author | wenzelm |
Tue, 27 Sep 2022 13:34:54 +0200 | |
changeset 76212 | f2094906e491 |
parent 76056 | c2fd8b88d262 |
child 76226 | 2aad8698f82f |
permissions | -rw-r--r-- |
28685 | 1 |
(* Title: HOL/Orderings.thy |
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
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*) |
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section \<open>Abstract orderings\<close> |
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theory Orderings |
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distributed theory Algebras to theories Groups and Lattices
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imports HOL |
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declare command keywords via theory header, including strict checking outside Pure;
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keywords "print_orders" :: diag |
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begin |
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ML_file \<open>~~/src/Provers/order_procedure.ML\<close> |
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ML_file \<open>~~/src/Provers/order_tac.ML\<close> |
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subsection \<open>Abstract ordering\<close> |
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locale partial_preordering = |
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fixes less_eq :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold>\<le>\<close> 50) |
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assumes refl: \<open>a \<^bold>\<le> a\<close> \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
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and trans: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c\<close> |
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locale preordering = partial_preordering + |
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fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50) |
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assumes strict_iff_not: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> |
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begin |
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lemma strict_implies_order: |
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\<open>a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b\<close> |
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by (simp add: strict_iff_not) |
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lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
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\<open>\<not> a \<^bold>< a\<close> |
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by (simp add: strict_iff_not) |
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|
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lemma asym: |
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\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False\<close> |
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by (auto simp add: strict_iff_not) |
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lemma strict_trans1: |
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\<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> |
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by (auto simp add: strict_iff_not intro: trans) |
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lemma strict_trans2: |
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\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c\<close> |
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by (auto simp add: strict_iff_not intro: trans) |
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lemma strict_trans: |
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\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> |
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by (auto intro: strict_trans1 strict_implies_order) |
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end |
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lemma preordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close> |
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fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50) |
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and less (infix \<open>\<^bold><\<close> 50) |
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assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close> |
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assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close> |
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assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close> |
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assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> |
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shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close> |
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proof |
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fix a b |
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show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> |
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by (auto simp add: less_eq_less asym irrefl) |
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next |
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fix a |
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show \<open>a \<^bold>\<le> a\<close> |
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by (auto simp add: less_eq_less) |
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next |
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fix a b c |
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assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close> |
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by (auto simp add: less_eq_less intro: trans) |
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qed |
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|
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lemma preordering_dualI: |
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fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50) |
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and less (infix \<open>\<^bold><\<close> 50) |
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assumes \<open>preordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close> |
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shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close> |
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proof - |
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from assms interpret preordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> . |
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show ?thesis |
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by standard (auto simp: strict_iff_not refl intro: trans) |
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qed |
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85 |
|
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locale ordering = partial_preordering + |
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fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50) |
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assumes strict_iff_order: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close> |
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assumes antisym: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b\<close> |
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90 |
begin |
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91 |
|
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sublocale preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close> |
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93 |
proof |
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show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> for a b |
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95 |
by (auto simp add: strict_iff_order intro: antisym) |
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qed |
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lemma strict_implies_not_eq: |
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\<open>a \<^bold>< b \<Longrightarrow> a \<noteq> b\<close> |
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by (simp add: strict_iff_order) |
101 |
||
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lemma not_eq_order_implies_strict: |
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\<open>a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b\<close> |
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by (simp add: strict_iff_order) |
105 |
||
106 |
lemma order_iff_strict: |
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\<open>a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close> |
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by (auto simp add: strict_iff_order refl) |
109 |
||
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lemma eq_iff: \<open>a = b \<longleftrightarrow> a \<^bold>\<le> b \<and> b \<^bold>\<le> a\<close> |
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by (auto simp add: refl intro: antisym) |
112 |
||
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end |
114 |
||
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lemma ordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close> |
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116 |
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50) |
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117 |
and less (infix \<open>\<^bold><\<close> 50) |
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assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close> |
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assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close> |
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assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close> |
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assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close> |
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shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close> |
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proof |
124 |
fix a b |
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125 |
show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close> |
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by (auto simp add: less_eq_less asym irrefl) |
127 |
next |
|
128 |
fix a |
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129 |
show \<open>a \<^bold>\<le> a\<close> |
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by (auto simp add: less_eq_less) |
131 |
next |
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132 |
fix a b c |
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133 |
assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close> |
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by (auto simp add: less_eq_less intro: trans) |
135 |
next |
|
136 |
fix a b |
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137 |
assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> a\<close> then show \<open>a = b\<close> |
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by (auto simp add: less_eq_less asym) |
139 |
qed |
|
140 |
||
141 |
lemma ordering_dualI: |
