 |
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88 lemma perm_append_swap: "xs @ ys <~~> ys @ xs" |
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89 apply (induct xs) |
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90 apply simp_all |
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91 apply (blast intro: perm_append_Cons) |
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92 done |
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93 |
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94 lemma perm_append_single: "a # xs <~~> xs @ [a]" |
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95 apply (rule perm.trans) |
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96 prefer 2 |
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97 apply (rule perm_append_swap) |
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98 apply simp |
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99 done |
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100 |
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101 lemma perm_rev: "rev xs <~~> xs" |
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102 apply (induct xs) |
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103 apply simp_all |
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104 apply (blast intro: perm_sym perm_append_single) |
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105 done |
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106 |
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107 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" |
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108 apply (induct l) |
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109 apply auto |
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110 done |
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111 |
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112 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" |
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113 apply (blast intro!: perm_append_swap perm_append1) |
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114 done |
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115 |
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116 |
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117 subsection {* Further results *} |
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118 |
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119 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])" |
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120 apply (blast intro: perm_empty_imp) |
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121 done |
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122 |
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123 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])" |
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124 apply auto |
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125 apply (erule perm_sym [THEN perm_empty_imp]) |
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126 done |
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127 |
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128 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]" |
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129 apply (erule perm.induct) |
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130 apply auto |
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131 done |
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132 |
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133 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])" |
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134 apply (blast intro: perm_sing_imp) |
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135 done |
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136 |
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137 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])" |
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138 apply (blast dest: perm_sym) |
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139 done |
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140 |
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141 |
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142 subsection {* Removing elements *} |
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143 |
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144 consts |
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145 remove :: "'a => 'a list => 'a list" |
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146 primrec |
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147 "remove x [] = []" |
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148 "remove x (y # ys) = (if x = y then ys else y # remove x ys)" |
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149 |
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150 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys" |
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151 apply (induct ys) |
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152 apply auto |
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153 done |
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154 |
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155 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" |
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156 apply (induct l) |
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157 apply auto |
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158 done |
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159 |
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160 |
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161 text {* \medskip Congruence rule *} |
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162 |
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163 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" |
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164 apply (erule perm.induct) |
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165 apply auto |
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166 done |
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167 |
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168 lemma remove_hd [simp]: "remove z (z # xs) = xs" |
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169 apply auto |
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170 done |
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171 |
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172 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" |
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173 apply (drule_tac z = z in perm_remove_perm) |
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174 apply auto |
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175 done |
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176 |
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177 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)" |
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178 apply (blast intro: cons_perm_imp_perm) |
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179 done |
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180 |
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181 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys" |
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182 apply (induct zs rule: rev_induct) |
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183 apply (simp_all (no_asm_use)) |
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184 apply blast |
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185 done |
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186 |
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187 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)" |
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188 apply (blast intro: append_perm_imp_perm perm_append1) |
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189 done |
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190 |
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191 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)" |
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192 apply (safe intro!: perm_append2) |
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193 apply (rule append_perm_imp_perm) |
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194 apply (rule perm_append_swap [THEN perm.trans]) |
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195 -- {* the previous step helps this @{text blast} call succeed quickly *} |
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196 apply (blast intro: perm_append_swap) |
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197 done |
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198 |
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199 end |