author | krauss |
Mon, 26 Feb 2007 21:34:16 +0100 | |
changeset 22359 | 94a794672c8b |
child 22371 | c9f5895972b0 |
permissions | -rw-r--r-- |
22359
94a794672c8b
Added formalization of size-change principle (experimental).
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1 |
theory SCT_Misc |
94a794672c8b
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2 |
imports Main |
94a794672c8b
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3 |
begin |
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4 |
|
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5 |
|
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6 |
subsection {* Searching in lists *} |
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7 |
|
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8 |
fun index_of :: "'a list \<Rightarrow> 'a \<Rightarrow> nat" |
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9 |
where |
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Added formalization of size-change principle (experimental).
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10 |
"index_of [] c = 0" |
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11 |
| "index_of (x#xs) c = (if x = c then 0 else Suc (index_of xs c))" |
94a794672c8b
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12 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
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13 |
lemma index_of_member: |
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14 |
"(x \<in> set l) \<Longrightarrow> (l ! index_of l x = x)" |
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15 |
by (induct l) auto |
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16 |
|
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Added formalization of size-change principle (experimental).
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17 |
lemma index_of_length: |
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18 |
"(x \<in> set l) = (index_of l x < length l)" |
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Added formalization of size-change principle (experimental).
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19 |
by (induct l) auto |
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20 |
|
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Added formalization of size-change principle (experimental).
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21 |
|
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Added formalization of size-change principle (experimental).
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22 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
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23 |
subsection {* Some reasoning tools *} |
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24 |
|
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Added formalization of size-change principle (experimental).
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25 |
lemma inc_induct[consumes 1]: |
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26 |
assumes less: "i \<le> j" |
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27 |
assumes base: "P j" |
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28 |
assumes step: "\<And>i. \<lbrakk>i < j; P (Suc i)\<rbrakk> \<Longrightarrow> P i" |
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29 |
shows "P i" |
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30 |
using less |
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31 |
proof (induct d\<equiv>"j - i" arbitrary: i) |
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32 |
case (0 i) |
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33 |
with `i \<le> j` have "i = j" by simp |
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34 |
with base show ?case by simp |
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35 |
next |
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36 |
case (Suc d i) |
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37 |
hence "i < j" "P (Suc i)" |
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38 |
by simp_all |
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thus "P i" by (rule step) |
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40 |
qed |
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41 |
|
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42 |
lemma strict_inc_induct[consumes 1]: |
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43 |
assumes less: "i < j" |
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44 |
assumes base: "\<And>i. j = Suc i \<Longrightarrow> P i" |
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45 |
assumes step: "\<And>i. \<lbrakk>i < j; P (Suc i)\<rbrakk> \<Longrightarrow> P i" |
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46 |
shows "P i" |
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47 |
using less |
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48 |
proof (induct d\<equiv>"j - i - 1" arbitrary: i) |
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49 |
case (0 i) |
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50 |
with `i < j` have "j = Suc i" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
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51 |
with base show ?case by simp |
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52 |
next |
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Added formalization of size-change principle (experimental).
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53 |
case (Suc d i) |
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54 |
hence "i < j" "P (Suc i)" |
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55 |
by simp_all |
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56 |
thus "P i" by (rule step) |
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57 |
qed |
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58 |
|
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59 |
|
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Added formalization of size-change principle (experimental).
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60 |
lemma three_cases: |
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61 |
assumes "a1 \<Longrightarrow> thesis" |
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62 |
assumes "a2 \<Longrightarrow> thesis" |
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63 |
assumes "a3 \<Longrightarrow> thesis" |
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64 |
assumes "\<And>R. \<lbrakk>a1 \<Longrightarrow> R; a2 \<Longrightarrow> R; a3 \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" |
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65 |
shows "thesis" |
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66 |
using prems |
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67 |
by auto |
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68 |
|
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69 |
|
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70 |
section {* Sequences *} |
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71 |
|
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72 |
types |
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73 |
'a sequence = "nat \<Rightarrow> 'a" |
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74 |
|
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Added formalization of size-change principle (experimental).
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75 |
subsection {* Increasing sequences *} |
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76 |
|
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77 |
definition increasing :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" |
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78 |
where |
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79 |
"increasing s = (\<forall>i j. i < j \<longrightarrow> s i < s j)" |
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80 |
|
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Added formalization of size-change principle (experimental).
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81 |
lemma increasing_strict: |
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82 |
assumes "increasing s" |
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83 |
assumes "i < j" |
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Added formalization of size-change principle (experimental).
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84 |
shows "s i < s j" |
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85 |
using prems |
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86 |
unfolding increasing_def by simp |
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87 |
|
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Added formalization of size-change principle (experimental).
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88 |
lemma increasing_weak: |
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89 |
assumes "increasing s" |
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90 |
assumes "i \<le> j" |
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91 |
shows "s i \<le> s j" |
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92 |
using prems increasing_strict[of s i j] |
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93 |
by (cases "i<j") auto |
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Added formalization of size-change principle (experimental).
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94 |
|
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Added formalization of size-change principle (experimental).
