author  haftmann 
Wed, 09 May 2007 07:53:06 +0200  
changeset 22886  cdff6ef76009 
parent 22841  83b9f2d3fb3c 
child 22916  8caf6da610e2 
permissions  rwrr 
15524  1 
(* Title: HOL/Orderings.thy 
2 
ID: $Id$ 

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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson 

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*) 

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21329  6 
header {* Syntactic and abstract orders *} 
15524  7 

8 
theory Orderings 

22886  9 
imports Code_Generator 
15524  10 
begin 
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21329  12 
subsection {* Order syntax *} 
15524  13 

22473  14 
class ord = type + 
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubseteq>" 50) 
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) 
21204  17 
begin 
18 

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notation 

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less_eq ("op \<^loc><=") and 
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less_eq ("(_/ \<^loc><= _)" [51, 51] 50) and 
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less ("op \<^loc><") and 
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less ("(_/ \<^loc>< _)" [51, 51] 50) 
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notation (xsymbols) 
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less_eq ("op \<^loc>\<le>") and 
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) 
15524  28 

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notation (HTML output) 
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less_eq ("op \<^loc>\<le>") and 
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) 
21204  32 

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abbreviation (input) 

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greater (infix "\<^loc>>" 50) where 
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"x \<^loc>> y \<equiv> y \<^loc>< x" 
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abbreviation (input) 
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greater_eq (infix "\<^loc>>=" 50) where 
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"x \<^loc>>= y \<equiv> y \<^loc><= x" 
21204  40 

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notation (input) 
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greater_eq (infix "\<^loc>\<ge>" 50) 
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22738  44 
text {* 
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syntactic min/max  these definitions reach 

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their usual semantics in class linorder ahead. 

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*} 

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definition 

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min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where 

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"min a b = (if a \<^loc>\<le> b then a else b)" 
22738  52 

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definition 

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max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where 

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"max a b = (if a \<^loc>\<le> b then b else a)" 
22738  56 

21204  57 
end 
58 

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notation 

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less_eq ("op <=") and 
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less_eq ("(_/ <= _)" [51, 51] 50) and 
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less ("op <") and 
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less ("(_/ < _)" [51, 51] 50) 
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notation (xsymbols) 

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less_eq ("op \<le>") and 
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less_eq ("(_/ \<le> _)" [51, 51] 50) 
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notation (HTML output) 
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less_eq ("op \<le>") and 
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less_eq ("(_/ \<le> _)" [51, 51] 50) 
20714  72 

19536  73 
abbreviation (input) 
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greater (infix ">" 50) where 
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"x > y \<equiv> y < x" 
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abbreviation (input) 
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greater_eq (infix ">=" 50) where 
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"x >= y \<equiv> y <= x" 
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notation (input) 
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greater_eq (infix "\<ge>" 50) 
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22738  84 
definition 
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min :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where 

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"min a b = (if a \<le> b then a else b)" 

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88 
definition 

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max :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where 

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"max a b = (if a \<le> b then b else a)" 

91 

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lemma min_linorder: 

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"ord.min (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = min" 

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by rule+ (simp add: min_def ord_class.min_def) 

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lemma max_linorder: 

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"ord.max (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = max" 

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by rule+ (simp add: max_def ord_class.max_def) 

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15524  100 

22841  101 
subsection {* Partial orders *} 
15524  102 

22841  103 
class order = ord + 
22316  104 
assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" 
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and order_refl [iff]: "x \<sqsubseteq> x" 
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and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z" 
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assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" 
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21248  109 
begin 
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15524  111 
text {* Reflexivity. *} 
112 

22841  113 
lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y" 
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 {* This form is useful with the classical reasoner. *} 
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by (erule ssubst) (rule order_refl) 
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22841  117 
lemma less_irrefl [iff]: "\<not> x \<^loc>< x" 
21248  118 
by (simp add: less_le) 
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22841  120 
lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y" 
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 {* NOT suitable for iff, since it can cause PROOF FAILED. *} 
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by (simp add: less_le) blast 
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lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y" 
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unfolding less_le by blast 
15524  126 