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142 |
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50) |
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and less (infix \<open>\<^bold><\<close> 50) |
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144 |
assumes \<open>ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close> |
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145 |
shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close> |
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proof - |
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147 |
from assms interpret ordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> . |
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show ?thesis |
149 |
by standard (auto simp: strict_iff_order refl intro: antisym trans) |
|
150 |
qed |
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151 |
||
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locale ordering_top = ordering + |
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|
153 |
fixes top :: \<open>'a\<close> (\<open>\<^bold>\<top>\<close>) |
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154 |
assumes extremum [simp]: \<open>a \<^bold>\<le> \<^bold>\<top>\<close> |
51487 | 155 |
begin |
156 |
||
157 |
lemma extremum_uniqueI: |
|
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158 |
\<open>\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>\<close> |
51487 | 159 |
by (rule antisym) auto |
160 |
||
161 |
lemma extremum_unique: |
|
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162 |
\<open>\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>\<close> |
51487 | 163 |
by (auto intro: antisym) |
164 |
||
165 |
lemma extremum_strict [simp]: |
|
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166 |
\<open>\<not> (\<^bold>\<top> \<^bold>< a)\<close> |
51487 | 167 |
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl) |
168 |
||
169 |
lemma not_eq_extremum: |
|
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170 |
\<open>a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>\<close> |
51487 | 171 |
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum) |
172 |
||
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173 |
end |
51487 | 174 |
|
175 |
||
60758 | 176 |
subsection \<open>Syntactic orders\<close> |
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177 |
|
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178 |
class ord = |
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179 |
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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181 |
begin |
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182 |
|
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183 |
notation |
67403 | 184 |
less_eq ("'(\<le>')") and |
185 |
less_eq ("(_/ \<le> _)" [51, 51] 50) and |
|
186 |
less ("'(<')") and |
|
187 |
less ("(_/ < _)" [51, 51] 50) |
|
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188 |
|
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189 |
abbreviation (input) |
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190 |
greater_eq (infix "\<ge>" 50) |
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|
191 |
where "x \<ge> y \<equiv> y \<le> x" |
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192 |
|
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193 |
abbreviation (input) |
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194 |
greater (infix ">" 50) |
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195 |
where "x > y \<equiv> y < x" |
e96292f32c3c
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196 |
|
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|
197 |
notation (ASCII) |
67403 | 198 |
less_eq ("'(<=')") and |
199 |
less_eq ("(_/ <= _)" [51, 51] 50) |
|
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200 |
|
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|
201 |
notation (input) |
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|
202 |
greater_eq (infix ">=" 50) |
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|
203 |
|
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204 |
end |
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205 |
|
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|
206 |
|
60758 | 207 |
subsection \<open>Quasi orders\<close> |
15524 | 208 |
|
27682 | 209 |
class preorder = ord + |
210 |
assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" |
|
25062 | 211 |
and order_refl [iff]: "x \<le> x" |
212 |
and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
|
21248 | 213 |
begin |
214 |
||
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215 |
sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater |
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|
216 |
proof - |
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|
217 |
interpret preordering less_eq less |
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218 |
by standard (auto intro: order_trans simp add: less_le_not_le) |
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219 |
show \<open>preordering less_eq less\<close> |
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220 |
by (fact preordering_axioms) |
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221 |
then show \<open>preordering greater_eq greater\<close> |
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222 |
by (rule preordering_dualI) |
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223 |
qed |
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|
224 |
|
60758 | 225 |
text \<open>Reflexivity.\<close> |
15524 | 226 |
|
25062 | 227 |
lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" |
61799 | 228 |
\<comment> \<open>This form is useful with the classical reasoner.\<close> |
23212 | 229 |
by (erule ssubst) (rule order_refl) |
15524 | 230 |
|
25062 | 231 |
lemma less_irrefl [iff]: "\<not> x < x" |
27682 | 232 |
by (simp add: less_le_not_le) |
233 |
||
234 |
lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
|
63172 | 235 |
by (simp add: less_le_not_le) |
27682 | 236 |
|
237 |
||
60758 | 238 |
text \<open>Asymmetry.\<close> |
27682 | 239 |
|
240 |
lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
|
241 |
by (simp add: less_le_not_le) |
|
242 |
||
243 |
lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" |
|
244 |
by (drule less_not_sym, erule contrapos_np) simp |
|
245 |
||
246 |
||
60758 | 247 |
text \<open>Transitivity.\<close> |
27682 | 248 |
|
249 |
lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
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250 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 251 |
|
252 |
lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
|
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|
253 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 254 |
|
255 |
lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" |
|
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|
256 |
by (auto simp add: less_le_not_le intro: order_trans) |
27682 | 257 |
|
258 |
||
60758 | 259 |
text \<open>Useful for simplification, but too risky to include by default.\<close> |
27682 | 260 |
|
261 |
lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" |
|
262 |
by (blast elim: less_asym) |
|
263 |
||
264 |
lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" |
|
265 |
by (blast elim: less_asym) |
|
266 |
||
267 |
||
60758 | 268 |
text \<open>Transitivity rules for calculational reasoning\<close> |
27682 | 269 |
|
270 |
lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" |
|
271 |
by (rule less_asym) |
|
272 |
||
273 |
||
60758 | 274 |
text \<open>Dual order\<close> |
27682 | 275 |
|
276 |
lemma dual_preorder: |
|
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277 |
\<open>class.preorder (\<ge>) (>)\<close> |
63819 | 278 |
by standard (auto simp add: less_le_not_le intro: order_trans) |
27682 | 279 |
|
280 |
end |
|
281 |
||
73794 | 282 |
lemma preordering_preorderI: |
283 |
\<open>class.preorder (\<^bold>\<le>) (\<^bold><)\<close> if \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close> |
|
284 |
for less_eq (infix \<open>\<^bold>\<le>\<close> 50) and less (infix \<open>\<^bold><\<close> 50) |
|
285 |
proof - |
|
286 |
from that interpret preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close> . |
|
287 |
show ?thesis |
|
288 |
by standard (auto simp add: strict_iff_not refl intro: trans) |
|
289 |
qed |
|
290 |
||
291 |
||
27682 | 292 |
|
60758 | 293 |
subsection \<open>Partial orders\<close> |
27682 | 294 |
|
295 |
class order = preorder + |
|
73411 | 296 |
assumes order_antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
27682 | 297 |
begin |
298 |
||
51487 | 299 |
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
73411 | 300 |
by (auto simp add: less_le_not_le intro: order_antisym) |
51487 | 301 |
|
63819 | 302 |
sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater |
303 |
proof - |
|
304 |
interpret ordering less_eq less |
|
73411 | 305 |
by standard (auto intro: order_antisym order_trans simp add: less_le) |
63819 | 306 |
show "ordering less_eq less" |
307 |
by (fact ordering_axioms) |
|
308 |
then show "ordering greater_eq greater" |
|
309 |
by (rule ordering_dualI) |
|
310 |
qed |
|
51487 | 311 |
|
73411 | 312 |
print_theorems |
313 |
||
60758 | 314 |
text \<open>Reflexivity.\<close> |
15524 | 315 |
|
25062 | 316 |
lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
61799 | 317 |
\<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> |
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haftmann
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|
318 |
by (fact order.order_iff_strict) |
15524 | 319 |
|
25062 | 320 |
lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
63172 | 321 |
by (simp add: less_le) |
15524 | 322 |
|
21329 | 323 |
|
60758 | 324 |
text \<open>Useful for simplification, but too risky to include by default.\<close> |
21329 | 325 |
|
25062 | 326 |
lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
23212 | 327 |
by auto |
21329 | 328 |
|
25062 | 329 |
lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
23212 | 330 |
by auto |
21329 | 331 |
|
332 |
||
60758 | 333 |
text \<open>Transitivity rules for calculational reasoning\<close> |
21329 | 334 |
|
25062 | 335 |
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" |
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|
336 |
by (fact order.not_eq_order_implies_strict) |
21329 | 337 |
|
25062 | 338 |
lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" |
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haftmann
parents:
51487
diff
changeset
|
339 |
by (rule order.not_eq_order_implies_strict) |
21329 | 340 |
|
15524 | 341 |
|
60758 | 342 |
text \<open>Asymmetry.\<close> |
15524 | 343 |
|
73411 | 344 |
lemma order_eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" |
71851 | 345 |
by (fact order.eq_iff) |
15524 | 346 |
|
25062 | 347 |
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
73411 | 348 |
by (simp add: order.eq_iff) |
15524 | 349 |
|
25062 | 350 |
lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" |
71851 | 351 |
by (fact order.strict_implies_not_eq) |
21248 | 352 |
|
70749
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|
353 |
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
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changeset
|
354 |
by (simp add: local.le_less) |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
|
355 |
|
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