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95 |
lemma increasing_inc: |
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96 |
assumes [simp]: "increasing s" |
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97 |
shows "n \<le> s n" |
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98 |
proof (induct n) |
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99 |
case (Suc n) |
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100 |
with increasing_strict[of s n "Suc n"] |
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101 |
show ?case by auto |
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102 |
qed auto |
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Added formalization of size-change principle (experimental).
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103 |
|
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Added formalization of size-change principle (experimental).
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104 |
|
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105 |
lemma increasing_bij: |
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106 |
assumes [simp]: "increasing s" |
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107 |
shows "(s i < s j) = (i < j)" |
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108 |
proof |
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109 |
assume "s i < s j" |
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110 |
show "i < j" |
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111 |
proof (rule classical) |
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112 |
assume "\<not> ?thesis" |
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113 |
hence "j \<le> i" by arith |
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114 |
with increasing_weak have "s j \<le> s i" by simp |
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115 |
with `s i < s j` show ?thesis by simp |
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116 |
qed |
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117 |
qed (simp add:increasing_strict) |
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118 |
|
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119 |
|
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Added formalization of size-change principle (experimental).
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120 |
|
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121 |
|
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122 |
subsection {* Sections induced by an increasing sequence *} |
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123 |
|
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124 |
abbreviation |
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125 |
"section s i == {s i ..< s (Suc i)}" |
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126 |
|
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127 |
definition |
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128 |
"section_of s n = (LEAST i. n < s (Suc i))" |
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129 |
|
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Added formalization of size-change principle (experimental).
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130 |
|
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Added formalization of size-change principle (experimental).
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131 |
lemma section_help: |
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132 |
assumes [intro, simp]: "increasing s" |
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133 |
shows "\<exists>i. n < s (Suc i)" |
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134 |
proof - |
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135 |
from increasing_inc have "n \<le> s n" . |
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136 |
also from increasing_strict have "\<dots> < s (Suc n)" by simp |
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137 |
finally show ?thesis .. |
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138 |
qed |
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139 |
|
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140 |
lemma section_of2: |
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141 |
assumes "increasing s" |
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142 |
shows "n < s (Suc (section_of s n))" |
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Added formalization of size-change principle (experimental).
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143 |
unfolding section_of_def |
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144 |
by (rule LeastI_ex) (rule section_help) |
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145 |
|
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146 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
147 |
lemma section_of1: |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
148 |
assumes [simp, intro]: "increasing s" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
149 |
assumes "s i \<le> n" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
150 |
shows "s (section_of s n) \<le> n" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
151 |
proof (rule classical) |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
152 |
let ?m = "section_of s n" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
153 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
154 |
assume "\<not> ?thesis" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
155 |
hence a: "n < s ?m" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
156 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
157 |
have nonzero: "?m \<noteq> 0" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
158 |
proof |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
159 |
assume "?m = 0" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
160 |
from increasing_weak have "s 0 \<le> s i" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
161 |
also note `\<dots> \<le> n` |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
162 |
finally show False using `?m = 0` `n < s ?m` by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
163 |
qed |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
164 |
with a have "n < s (Suc (?m - 1))" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
165 |
with Least_le have "?m \<le> ?m - 1" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
166 |
unfolding section_of_def . |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
167 |
with nonzero show ?thesis by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
168 |
qed |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
169 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
170 |
lemma section_of_known: |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
171 |
assumes [simp]: "increasing s" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
172 |
assumes in_sect: "k \<in> section s i" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
173 |
shows "section_of s k = i" (is "?s = i") |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
174 |
proof (rule classical) |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
175 |
assume "\<not> ?thesis" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
176 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
177 |
hence "?s < i \<or> ?s > i" by arith |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
178 |
thus ?thesis |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
179 |
proof |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
180 |
assume "?s < i" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
181 |
hence "Suc ?s \<le> i" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
182 |
with increasing_weak have "s (Suc ?s) \<le> s i" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
183 |
moreover have "k < s (Suc ?s)" using section_of2 by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
184 |
moreover from in_sect have "s i \<le> k" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
185 |
ultimately show ?thesis by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
186 |
next |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
187 |
assume "i < ?s" hence "Suc i \<le> ?s" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
188 |
with increasing_weak have "s (Suc i) \<le> s ?s" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
189 |
moreover |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
190 |
from in_sect have "s i \<le> k" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
191 |
with section_of1 have "s ?s \<le> k" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
192 |
moreover from in_sect have "k < s (Suc i)" by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
193 |
ultimately show ?thesis by simp |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
194 |
qed |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
195 |
qed |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
196 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
197 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
198 |
lemma in_section_of: |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
199 |
assumes [simp, intro]: "increasing s" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
200 |
assumes "s i \<le> k" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
201 |
shows "k \<in> section s (section_of s k)" |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
202 |
using prems |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
203 |
by (auto intro:section_of1 section_of2) |
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
204 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
205 |
|
94a794672c8b
Added formalization of size-change principle (experimental).
krauss
parents:
diff
changeset
|
206 |
end |