22841  127 
lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y" 
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unfolding less_le by blast 
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22841  130 
lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" 
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by (erule contrapos_pn, erule subst, rule less_irrefl) 
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133 

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text {* Useful for simplification, but too risky to include by default. *} 

135 

22841  136 
lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False" 
21329  137 
by auto 
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lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False" 
21329  140 
by auto 
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142 

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text {* Transitivity rules for calculational reasoning *} 

144 

22841  145 
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b" 
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by (simp add: less_le) 
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lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b" 
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by (simp add: less_le) 

21329  150 

15524  151 

152 
text {* Asymmetry. *} 

153 

22841  154 
lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)" 
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by (simp add: less_le antisym) 
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22841  157 
lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P" 
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by (drule less_not_sym, erule contrapos_np) simp 
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lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x" 
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by (blast intro: antisym) 
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lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym) 
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" 
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by (erule contrapos_pn, erule subst, rule less_irrefl) 
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21083  169 

15524  170 
text {* Transitivity. *} 
171 

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lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" 
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by (simp add: less_le) (blast intro: order_trans antisym) 
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lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" 
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by (simp add: less_le) (blast intro: order_trans antisym) 
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lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z" 
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by (simp add: less_le) (blast intro: order_trans antisym) 
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181 

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text {* Useful for simplification, but too risky to include by default. *} 

183 

22841  184 
lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True" 
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by (blast elim: less_asym) 
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22841  187 
lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True" 
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by (blast elim: less_asym) 
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21248  190 

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text {* Transitivity rules for calculational reasoning *} 
15524  192 

22841  193 
lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P" 
21248  194 
by (rule less_asym) 
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196 
end 

15524  197 

21329  198 

199 
subsection {* Linear (total) orders *} 

200 

22316  201 
class linorder = order + 
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assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" 
21248  203 
begin 
204 

22841  205 
lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x" 
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unfolding less_le using less_le linear by blast 
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22841  208 
lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x" 
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by (simp add: le_less less_linear) 
21248  210 

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lemma le_cases [case_names le ge]: 

22841  212 
"(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P" 
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using linear by blast 
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lemma linorder_cases [case_names less equal greater]: 
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"(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P" 
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using less_linear by blast 
21248  218 

22841  219 
lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x" 
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apply (simp add: less_le) 
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using linear apply (blast intro: antisym) 

15524  222 
done 
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22841  224 
lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x" 
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apply (simp add: less_le) 
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using linear apply (blast intro: antisym) 

15524  227 
done 
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22841  229 
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x" 
21412  230 
by (cut_tac x = x and y = y in less_linear, auto) 
15524  231 

22841  232 
lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R" 
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by (simp add: neq_iff) blast 
15524  234 

22841  235 
lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 
15524  237 

22841  238 
lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 
15524  240 

22841  241 
lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 
15524  243 

16796  244 
text{*Replacing the old Nat.leI*} 
22841  245 
lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x" 
21248  246 
unfolding not_less . 
16796  247 

22841  248 
lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y" 
21248  249 
unfolding not_less . 
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(*FIXME inappropriate name (or delete altogether)*) 

22841  252 
lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y" 
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unfolding not_le . 
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22738  255 
text {* min/max properties *} 
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lemma min_le_iff_disj: 
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"min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z" 
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unfolding min_def using linear by (auto intro: order_trans) 
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lemma le_max_iff_disj: 
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"z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y" 
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unfolding max_def using linear by (auto intro: order_trans) 
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lemma min_less_iff_disj: 
22841  266 
"min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z" 
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unfolding min_def le_less using less_linear by (auto intro: less_trans) 
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lemma less_max_iff_disj: 
22841  270 
"z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y" 
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unfolding max_def le_less using less_linear by (auto intro: less_trans) 
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lemma min_less_iff_conj [simp]: 
22841  274 
"z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y" 
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lemma max_less_iff_conj [simp]: 
22841  278 
"max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z" 
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lemma split_min: 
22841  282 
"P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)" 
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by (simp add: min_def) 
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lemma split_max: 
22841  286 
"P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)" 
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by (simp add: max_def) 
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21248  289 
end 
290 