356 |
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
|
357 |
by (simp add: local.less_le) |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
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diff
changeset
|
358 |
|
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
359 |
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" |
73411 | 360 |
by (auto simp: less_le order.antisym) |
21083 | 361 |
|
60758 | 362 |
text \<open>Least value operator\<close> |
27107 | 363 |
|
27299 | 364 |
definition (in ord) |
27107 | 365 |
Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where |
366 |
"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" |
|
367 |
||
368 |
lemma Least_equality: |
|
369 |
assumes "P x" |
|
370 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
371 |
shows "Least P = x" |
|
372 |
unfolding Least_def by (rule the_equality) |
|
73411 | 373 |
(blast intro: assms order.antisym)+ |
27107 | 374 |
|
375 |
lemma LeastI2_order: |
|
376 |
assumes "P x" |
|
377 |
and "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
378 |
and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" |
|
379 |
shows "Q (Least P)" |
|
380 |
unfolding Least_def by (rule theI2) |
|
73411 | 381 |
(blast intro: assms order.antisym)+ |
27107 | 382 |
|
69791
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Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
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diff
changeset
|
383 |
lemma Least_ex1: |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
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diff
changeset
|
384 |
assumes "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y)" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
385 |
shows Least1I: "P (Least P)" and Least1_le: "P z \<Longrightarrow> Least P \<le> z" |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
386 |
using theI'[OF assms] |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
387 |
unfolding Least_def |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
388 |
by auto |
195aeee8b30a
Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents:
69605
diff
changeset
|
389 |
|
65963 | 390 |
text \<open>Greatest value operator\<close> |
391 |
||
392 |
definition Greatest :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10) where |
|
393 |
"Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<ge> y))" |
|
394 |
||
395 |
lemma GreatestI2_order: |
|
396 |
"\<lbrakk> P x; |
|
397 |
\<And>y. P y \<Longrightarrow> x \<ge> y; |
|
398 |
\<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk> |
|
399 |
\<Longrightarrow> Q (Greatest P)" |
|
400 |
unfolding Greatest_def |
|
73411 | 401 |
by (rule theI2) (blast intro: order.antisym)+ |
65963 | 402 |
|
403 |
lemma Greatest_equality: |
|
404 |
"\<lbrakk> P x; \<And>y. P y \<Longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Greatest P = x" |
|
405 |
unfolding Greatest_def |
|
73411 | 406 |
by (rule the_equality) (blast intro: order.antisym)+ |
65963 | 407 |
|
21248 | 408 |
end |
15524 | 409 |
|
63819 | 410 |
lemma ordering_orderI: |
411 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
|
412 |
and less (infix "\<^bold><" 50) |
|
413 |
assumes "ordering less_eq less" |
|
414 |
shows "class.order less_eq less" |
|
415 |
proof - |
|
416 |
from assms interpret ordering less_eq less . |
|
417 |
show ?thesis |
|
418 |
by standard (auto intro: antisym trans simp add: refl strict_iff_order) |
|
419 |
qed |
|
56545 | 420 |
|
421 |
lemma order_strictI: |
|
73794 | 422 |
fixes less (infix "\<^bold><" 50) |
423 |
and less_eq (infix "\<^bold>\<le>" 50) |
|
424 |
assumes "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b" |
|
425 |
assumes "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a" |
|
426 |
assumes "\<And>a. \<not> a \<^bold>< a" |
|
427 |
assumes "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c" |
|
56545 | 428 |
shows "class.order less_eq less" |
63819 | 429 |
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+) |
430 |
||
431 |
context order |
|
432 |
begin |
|
433 |
||
434 |
text \<open>Dual order\<close> |
|
435 |
||
436 |
lemma dual_order: |
|
67398 | 437 |
"class.order (\<ge>) (>)" |
63819 | 438 |
using dual_order.ordering_axioms by (rule ordering_orderI) |
439 |
||
440 |
end |
|
56545 | 441 |
|
442 |
||
60758 | 443 |
subsection \<open>Linear (total) orders\<close> |
21329 | 444 |
|
22316 | 445 |
class linorder = order + |
25207 | 446 |
assumes linear: "x \<le> y \<or> y \<le> x" |
21248 | 447 |
begin |
448 |
||
25062 | 449 |
lemma less_linear: "x < y \<or> x = y \<or> y < x" |
23212 | 450 |
unfolding less_le using less_le linear by blast |
21248 | 451 |
|
25062 | 452 |
lemma le_less_linear: "x \<le> y \<or> y < x" |
23212 | 453 |
by (simp add: le_less less_linear) |
21248 | 454 |
|
455 |
lemma le_cases [case_names le ge]: |
|
25062 | 456 |
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 457 |
using linear by blast |
21248 | 458 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
459 |
lemma (in linorder) le_cases3: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
460 |
"\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P; |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
461 |
\<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
462 |
by (blast intro: le_cases) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
463 |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22377
diff
changeset
|
464 |
lemma linorder_cases [case_names less equal greater]: |
25062 | 465 |
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 466 |
using less_linear by blast |
21248 | 467 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
468 |
lemma linorder_wlog[case_names le sym]: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
469 |
"(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
470 |
by (cases rule: le_cases[of a b]) blast+ |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
471 |
|
25062 | 472 |
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" |
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
473 |
unfolding less_le |
73411 | 474 |
using linear by (blast intro: order.antisym) |
23212 | 475 |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
476 |
lemma not_less_iff_gr_or_eq: "\<not>(x < y) \<longleftrightarrow> (x > y \<or> x = y)" |
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
477 |
by (auto simp add:not_less le_less) |
15524 | 478 |
|
25062 | 479 |
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" |
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
480 |
unfolding less_le |
73411 | 481 |
using linear by (blast intro: order.antisym) |
15524 | 482 |
|
25062 | 483 |
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" |
23212 | 484 |
by (cut_tac x = x and y = y in less_linear, auto) |
15524 | 485 |
|
25062 | 486 |
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" |
23212 | 487 |
by (simp add: neq_iff) blast |
15524 | 488 |
|
25062 | 489 |
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
73411 | 490 |
by (blast intro: order.antisym dest: not_less [THEN iffD1]) |
15524 | 491 |
|
25062 | 492 |
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" |
23212 | 493 |
unfolding not_less . |
16796 | 494 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
495 |
lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y" |
23212 | 496 |
unfolding not_le . |
21248 | 497 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
498 |
lemma linorder_less_wlog[case_names less refl sym]: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
499 |
"\<lbrakk>\<And>a b. a < b \<Longrightarrow> P a b; \<And>a. P a a; \<And>a b. P b a \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
500 |
using antisym_conv3 by blast |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64287
diff
changeset
|
501 |
|
60758 | 502 |
text \<open>Dual order\<close> |
22916 | 503 |
|
26014 | 504 |
lemma dual_linorder: |
67398 | 505 |
"class.linorder (\<ge>) (>)" |
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
35828
diff
changeset
|
506 |
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear) |
22916 | 507 |
|
21248 | 508 |
end |
509 |
||
23948 | 510 |
|
60758 | 511 |
text \<open>Alternative introduction rule with bias towards strict order\<close> |
56545 | 512 |
|
513 |
lemma linorder_strictI: |
|
63819 | 514 |
fixes less_eq (infix "\<^bold>\<le>" 50) |
515 |
and less (infix "\<^bold><" 50) |
|
56545 | 516 |
assumes "class.order less_eq less" |
63819 | 517 |
assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> b \<^bold>< a" |
56545 | 518 |
shows "class.linorder less_eq less" |
519 |
proof - |
|
520 |
interpret order less_eq less |
|
60758 | 521 |
by (fact \<open>class.order less_eq less\<close>) |
56545 | 522 |
show ?thesis |
523 |
proof |
|
524 |
fix a b |
|
63819 | 525 |
show "a \<^bold>\<le> b \<or> b \<^bold>\<le> a" |
56545 | 526 |
using trichotomy by (auto simp add: le_less) |
527 |
qed |
|
528 |
qed |
|
529 |
||
530 |
||
60758 | 531 |
subsection \<open>Reasoning tools setup\<close> |
21083 | 532 |
|
60758 | 533 |
ML \<open> |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
534 |
structure Logic_Signature : LOGIC_SIGNATURE = struct |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
535 |
val mk_Trueprop = HOLogic.mk_Trueprop |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
536 |
val dest_Trueprop = HOLogic.dest_Trueprop |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
537 |
val Trueprop_conv = HOLogic.Trueprop_conv |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
538 |
val Not = HOLogic.Not |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
539 |
val conj = HOLogic.conj |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
540 |
val disj = HOLogic.disj |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
541 |
|
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
542 |
val notI = @{thm notI} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
543 |
val ccontr = @{thm ccontr} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
544 |
val conjI = @{thm conjI} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
545 |
val conjE = @{thm conjE} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
546 |
val disjE = @{thm disjE} |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
547 |
|
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
548 |
val not_not_conv = Conv.rewr_conv @{thm eq_reflection[OF not_not]} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
549 |
val de_Morgan_conj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
550 |
val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_disj]} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
551 |
val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
552 |
val conj_disj_distribR_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribR]} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
553 |
end |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
554 |
|
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
555 |
structure HOL_Base_Order_Tac = Base_Order_Tac( |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
556 |
structure Logic_Sig = Logic_Signature; |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
557 |
(* Exclude types with specialised solvers. *) |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
558 |
val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic.realT] |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
559 |
) |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
560 |
|