291 

292 
subsection {* Name duplicates *} 

293 

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lemmas order_less_le = less_le 
22841  295 
lemmas order_eq_refl = order_class.eq_refl 
296 
lemmas order_less_irrefl = order_class.less_irrefl 

297 
lemmas order_le_less = order_class.le_less 

298 
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq 

299 
lemmas order_less_imp_le = order_class.less_imp_le 

300 
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq 

301 
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 

302 
lemmas order_neq_le_trans = order_class.neq_le_trans 

303 
lemmas order_le_neq_trans = order_class.le_neq_trans 

22316  304 

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305 
lemmas order_antisym = antisym 
22316  306 
lemmas order_less_not_sym = order_class.less_not_sym 
307 
lemmas order_less_asym = order_class.less_asym 

308 
lemmas order_eq_iff = order_class.eq_iff 

309 
lemmas order_antisym_conv = order_class.antisym_conv 

310 
lemmas less_imp_neq = order_class.less_imp_neq 

311 
lemmas order_less_trans = order_class.less_trans 

312 
lemmas order_le_less_trans = order_class.le_less_trans 

313 
lemmas order_less_le_trans = order_class.less_le_trans 

314 
lemmas order_less_imp_not_less = order_class.less_imp_not_less 

315 
lemmas order_less_imp_triv = order_class.less_imp_triv 

316 
lemmas order_less_asym' = order_class.less_asym' 

317 

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318 
lemmas linorder_linear = linear 
22316  319 
lemmas linorder_less_linear = linorder_class.less_linear 
320 
lemmas linorder_le_less_linear = linorder_class.le_less_linear 

321 
lemmas linorder_le_cases = linorder_class.le_cases 

322 
lemmas linorder_not_less = linorder_class.not_less 

323 
lemmas linorder_not_le = linorder_class.not_le 

324 
lemmas linorder_neq_iff = linorder_class.neq_iff 

325 
lemmas linorder_neqE = linorder_class.neqE 

326 
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 

327 
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 

328 
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 

329 
lemmas leI = linorder_class.leI 

330 
lemmas leD = linorder_class.leD 

331 
lemmas not_leE = linorder_class.not_leE 

16796  332 

21083  333 

334 
subsection {* Reasoning tools setup *} 

335 

21091  336 
ML {* 
337 
local 

338 

339 
fun decomp_gen sort thy (Trueprop $ t) = 

21248  340 
let 
341 
fun of_sort t = 

342 
let 

343 
val T = type_of t 

344 
in 

21091  345 
(* exclude numeric types: linear arithmetic subsumes transitivity *) 
21248  346 
T <> HOLogic.natT andalso T <> HOLogic.intT 
347 
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) 

348 
end; 

349 
fun dec (Const ("Not", _) $ t) = (case dec t 

350 
of NONE => NONE 

351 
 SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) 

352 
 dec (Const ("op =", _) $ t1 $ t2) = 

353 
if of_sort t1 

354 
then SOME (t1, "=", t2) 

355 
else NONE 

356 
 dec (Const ("Orderings.less_eq", _) $ t1 $ t2) = 

357 
if of_sort t1 

358 
then SOME (t1, "<=", t2) 

359 
else NONE 

360 
 dec (Const ("Orderings.less", _) $ t1 $ t2) = 

361 
if of_sort t1 

362 
then SOME (t1, "<", t2) 

363 
else NONE 

364 
 dec _ = NONE; 

21091  365 
in dec t end; 
366 

367 
in 

368 

22841  369 
(* sorry  there is no preorder class 
21248  370 
structure Quasi_Tac = Quasi_Tac_Fun ( 
371 
struct 

372 
val le_trans = thm "order_trans"; 

373 
val le_refl = thm "order_refl"; 

374 
val eqD1 = thm "order_eq_refl"; 

375 
val eqD2 = thm "sym" RS thm "order_eq_refl"; 

376 
val less_reflE = thm "order_less_irrefl" RS thm "notE"; 

377 
val less_imp_le = thm "order_less_imp_le"; 

378 
val le_neq_trans = thm "order_le_neq_trans"; 

379 
val neq_le_trans = thm "order_neq_le_trans"; 

380 
val less_imp_neq = thm "less_imp_neq"; 

22738  381 
val decomp_trans = decomp_gen ["Orderings.preorder"]; 
382 
val decomp_quasi = decomp_gen ["Orderings.preorder"]; 

22841  383 
end);*) 
21091  384 

385 
structure Order_Tac = Order_Tac_Fun ( 

21248  386 
struct 
387 
val less_reflE = thm "order_less_irrefl" RS thm "notE"; 

388 
val le_refl = thm "order_refl"; 

389 
val less_imp_le = thm "order_less_imp_le"; 

390 
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; 

391 
val not_leI = thm "linorder_not_le" RS thm "iffD2"; 

392 
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; 

393 
val not_leD = thm "linorder_not_le" RS thm "iffD1"; 

394 
val eqI = thm "order_antisym"; 

395 
val eqD1 = thm "order_eq_refl"; 

396 
val eqD2 = thm "sym" RS thm "order_eq_refl"; 

397 
val less_trans = thm "order_less_trans"; 

398 
val less_le_trans = thm "order_less_le_trans"; 

399 
val le_less_trans = thm "order_le_less_trans"; 

400 
val le_trans = thm "order_trans"; 

401 
val le_neq_trans = thm "order_le_neq_trans"; 

402 
val neq_le_trans = thm "order_neq_le_trans"; 

403 
val less_imp_neq = thm "less_imp_neq"; 

404 
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; 

405 
val not_sym = thm "not_sym"; 

406 
val decomp_part = decomp_gen ["Orderings.order"]; 

407 
val decomp_lin = decomp_gen ["Orderings.linorder"]; 

408 
end); 

21091  409 

410 
end; 

411 
*} 

412 

21083  413 
setup {* 
414 
let 

415 

416 
fun prp t thm = (#prop (rep_thm thm) = t); 

15524  417 

21083  418 
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = 
419 
let val prems = prems_of_ss ss; 

420 
val less = Const("Orderings.less",T); 

421 
val t = HOLogic.mk_Trueprop(le $ s $ r); 

422 
in case find_first (prp t) prems of 

423 
NONE => 

424 
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) 

425 
in case find_first (prp t) prems of 

426 
NONE => NONE 

22738  427 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1})) 
21083  428 
end 
22738  429 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv})) 
21083  430 
end 
431 
handle THM _ => NONE; 

15524  432 

21083  433 
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = 
434 
let val prems = prems_of_ss ss; 

435 
val le = Const("Orderings.less_eq",T); 

436 
val t = HOLogic.mk_Trueprop(le $ r $ s); 

437 
in case find_first (prp t) prems of 

438 
NONE => 

439 
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) 

440 
in case find_first (prp t) prems of 

441 
NONE => NONE 

22738  442 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3})) 
21083  443 
end 
22738  444 
 SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2})) 
21083  445 
end 
446 
handle THM _ => NONE; 

15524  447 

21248  448 
fun add_simprocs procs thy = 
449 
(Simplifier.change_simpset_of thy (fn ss => ss 

450 
addsimprocs (map (fn (name, raw_ts, proc) => 

451 
Simplifier.simproc thy name raw_ts proc)) procs); thy); 

452 
fun add_solver name tac thy = 

453 
(Simplifier.change_simpset_of thy (fn ss => ss addSolver 

454 
(mk_solver name (K tac))); thy); 

21083  455 

456 
in 

21248  457 
add_simprocs [ 
458 
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), 

459 
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) 

460 
] 

461 
#> add_solver "Trans_linear" Order_Tac.linear_tac 

462 
#> add_solver "Trans_partial" Order_Tac.partial_tac 

463 
(* Adding the transitivity reasoners also as safe solvers showed a slight 

464 
speed up, but the reasoning strength appears to be not higher (at least 

465 
no breaking of additional proofs in the entire HOL distribution, as 

466 
of 5 March 2004, was observed). *) 