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
561 |
structure HOL_Order_Tac = Order_Tac(structure Base_Tac = HOL_Base_Order_Tac) |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
562 |
|
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
563 |
fun print_orders ctxt0 = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
564 |
let |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
565 |
val ctxt = Config.put show_sorts true ctxt0 |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
566 |
val orders = HOL_Order_Tac.Data.get (Context.Proof ctxt) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
567 |
fun pretty_term t = Pretty.block |
24920 | 568 |
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1, |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
569 |
Pretty.str "::", Pretty.brk 1, |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
570 |
Pretty.quote (Syntax.pretty_typ ctxt (type_of t)), Pretty.brk 1] |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
571 |
fun pretty_order ({kind = kind, ops = ops, ...}, _) = |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
572 |
Pretty.block ([Pretty.str (@{make_string} kind), Pretty.str ":", Pretty.brk 1] |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
573 |
@ map pretty_term ops) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
574 |
in |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
575 |
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders)) |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
576 |
end |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
577 |
|
56508 | 578 |
val _ = |
69593 | 579 |
Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close> |
56508 | 580 |
"print order structures available to transitivity reasoner" |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
581 |
(Scan.succeed (Toplevel.keep (print_orders o Toplevel.context_of))) |
56508 | 582 |
|
60758 | 583 |
\<close> |
21091 | 584 |
|
60758 | 585 |
method_setup order = \<open> |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
586 |
Scan.succeed (fn ctxt => SIMPLE_METHOD' (HOL_Order_Tac.tac [] ctxt)) |
60758 | 587 |
\<close> "transitivity reasoner" |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
588 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
589 |
|
60758 | 590 |
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
591 |
|
25076 | 592 |
context order |
593 |
begin |
|
594 |
||
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
595 |
lemma nless_le: "(\<not> a < b) \<longleftrightarrow> (\<not> a \<le> b) \<or> a = b" |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
596 |
using local.dual_order.order_iff_strict by blast |
27689 | 597 |
|
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
598 |
local_setup \<open> |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
599 |
HOL_Order_Tac.declare_order { |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
600 |
ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
601 |
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
602 |
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
603 |
conv_thms = {less_le = @{thm eq_reflection[OF less_le]}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
604 |
nless_le = @{thm eq_reflection[OF nless_le]}} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
605 |
} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
606 |
\<close> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
607 |
|
25076 | 608 |
end |
609 |
||
610 |
context linorder |
|
611 |
begin |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
612 |
|
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
613 |
lemma nle_le: "(\<not> a \<le> b) \<longleftrightarrow> b \<le> a \<and> b \<noteq> a" |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
614 |
using not_le less_le by simp |
25076 | 615 |
|
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
616 |
local_setup \<open> |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
617 |
HOL_Order_Tac.declare_linorder { |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
618 |
ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
619 |
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
620 |
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
621 |
conv_thms = {less_le = @{thm eq_reflection[OF less_le]}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
622 |
nless_le = @{thm eq_reflection[OF not_less]}, |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
623 |
nle_le = @{thm eq_reflection[OF nle_le]}} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
624 |
} |
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
625 |
\<close> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
626 |
|
25076 | 627 |
end |
628 |
||
60758 | 629 |
setup \<open> |
56509 | 630 |
map_theory_simpset (fn ctxt0 => ctxt0 addSolver |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
631 |
mk_solver "Transitivity" (fn ctxt => HOL_Order_Tac.tac (Simplifier.prems_of ctxt) ctxt)) |
60758 | 632 |
\<close> |
15524 | 633 |
|
60758 | 634 |
ML \<open> |
56509 | 635 |
local |
636 |
fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *) |
|
637 |
in |
|
15524 | 638 |
|
56509 | 639 |
fun antisym_le_simproc ctxt ct = |
59582 | 640 |
(case Thm.term_of ct of |
56509 | 641 |
(le as Const (_, T)) $ r $ s => |
642 |
(let |
|
643 |
val prems = Simplifier.prems_of ctxt; |
|
69593 | 644 |
val less = Const (\<^const_name>\<open>less\<close>, T); |
56509 | 645 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
646 |
in |
|
647 |
(case find_first (prp t) prems of |
|
648 |
NONE => |
|
649 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in |
|
650 |
(case find_first (prp t) prems of |
|
651 |
NONE => NONE |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
652 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1}))) |
56509 | 653 |
end |
654 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv}))) |
|
655 |
end handle THM _ => NONE) |
|
656 |
| _ => NONE); |
|
15524 | 657 |
|
56509 | 658 |
fun antisym_less_simproc ctxt ct = |
59582 | 659 |
(case Thm.term_of ct of |
56509 | 660 |
NotC $ ((less as Const(_,T)) $ r $ s) => |
661 |
(let |
|
662 |
val prems = Simplifier.prems_of ctxt; |
|
69593 | 663 |
val le = Const (\<^const_name>\<open>less_eq\<close>, T); |
56509 | 664 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
665 |
in |
|
666 |
(case find_first (prp t) prems of |
|
667 |
NONE => |
|
668 |
let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in |
|
669 |
(case find_first (prp t) prems of |
|
670 |
NONE => NONE |
|
671 |
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))) |
|
672 |
end |
|
70749
5d06b7bb9d22
More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents:
69815
diff
changeset
|
673 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2}))) |
73526
a3cc9fa1295d
new automatic order prover: stateless, complete, verified
nipkow
parents:
73411
diff
changeset
|
674 |
end handle THM _ => NONE) |
56509 | 675 |
| _ => NONE); |
21083 | 676 |
|
56509 | 677 |
end; |
60758 | 678 |
\<close> |
15524 | 679 |
|
56509 | 680 |
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc" |
681 |
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc" |
|
682 |
||
15524 | 683 |
|
60758 | 684 |
subsection \<open>Bounded quantifiers\<close> |
21083 | 685 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
686 |
syntax (ASCII) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
687 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
688 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
689 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
690 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 691 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
692 |
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
693 |
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
694 |
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
695 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 696 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
697 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3ALL _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
698 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3EX _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
699 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61824
diff
changeset
|
700 |
syntax |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
701 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
702 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
703 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
704 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 705 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
706 |
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
707 |
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
708 |
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
709 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 710 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
711 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<noteq>_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
712 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<noteq>_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
713 |
|
62521 | 714 |
syntax (input) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
715 |
"_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
716 |
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
717 |
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
718 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
719 |
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3! _~=_./ _)" [0, 0, 10] 10) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
720 |
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3? _~=_./ _)" [0, 0, 10] 10) |
21083 | 721 |
|
722 |
translations |
|
67091 | 723 |
"\<forall>x<y. P" \<rightharpoonup> "\<forall>x. x < y \<longrightarrow> P" |
724 |
"\<exists>x<y. P" \<rightharpoonup> "\<exists>x. x < y \<and> P" |
|
725 |
"\<forall>x\<le>y. P" \<rightharpoonup> "\<forall>x. x \<le> y \<longrightarrow> P" |
|
726 |
"\<exists>x\<le>y. P" \<rightharpoonup> "\<exists>x. x \<le> y \<and> P" |
|
727 |
"\<forall>x>y. P" \<rightharpoonup> "\<forall>x. x > y \<longrightarrow> P" |
|
728 |
"\<exists>x>y. P" \<rightharpoonup> "\<exists>x. x > y \<and> P" |
|
729 |
"\<forall>x\<ge>y. P" \<rightharpoonup> "\<forall>x. x \<ge> y \<longrightarrow> P" |
|
730 |
"\<exists>x\<ge>y. P" \<rightharpoonup> "\<exists>x. x \<ge> y \<and> P" |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
731 |
"\<forall>x\<noteq>y. P" \<rightharpoonup> "\<forall>x. x \<noteq> y \<longrightarrow> P" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67452
diff
changeset
|
732 |
"\<exists>x\<noteq>y. P" \<rightharpoonup> "\<exists>x. x \<noteq> y \<and> P" |
21083 | 733 |
|
60758 | 734 |
print_translation \<open> |
21083 | 735 |
let |
69593 | 736 |
val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>; |
737 |
val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>; |
|
738 |
val impl = \<^const_syntax>\<open>HOL.implies\<close>; |
|
739 |
val conj = \<^const_syntax>\<open>HOL.conj\<close>; |
|
740 |
val less = \<^const_syntax>\<open>less\<close>; |
|
741 |
val less_eq = \<^const_syntax>\<open>less_eq\<close>; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
742 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
743 |
val trans = |
35115 | 744 |
[((All_binder, impl, less), |
69593 | 745 |
(\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)), |