21083  467 
end 
468 
*} 

15524  469 

470 

21083  471 
subsection {* Bounded quantifiers *} 
472 

473 
syntax 

21180
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474 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
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475 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
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diff
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476 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
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wenzelm
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21091
diff
changeset

477 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
21083  478 

21180
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diff
changeset

479 
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) 
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fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
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diff
changeset

480 
"_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:
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diff
changeset

481 
"_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
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21091
diff
changeset

482 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) 
21083  483 

484 
syntax (xsymbols) 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
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21091
diff
changeset

485 
"_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

486 
"_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

487 
"_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

488 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  489 

21180
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fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

490 
"_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

491 
"_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

492 
"_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

493 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  494 

495 
syntax (HOL) 

21180
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wenzelm
parents:
21091
diff
changeset

496 
"_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

497 
"_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

498 
"_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

499 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
21083  500 

501 
syntax (HTML output) 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

502 
"_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

503 
"_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

504 
"_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

505 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  506 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

507 
"_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

508 
"_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

509 
"_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

510 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  511 

512 
translations 

513 
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" 

514 
"EX x<y. P" => "EX x. x < y \<and> P" 

515 
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" 

516 
"EX x<=y. P" => "EX x. x <= y \<and> P" 

517 
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" 

518 
"EX x>y. P" => "EX x. x > y \<and> P" 

519 
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" 

520 
"EX x>=y. P" => "EX x. x >= y \<and> P" 

521 

522 
print_translation {* 

523 
let 

22377  524 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 
525 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

526 
val impl = @{const_syntax "op >"}; 

527 
val conj = @{const_syntax "op &"}; 

528 
val less = @{const_syntax "less"}; 

529 
val less_eq = @{const_syntax "less_eq"}; 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

530 

f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

531 
val trans = 
21524  532 
[((All_binder, impl, less), ("_All_less", "_All_greater")), 
533 
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), 

534 
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), 

535 
((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

536 

22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

537 
fun matches_bound v t = 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

538 
case t of (Const ("_bound", _) $ Free (v', _)) => (v = v') 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

539 
 _ => false 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

540 
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v  _ => false) 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

541 
fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P 
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

542 

f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

543 
fun tr' q = (q, 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

544 
fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] => 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

545 
(case AList.lookup (op =) trans (q, c, d) of 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

546 
NONE => raise Match 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

547 
 SOME (l, g) => 
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

548 
if matches_bound v t andalso not (contains_var v u) then mk v l u P 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

549 
else if matches_bound v u andalso not (contains_var v t) then mk v g t P 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

550 
else raise Match) 
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

551 
 _ => raise Match); 
21524  552 
in [tr' All_binder, tr' Ex_binder] end 
21083  553 
*} 
554 

555 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

556 
subsection {* Transitivity reasoning *} 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

557 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

558 
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

559 
by (rule subst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

560 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

561 
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

562 
by (rule ssubst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

563 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

564 
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

565 
by (rule subst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

566 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

567 
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

568 
by (rule ssubst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

569 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

570 
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

571 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

572 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

573 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

574 
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

575 
also assume "f b < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

576 
finally (order_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

577 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

578 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

579 
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

580 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

581 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

582 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

583 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

584 
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

585 
finally (order_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

586 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

587 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

588 
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

589 
(!!x y. x <= y ==> f x <= f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

590 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

591 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

592 
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

593 
also assume "f b < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

594 
finally (order_le_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

595 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

596 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

597 
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

598 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

599 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

600 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

601 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

602 
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

603 
finally (order_le_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

604 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

605 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

606 
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

607 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

608 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

609 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

610 
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

611 
also assume "f b <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

612 
finally (order_less_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

613 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

614 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

615 
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

616 
(!!x y. x <= y ==> f x <= f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

617 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

618 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

619 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

620 
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

621 
finally (order_less_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

622 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

623 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

624 
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

625 
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

626 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

627 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

628 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

629 
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

630 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

631 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

632 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

633 
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

634 
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

635 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

636 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

637 
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

638 
also assume "f b <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

639 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

640 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

641 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

642 
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

643 
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

644 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

645 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

646 
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

647 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

648 
finally (ord_le_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

649 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

650 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

651 
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

652 
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

653 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

654 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

655 
assume "a = f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

656 
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

657 
finally (ord_eq_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

658 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

659 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

660 
lemma ord_less_eq_subst: "a < b ==> f b = c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

661 
(!!x y. x < y ==> f x < f y) ==> f a < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

662 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

663 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

664 
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

665 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

666 
finally (ord_less_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

667 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

668 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

669 
lemma ord_eq_less_subst: "a = f b ==> b < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

670 
(!!x y. x < y ==> f x < f y) ==> a < f c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

671 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

672 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

673 
assume "a = f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

674 
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

675 
finally (ord_eq_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

676 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

677 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

678 
text {* 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

679 
Note that this list of rules is in reverse order of priorities. 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

680 
*} 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

681 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

682 
lemmas order_trans_rules [trans] = 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

683 
order_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

684 
order_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

685 
order_le_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

686 
order_le_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

687 
order_less_le_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

688 
order_less_le_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

689 
order_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

690 
order_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

691 
ord_le_eq_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

692 
ord_eq_le_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

693 
ord_less_eq_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

694 
ord_eq_less_subst 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

695 
forw_subst 
17e6275e13f5
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haftmann
parents:
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diff
changeset

696 
back_subst 
17e6275e13f5
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parents:
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diff
changeset

697 
rev_mp 
17e6275e13f5
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haftmann
parents:
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diff
changeset

698 
mp 
17e6275e13f5
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parents:
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diff
changeset

699 
order_neq_le_trans 
17e6275e13f5
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parents:
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diff
changeset

700 
order_le_neq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
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parents:
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diff
changeset

701 
order_less_trans 
17e6275e13f5
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parents:
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diff
changeset

702 
order_less_asym' 
17e6275e13f5
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haftmann
parents:
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diff
changeset

703 
order_le_less_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

704 
order_less_le_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

705 
order_trans 
17e6275e13f5
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parents:
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diff
changeset

706 
order_antisym 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

707 
ord_le_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

708 
ord_eq_le_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
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parents:
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diff
changeset

709 
ord_less_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

710 
ord_eq_less_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

711 
trans 
17e6275e13f5
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parents:
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diff
changeset

712 

21083  713 

21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

714 
(* FIXME cleanup *) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

715 

21083  716 
text {* These support proving chains of decreasing inequalities 
717 
a >= b >= c ... in Isar proofs. *} 

718 

719 
lemma xt1: 

720 
"a = b ==> b > c ==> a > c" 

721 
"a > b ==> b = c ==> a > c" 

722 
"a = b ==> b >= c ==> a >= c" 

723 
"a >= b ==> b = c ==> a >= c" 

724 
"(x::'a::order) >= y ==> y >= x ==> x = y" 

725 
"(x::'a::order) >= y ==> y >= z ==> x >= z" 

726 
"(x::'a::order) > y ==> y >= z ==> x > z" 

727 
"(x::'a::order) >= y ==> y > z ==> x > z" 

728 
"(a::'a::order) > b ==> b > a ==> ?P" 

729 
"(x::'a::order) > y ==> y > z ==> x > z" 

730 
"(a::'a::order) >= b ==> a ~= b ==> a > b" 

731 
"(a::'a::order) ~= b ==> a >= b ==> a > b" 

732 
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 

733 
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" 

734 
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" 

735 
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" 

736 
by auto 

737 

738 
lemma xt2: 

739 
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" 

740 
by (subgoal_tac "f b >= f c", force, force) 

741 

742 
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 

743 
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" 

744 
by (subgoal_tac "f a >= f b", force, force) 

745 

746 
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> 

747 
(!!x y. x >= y ==> f x >= f y) ==> a > f c" 

748 
by (subgoal_tac "f b >= f c", force, force) 

749 

750 
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> 

751 
(!!x y. x > y ==> f x > f y) ==> f a > c" 

752 
by (subgoal_tac "f a > f b", force, force) 