35115 | 746 |
((All_binder, impl, less_eq), |
69593 | 747 |
(\<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)), |
35115 | 748 |
((Ex_binder, conj, less), |
69593 | 749 |
(\<^syntax_const>\<open>_Ex_less\<close>, \<^syntax_const>\<open>_Ex_greater\<close>)), |
35115 | 750 |
((Ex_binder, conj, less_eq), |
69593 | 751 |
(\<^syntax_const>\<open>_Ex_less_eq\<close>, \<^syntax_const>\<open>_Ex_greater_eq\<close>))]; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
752 |
|
35115 | 753 |
fun matches_bound v t = |
754 |
(case t of |
|
69593 | 755 |
Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', _) => v = v' |
35115 | 756 |
| _ => false); |
757 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false); |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
758 |
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
759 |
|
52143 | 760 |
fun tr' q = (q, fn _ => |
69593 | 761 |
(fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T), |
35364 | 762 |
Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
67398 | 763 |
(case AList.lookup (=) trans (q, c, d) of |
35115 | 764 |
NONE => raise Match |
765 |
| SOME (l, g) => |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
766 |
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P |
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
767 |
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P |
35115 | 768 |
else raise Match) |
52143 | 769 |
| _ => raise Match)); |
21524 | 770 |
in [tr' All_binder, tr' Ex_binder] end |
60758 | 771 |
\<close> |
21083 | 772 |
|
773 |
||
60758 | 774 |
subsection \<open>Transitivity reasoning\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
775 |
|
25193 | 776 |
context ord |
777 |
begin |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
778 |
|
25193 | 779 |
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" |
780 |
by (rule subst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
781 |
|
25193 | 782 |
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
783 |
by (rule ssubst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
784 |
|
25193 | 785 |
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" |
786 |
by (rule subst) |
|
787 |
||
788 |
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" |
|
789 |
by (rule ssubst) |
|
790 |
||
791 |
end |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
792 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
793 |
lemma order_less_subst2: "(a::'a::order) < b \<Longrightarrow> f b < (c::'c::order) \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
794 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
795 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
796 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
797 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
798 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
799 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
800 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
801 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
802 |
lemma order_less_subst1: "(a::'a::order) < f b \<Longrightarrow> (b::'b::order) < c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
803 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
804 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
805 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
806 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
807 |
also assume "b < c" hence "f b < f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
808 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
809 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
810 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
811 |
lemma order_le_less_subst2: "(a::'a::order) <= b \<Longrightarrow> f b < (c::'c::order) \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
812 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a < c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
813 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
814 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
815 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
816 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
817 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
818 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
819 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
820 |
lemma order_le_less_subst1: "(a::'a::order) <= f b \<Longrightarrow> (b::'b::order) < c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
821 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
822 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
823 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
824 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
825 |
also assume "b < c" hence "f b < f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
826 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
827 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
828 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
829 |
lemma order_less_le_subst2: "(a::'a::order) < b \<Longrightarrow> f b <= (c::'c::order) \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
830 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
831 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
832 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
833 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
834 |
also assume "f b <= c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
835 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
836 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
837 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
838 |
lemma order_less_le_subst1: "(a::'a::order) < f b \<Longrightarrow> (b::'b::order) <= c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
839 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a < f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
840 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
841 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
842 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
843 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
844 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
845 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
846 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
847 |
lemma order_subst1: "(a::'a::order) <= f b \<Longrightarrow> (b::'b::order) <= c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
848 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a <= f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
849 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
850 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
851 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
852 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
853 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
854 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
855 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
856 |
lemma order_subst2: "(a::'a::order) <= b \<Longrightarrow> f b <= (c::'c::order) \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
857 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
858 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
859 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
860 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
862 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
865 |
lemma ord_le_eq_subst: "a <= b \<Longrightarrow> f b = c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
866 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
867 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
868 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
869 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
870 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
871 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
872 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
873 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
874 |
lemma ord_eq_le_subst: "a = f b \<Longrightarrow> b <= c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
875 |
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a <= f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
876 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
877 |
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
878 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
879 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
880 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
881 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
882 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
883 |
lemma ord_less_eq_subst: "a < b \<Longrightarrow> f b = c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
884 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
885 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
886 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
887 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
888 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
889 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
890 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
891 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
892 |
lemma ord_eq_less_subst: "a = f b \<Longrightarrow> b < c \<Longrightarrow> |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
893 |
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
894 |
proof - |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
895 |
assume r: "!!x y. x < y \<Longrightarrow> f x < f y" |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
896 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
897 |
also assume "b < c" hence "f b < f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
898 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
899 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
900 |
|
60758 | 901 |
text \<open> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
902 |
Note that this list of rules is in reverse order of priorities. |
60758 | 903 |
\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
904 |
|
27682 | 905 |
lemmas [trans] = |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
906 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
907 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
908 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
909 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
910 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
911 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
912 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
913 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
914 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
915 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
916 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
917 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
918 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
919 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
920 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
921 |
mp |
27682 | 922 |
|
923 |
lemmas (in order) [trans] = |
|
924 |
neq_le_trans |
|
925 |
le_neq_trans |
|
926 |
||
927 |
lemmas (in preorder) [trans] = |
|
928 |
less_trans |
|
929 |
less_asym' |
|
930 |
le_less_trans |
|
931 |
less_le_trans |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
932 |
order_trans |
27682 | 933 |
|
934 |
lemmas (in order) [trans] = |
|
73411 | 935 |
order.antisym |
27682 | 936 |
|
937 |
lemmas (in ord) [trans] = |
|
938 |
ord_le_eq_trans |
|
939 |
ord_eq_le_trans |
|
940 |
ord_less_eq_trans |
|
941 |
ord_eq_less_trans |
|
942 |
||
943 |
lemmas [trans] = |
|
944 |
trans |
|
945 |
||
946 |
lemmas order_trans_rules = |
|
947 |
order_less_subst2 |
|
948 |
order_less_subst1 |
|
949 |
order_le_less_subst2 |
|
950 |
order_le_less_subst1 |
|
951 |
order_less_le_subst2 |
|
952 |
order_less_le_subst1 |
|
953 |
order_subst2 |
|
954 |
order_subst1 |
|
955 |
ord_le_eq_subst |
|
956 |
ord_eq_le_subst |
|
957 |
ord_less_eq_subst |
|
958 |
ord_eq_less_subst |
|
959 |
forw_subst |
|
960 |
back_subst |
|
961 |
rev_mp |
|
962 |
mp |
|
963 |
neq_le_trans |
|
964 |
le_neq_trans |
|
965 |
less_trans |
|
966 |
less_asym' |
|
967 |
le_less_trans |
|
968 |
less_le_trans |
|
969 |
order_trans |
|
73411 | 970 |
order.antisym |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
971 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
972 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
973 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
974 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
975 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
976 |
|
60758 | 977 |
text \<open>These support proving chains of decreasing inequalities |
75670
acf86c9f7698
fix document build error
Lukas Stevens <mail@lukas-stevens.de>
parents:
75669
diff
changeset
|
978 |
a \<open>\<ge>\<close> b \<open>\<ge>\<close> c ... in Isar proofs.\<close> |
21083 | 979 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
980 |
lemma xt1 [no_atp]: |
67091 | 981 |
"a = b \<Longrightarrow> b > c \<Longrightarrow> a > c" |
982 |
"a > b \<Longrightarrow> b = c \<Longrightarrow> a > c" |
|
983 |
"a = b \<Longrightarrow> b \<ge> c \<Longrightarrow> a \<ge> c" |
|
984 |
"a \<ge> b \<Longrightarrow> b = c \<Longrightarrow> a \<ge> c" |
|
985 |
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> x \<Longrightarrow> x = y" |
|
986 |
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> z \<Longrightarrow> x \<ge> z" |
|
987 |
"(x::'a::order) > y \<Longrightarrow> y \<ge> z \<Longrightarrow> x > z" |
|
988 |
"(x::'a::order) \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z" |
|
989 |
"(a::'a::order) > b \<Longrightarrow> b > a \<Longrightarrow> P" |
|
990 |
"(x::'a::order) > y \<Longrightarrow> y > z \<Longrightarrow> x > z" |
|
991 |
"(a::'a::order) \<ge> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a > b" |
|
992 |
"(a::'a::order) \<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b" |
|
993 |
"a = f b \<Longrightarrow> b > c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> a > f c" |
|
994 |
"a > b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> f a > c" |
|
995 |
"a = f b \<Longrightarrow> b \<ge> c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> a \<ge> f c" |
|
996 |
"a \<ge> b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> f a \<ge> c" |
|
25076 | 997 |
by auto |
21083 | 998 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
999 |
lemma xt2 [no_atp]: |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1000 |
assumes "(a::'a::order) \<ge> f b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1001 |
and "b \<ge> c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1002 |
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1003 |
shows "a \<ge> f c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1004 |
using assms by force |
21083 | 1005 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1006 |
lemma xt3 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1007 |
assumes "(a::'a::order) \<ge> b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1008 |
and "(f b::'b::order) \<ge> c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1009 |
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1010 |
shows "f a \<ge> c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1011 |
using assms by force |
21083 | 1012 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1013 |
lemma xt4 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1014 |
assumes "(a::'a::order) > f b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1015 |
and "(b::'b::order) \<ge> c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1016 |
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1017 |
shows "a > f c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1018 |
using assms by force |
21083 | 1019 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1020 |
lemma xt5 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1021 |
assumes "(a::'a::order) > b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1022 |
and "(f b::'b::order) \<ge> c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1023 |
and "\<And>x y. x > y \<Longrightarrow> f x > f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1024 |
shows "f a > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1025 |
using assms by force |
21083 | 1026 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1027 |
lemma xt6 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1028 |
assumes "(a::'a::order) \<ge> f b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1029 |
and "b > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1030 |
and "\<And>x y. x > y \<Longrightarrow> f x > f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1031 |
shows "a > f c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1032 |
using assms by force |
21083 | 1033 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1034 |
lemma xt7 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1035 |
assumes "(a::'a::order) \<ge> b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1036 |
and "(f b::'b::order) > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1037 |
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1038 |
shows "f a > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1039 |
using assms by force |
21083 | 1040 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1041 |
lemma xt8 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1042 |
assumes "(a::'a::order) > f b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1043 |
and "(b::'b::order) > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1044 |
and "\<And>x y. x > y \<Longrightarrow> f x > f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1045 |
shows "a > f c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1046 |
using assms by force |
21083 | 1047 |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1048 |
lemma xt9 [no_atp]: |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1049 |
assumes "(a::'a::order) > b" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1050 |
and "(f b::'b::order) > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1051 |
and "\<And>x y. x > y \<Longrightarrow> f x > f y" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1052 |
shows "f a > c" |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1053 |
using assms by force |
21083 | 1054 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53216
diff
changeset
|
1055 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
21083 | 1056 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1057 |
(* |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1058 |
Since "a \<ge> b" abbreviates "b \<le> a", the abbreviation "..." stands |
21083 | 1059 |
for the wrong thing in an Isar proof. |
1060 |
||
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1061 |
The extra transitivity rules can be used as follows: |
21083 | 1062 |
|
1063 |
lemma "(a::'a::order) > z" |
|
1064 |
proof - |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1065 |
have "a \<ge> b" (is "_ \<ge> ?rhs") |
21083 | 1066 |
sorry |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1067 |
also have "?rhs \<ge> c" (is "_ \<ge> ?rhs") |
21083 | 1068 |
sorry |
1069 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
1070 |
sorry |
|
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
75582
diff
changeset
|
1071 |
also (xtrans) have "?rhs \<ge> e" (is "_ \<ge> ?rhs") |
21083 | 1072 |
sorry |
1073 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
1074 |
sorry |
|
1075 |
also (xtrans) have "?rhs > z" |
|
1076 |
sorry |
|
1077 |
finally (xtrans) show ?thesis . |
|
1078 |
qed |
|
1079 |
||
1080 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
1081 |
leave out the "(xtrans)" above. |
|
1082 |
*) |
|
1083 |
||
23881 | 1084 |
|
60758 | 1085 |
subsection \<open>min and max -- fundamental\<close> |
54860 | 1086 |
|
1087 |
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1088 |
"min a b = (if a \<le> b then a else b)" |
|
1089 |
||
1090 |
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1091 |
"max a b = (if a \<le> b then b else a)" |
|
1092 |
||
45931 | 1093 |
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1094 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1095 |
|
54857 | 1096 |
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1097 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1098 |
|
61076 | 1099 |
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1100 |
by (simp add:min_def) |
45893 | 1101 |
|
61076 | 1102 |
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1103 |
by (simp add: max_def) |
45893 | 1104 |
|
61630 | 1105 |
lemma max_min_same [simp]: |
1106 |
fixes x y :: "'a :: linorder" |
|
1107 |
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y" |
|
1108 |
by(auto simp add: max_def min_def) |
|
45893 | 1109 |
|
66936 | 1110 |
|
60758 | 1111 |
subsection \<open>(Unique) top and bottom elements\<close> |
28685 | 1112 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1113 |
class bot = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1114 |
fixes bot :: 'a ("\<bottom>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1115 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1116 |
class order_bot = order + bot + |
51487 | 1117 |
assumes bot_least: "\<bottom> \<le> a" |
54868 | 1118 |
begin |
51487 | 1119 |
|
61605 | 1120 |
sublocale bot: ordering_top greater_eq greater bot |
61169 | 1121 |
by standard (fact bot_least) |
51487 | 1122 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1123 |
lemma le_bot: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1124 |
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>" |
51487 | 1125 |
by (fact bot.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1126 |
|
43816 | 1127 |
lemma bot_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1128 |
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>" |
51487 | 1129 |
by (fact bot.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1130 |
|
51487 | 1131 |
lemma not_less_bot: |
1132 |
"\<not> a < \<bottom>" |
|
1133 |
by (fact bot.extremum_strict) |
|
43816 | 1134 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1135 |
lemma bot_less: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1136 |
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a" |
51487 | 1137 |
by (fact bot.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1138 |
|
67452 | 1139 |
lemma max_bot[simp]: "max bot x = x" |
1140 |
by(simp add: max_def bot_unique) |
|
1141 |
||
1142 |
lemma max_bot2[simp]: "max x bot = x" |
|
1143 |
by(simp add: max_def bot_unique) |
|
1144 |
||
1145 |
lemma min_bot[simp]: "min bot x = bot" |
|
1146 |
by(simp add: min_def bot_unique) |
|
1147 |
||
1148 |
lemma min_bot2[simp]: "min x bot = bot" |
|
1149 |
by(simp add: min_def bot_unique) |
|
1150 |
||
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1151 |
end |
41082 | 1152 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1153 |
class top = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1154 |
fixes top :: 'a ("\<top>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1155 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1156 |
class order_top = order + top + |
51487 | 1157 |
assumes top_greatest: "a \<le> \<top>" |
54868 | 1158 |
begin |
51487 | 1159 |
|
61605 | 1160 |
sublocale top: ordering_top less_eq less top |
61169 | 1161 |
by standard (fact top_greatest) |
51487 | 1162 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1163 |
lemma top_le: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1164 |
"\<top> \<le> a \<Longrightarrow> a = \<top>" |
51487 | 1165 |
by (fact top.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1166 |
|
43816 | 1167 |
lemma top_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1168 |
"\<top> \<le> a \<longleftrightarrow> a = \<top>" |
51487 | 1169 |
by (fact top.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1170 |
|
51487 | 1171 |
lemma not_top_less: |
1172 |
"\<not> \<top> < a" |
|
1173 |
by (fact top.extremum_strict) |
|
43816 | 1174 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1175 |
lemma less_top: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1176 |
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>" |
51487 | 1177 |
by (fact top.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1178 |
|
67452 | 1179 |
lemma max_top[simp]: "max top x = top" |
1180 |
by(simp add: max_def top_unique) |
|
1181 |
||
1182 |
lemma max_top2[simp]: "max x top = top" |
|
1183 |
by(simp add: max_def top_unique) |
|
1184 |
||
1185 |
lemma min_top[simp]: "min top x = x" |
|
1186 |
by(simp add: min_def top_unique) |
|
1187 |
||
1188 |
lemma min_top2[simp]: "min x top = x" |
|
1189 |
by(simp add: min_def top_unique) |
|
1190 |
||
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1191 |
end |
28685 | 1192 |
|
1193 |
||
60758 | 1194 |
subsection \<open>Dense orders\<close> |
27823 | 1195 |
|
53216 | 1196 |
class dense_order = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1197 |
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1198 |
|
53216 | 1199 |
class dense_linorder = linorder + dense_order |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1200 |
begin |
27823 | 1201 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1202 |
lemma dense_le: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1203 |
fixes y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1204 |
assumes "\<And>x. x < y \<Longrightarrow> x \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1205 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1206 |
proof (rule ccontr) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1207 |
assume "\<not> ?thesis" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1208 |
hence "z < y" by simp |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1209 |
from dense[OF this] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1210 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1211 |
moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] . |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1212 |
ultimately show False by auto |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1213 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1214 |
|
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1215 |
lemma dense_le_bounded: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1216 |
fixes x y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1217 |
assumes "x < y" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1218 |
assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1219 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1220 |
proof (rule dense_le) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1221 |
fix w assume "w < y" |
60758 | 1222 |
from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1223 |
from linear[of u w] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1224 |
show "w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1225 |
proof (rule disjE) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1226 |
assume "u \<le> w" |
60758 | 1227 |
from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close> |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1228 |
show "w \<le> z" by (rule *) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1229 |
next |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1230 |
assume "w \<le> u" |
60758 | 1231 |
from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>] |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1232 |
show "w \<le> z" by (rule order_trans) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1233 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1234 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1235 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1236 |
lemma dense_ge: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1237 |
fixes y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1238 |
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1239 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1240 |
proof (rule ccontr) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1241 |
assume "\<not> ?thesis" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1242 |
hence "z < y" by simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1243 |
from dense[OF this] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1244 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1245 |
moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] . |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1246 |
ultimately show False by auto |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1247 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1248 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1249 |
lemma dense_ge_bounded: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1250 |
fixes x y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1251 |
assumes "z < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1252 |
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1253 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1254 |
proof (rule dense_ge) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1255 |
fix w assume "z < w" |
60758 | 1256 |
from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1257 |
from linear[of u w] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1258 |
show "y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1259 |
proof (rule disjE) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1260 |
assume "w \<le> u" |
60758 | 1261 |
from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>] |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1262 |
show "y \<le> w" by (rule *) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1263 |
next |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1264 |
assume "u \<le> w" |
60758 | 1265 |
from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close> |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1266 |
show "y \<le> w" by (rule order_trans) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1267 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1268 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1269 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1270 |
end |
27823 | 1271 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1272 |
class no_top = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1273 |
assumes gt_ex: "\<exists>y. x < y" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1274 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1275 |
class no_bot = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1276 |
assumes lt_ex: "\<exists>y. y < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1277 |
|
53216 | 1278 |
class unbounded_dense_linorder = dense_linorder + no_top + no_bot |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1279 |
|
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
1280 |
|
60758 | 1281 |
subsection \<open>Wellorders\<close> |
27823 | 1282 |
|
1283 |
class wellorder = linorder + |
|
1284 |
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" |
|
1285 |
begin |
|
1286 |
||
1287 |
lemma wellorder_Least_lemma: |
|
1288 |
fixes k :: 'a |
|
1289 |
assumes "P k" |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1290 |
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k" |
27823 | 1291 |
proof - |
1292 |
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" |
|
1293 |
using assms proof (induct k rule: less_induct) |
|
1294 |
case (less x) then have "P x" by simp |
|
1295 |
show ?case proof (rule classical) |
|
1296 |
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" |
|
1297 |
have "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
1298 |
proof (rule classical) |
|
1299 |
fix y |
|
38705 | 1300 |
assume "P y" and "\<not> x \<le> y" |
27823 | 1301 |
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
1302 |
by (auto simp add: not_le) |
|
1303 |
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1304 |
by auto |
|
1305 |
then show "x \<le> y" by auto |
|
1306 |
qed |
|
60758 | 1307 |
with \<open>P x\<close> have Least: "(LEAST a. P a) = x" |
27823 | 1308 |
by (rule Least_equality) |
60758 | 1309 |
with \<open>P x\<close> show ?thesis by simp |
27823 | 1310 |
qed |
1311 |
qed |
|
1312 |
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto |
|
1313 |
qed |
|
1314 |
||
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67405
diff
changeset
|
1315 |
\<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close> |
27823 | 1316 |
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)" |
1317 |
by (erule exE) (erule LeastI) |
|
1318 |
||
1319 |
lemma LeastI2: |
|
1320 |
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1321 |
by (blast intro: LeastI) |
|
1322 |
||
1323 |
lemma LeastI2_ex: |
|
1324 |
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1325 |
by (blast intro: LeastI_ex) |
|
1326 |
||
38705 | 1327 |
lemma LeastI2_wellorder: |
1328 |
assumes "P a" |
|
1329 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
|
1330 |
shows "Q (Least P)" |
|
1331 |
proof (rule LeastI2_order) |
|
60758 | 1332 |
show "P (Least P)" using \<open>P a\<close> by (rule LeastI) |
38705 | 1333 |
next |
1334 |
fix y assume "P y" thus "Least P \<le> y" by (rule Least_le) |
|
1335 |
next |
|
1336 |
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2)) |
|
1337 |
qed |
|
1338 |
||
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1339 |
lemma LeastI2_wellorder_ex: |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1340 |
assumes "\<exists>x. P x" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1341 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1342 |
shows "Q (Least P)" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1343 |
using assms by clarify (blast intro!: LeastI2_wellorder) |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1344 |
|
27823 | 1345 |
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k" |
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1346 |
apply (simp add: not_le [symmetric]) |
27823 | 1347 |
apply (erule contrapos_nn) |
1348 |
apply (erule Least_le) |
|
1349 |
done |
|
1350 |
||
64287 | 1351 |
lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs") |
1352 |
proof |
|
1353 |
assume ?rhs thus ?lhs by blast |
|
1354 |
next |
|
1355 |
assume H: ?lhs then obtain n where n: "P n" by blast |
|
1356 |
let ?x = "Least P" |
|
1357 |
{ fix m assume m: "m < ?x" |
|
1358 |
from not_less_Least[OF m] have "\<not> P m" . } |
|
1359 |
with LeastI_ex[OF H] show ?rhs by blast |
|
1360 |
qed |
|
1361 |
||
38705 | 1362 |
end |
27823 | 1363 |
|
28685 | 1364 |
|
69593 | 1365 |
subsection \<open>Order on \<^typ>\<open>bool\<close>\<close> |
28685 | 1366 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1367 |
instantiation bool :: "{order_bot, order_top, linorder}" |
28685 | 1368 |
begin |
1369 |
||
1370 |
definition |
|
41080 | 1371 |
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q" |
28685 | 1372 |
|
1373 |
definition |
|
61076 | 1374 |
[simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q" |
28685 | 1375 |
|
1376 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1377 |
[simp]: "\<bottom> \<longleftrightarrow> False" |
28685 | 1378 |
|
1379 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1380 |
[simp]: "\<top> \<longleftrightarrow> True" |
28685 | 1381 |
|
1382 |
instance proof |
|
41080 | 1383 |
qed auto |
28685 | 1384 |
|
15524 | 1385 |
end |
28685 | 1386 |
|
1387 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
41080 | 1388 |
by simp |
28685 | 1389 |
|
1390 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
41080 | 1391 |
by simp |
28685 | 1392 |
|
1393 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
41080 | 1394 |
by simp |
28685 | 1395 |
|
1396 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
41080 | 1397 |
by simp |
32899 | 1398 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1399 |
lemma bot_boolE: "\<bottom> \<Longrightarrow> P" |
41080 | 1400 |
by simp |
32899 | 1401 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1402 |
lemma top_boolI: \<top> |
41080 | 1403 |
by simp |
28685 | 1404 |
|
1405 |
lemma [code]: |
|
1406 |
"False \<le> b \<longleftrightarrow> True" |
|
1407 |
"True \<le> b \<longleftrightarrow> b" |
|
1408 |
"False < b \<longleftrightarrow> b" |
|
1409 |
"True < b \<longleftrightarrow> False" |
|
41080 | 1410 |
by simp_all |
28685 | 1411 |
|
1412 |
||
69593 | 1413 |
subsection \<open>Order on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close> |
28685 | 1414 |
|
1415 |
instantiation "fun" :: (type, ord) ord |
|
1416 |
begin |
|
1417 |
||
1418 |
definition |
|
37767 | 1419 |
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)" |
28685 | 1420 |
|
1421 |
definition |
|
61076 | 1422 |
"(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)" |
28685 | 1423 |
|
1424 |
instance .. |
|
1425 |
||
1426 |
end |
|
1427 |
||
1428 |
instance "fun" :: (type, preorder) preorder proof |
|
1429 |
qed (auto simp add: le_fun_def less_fun_def |
|
73411 | 1430 |
intro: order_trans order.antisym) |
28685 | 1431 |
|
1432 |
instance "fun" :: (type, order) order proof |
|
73411 | 1433 |
qed (auto simp add: le_fun_def intro: order.antisym) |
28685 | 1434 |
|
41082 | 1435 |
instantiation "fun" :: (type, bot) bot |
1436 |
begin |
|
1437 |
||
1438 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1439 |
"\<bottom> = (\<lambda>x. \<bottom>)" |
41082 | 1440 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1441 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1442 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1443 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1444 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1445 |
instantiation "fun" :: (type, order_bot) order_bot |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1446 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1447 |
|
49769 | 1448 |
lemma bot_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1449 |
"\<bottom> x = \<bottom>" |
41082 | 1450 |
by (simp add: bot_fun_def) |
1451 |
||
1452 |
instance proof |
|
46884 | 1453 |
qed (simp add: le_fun_def) |
41082 | 1454 |
|
1455 |
end |
|
1456 |
||
28685 | 1457 |
instantiation "fun" :: (type, top) top |
1458 |
begin |
|
1459 |
||
1460 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1461 |
[no_atp]: "\<top> = (\<lambda>x. \<top>)" |
28685 | 1462 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1463 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1464 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1465 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1466 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1467 |
instantiation "fun" :: (type, order_top) order_top |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1468 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1469 |
|
49769 | 1470 |
lemma top_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1471 |
"\<top> x = \<top>" |
41080 | 1472 |
by (simp add: top_fun_def) |
1473 |
||
28685 | 1474 |
instance proof |
46884 | 1475 |
qed (simp add: le_fun_def) |
28685 | 1476 |
|
1477 |
end |
|
1478 |
||
1479 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
1480 |
unfolding le_fun_def by simp |
|
1481 |
||
1482 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
1483 |
unfolding le_fun_def by simp |
|
1484 |
||
1485 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
54860 | 1486 |
by (rule le_funE) |
28685 | 1487 |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1488 |
|
60758 | 1489 |
subsection \<open>Order on unary and binary predicates\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1490 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1491 |
lemma predicate1I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1492 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1493 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1494 |
apply (rule le_funI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1495 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1496 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1497 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1498 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1499 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1500 |
lemma predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1501 |
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1502 |
apply (erule le_funE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1503 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1504 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1505 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1506 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1507 |
lemma rev_predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1508 |
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1509 |
by (rule predicate1D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1510 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1511 |
lemma predicate2I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1512 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1513 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1514 |
apply (rule le_funI)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1515 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1516 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1517 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1518 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1519 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1520 |
lemma predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1521 |
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1522 |
apply (erule le_funE)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1523 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1524 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1525 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1526 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1527 |
lemma rev_predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1528 |
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1529 |
by (rule predicate2D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1530 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1531 |
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1532 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1533 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1534 |
lemma bot2E: "\<bottom> x y \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1535 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1536 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1537 |
lemma top1I: "\<top> x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1538 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1539 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1540 |
lemma top2I: "\<top> x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1541 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1542 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1543 |
|
60758 | 1544 |
subsection \<open>Name duplicates\<close> |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1545 |
|
73411 | 1546 |
lemmas antisym = order.antisym |
1547 |
lemmas eq_iff = order.eq_iff |
|
1548 |
||
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1549 |
lemmas order_eq_refl = preorder_class.eq_refl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1550 |
lemmas order_less_irrefl = preorder_class.less_irrefl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1551 |
lemmas order_less_imp_le = preorder_class.less_imp_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1552 |
lemmas order_less_not_sym = preorder_class.less_not_sym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1553 |
lemmas order_less_asym = preorder_class.less_asym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1554 |
lemmas order_less_trans = preorder_class.less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1555 |
lemmas order_le_less_trans = preorder_class.le_less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1556 |
lemmas order_less_le_trans = preorder_class.less_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1557 |
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1558 |
lemmas order_less_imp_triv = preorder_class.less_imp_triv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1559 |
lemmas order_less_asym' = preorder_class.less_asym' |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1560 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1561 |
lemmas order_less_le = order_class.less_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1562 |
lemmas order_le_less = order_class.le_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1563 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1564 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1565 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1566 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1567 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
73411 | 1568 |
lemmas order_eq_iff = order_class.order.eq_iff |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1569 |
lemmas order_antisym_conv = order_class.antisym_conv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1570 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1571 |
lemmas linorder_linear = linorder_class.linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1572 |
lemmas linorder_less_linear = linorder_class.less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1573 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1574 |
lemmas linorder_le_cases = linorder_class.le_cases |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1575 |
lemmas linorder_not_less = linorder_class.not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1576 |
lemmas linorder_not_le = linorder_class.not_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1577 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1578 |
lemmas linorder_neqE = linorder_class.neqE |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1579 |
|
28685 | 1580 |
end |