753 

754 
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> 

755 
(!!x y. x > y ==> f x > f y) ==> a > f c" 

756 
by (subgoal_tac "f b > f c", force, force) 

757 

758 
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> 

759 
(!!x y. x >= y ==> f x >= f y) ==> f a > c" 

760 
by (subgoal_tac "f a >= f b", force, force) 

761 

762 
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> 

763 
(!!x y. x > y ==> f x > f y) ==> a > f c" 

764 
by (subgoal_tac "f b > f c", force, force) 

765 

766 
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> 

767 
(!!x y. x > y ==> f x > f y) ==> f a > c" 

768 
by (subgoal_tac "f a > f b", force, force) 

769 

770 
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 

771 

772 
(* 

773 
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands 

774 
for the wrong thing in an Isar proof. 

775 

776 
The extra transitivity rules can be used as follows: 

777 

778 
lemma "(a::'a::order) > z" 

779 
proof  

780 
have "a >= b" (is "_ >= ?rhs") 

781 
sorry 

782 
also have "?rhs >= c" (is "_ >= ?rhs") 

783 
sorry 

784 
also (xtrans) have "?rhs = d" (is "_ = ?rhs") 

785 
sorry 

786 
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") 

787 
sorry 

788 
also (xtrans) have "?rhs > f" (is "_ > ?rhs") 

789 
sorry 

790 
also (xtrans) have "?rhs > z" 

791 
sorry 

792 
finally (xtrans) show ?thesis . 

793 
qed 

794 

795 
Alternatively, one can use "declare xtrans [trans]" and then 

796 
leave out the "(xtrans)" above. 

797 
*) 

798 

21546  799 
subsection {* Order on bool *} 
800 

22886  801 
instance bool :: order 
21546  802 
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" 
803 
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" 

804 
by default (auto simp add: le_bool_def less_bool_def) 

805 

806 
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" 

807 
by (simp add: le_bool_def) 

808 

809 
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" 

810 
by (simp add: le_bool_def) 

811 

812 
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" 

813 
by (simp add: le_bool_def) 

814 

815 
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" 

816 
by (simp add: le_bool_def) 

817 

22348  818 
lemma [code func]: 
819 
"False \<le> b \<longleftrightarrow> True" 

820 
"True \<le> b \<longleftrightarrow> b" 

821 
"False < b \<longleftrightarrow> b" 

822 
"True < b \<longleftrightarrow> False" 

823 
unfolding le_bool_def less_bool_def by simp_all 

824 

22424  825 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
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diff
changeset

826 
subsection {* Monotonicity, syntactic least value operator and min/max *} 
21083  827 

21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

828 
locale mono = 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

829 
fixes f 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

830 
assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B" 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

831 

1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

832 
lemmas monoI [intro?] = mono.intro 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

833 
and monoD [dest?] = mono.mono 
21083  834 

835 
constdefs 

836 
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) 

837 
"Least P == THE x. P x & (ALL y. P y > x <= y)" 

838 
 {* We can no longer use LeastM because the latter requires HilbertAC. *} 

839 

21383
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diff
changeset

840 
lemma LeastI2_order: 
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diff
changeset

841 
"[ P (x::'a::order); 
17e6275e13f5
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parents:
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diff
changeset

842 
!!y. P y ==> x <= y; 
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changeset

843 
!!x. [ P x; ALL y. P y > x \<le> y ] ==> Q x ] 
17e6275e13f5
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diff
changeset

844 
==> Q (Least P)" 
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diff
changeset

845 
apply (unfold Least_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

846 
apply (rule theI2) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

847 
apply (blast intro: order_antisym)+ 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

848 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

849 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

850 
lemma Least_equality: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

851 
"[ P (k::'a::order); !!x. P x ==> k <= x ] ==> (LEAST x. P x) = k" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

852 
apply (simp add: Least_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

853 
apply (rule the_equality) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

854 
apply (auto intro!: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

855 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

856 

22316  857 
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder] 
858 
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder] 

859 
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder] 

860 
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder] 

861 
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder] 

862 
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder] 

863 
lemmas split_min = linorder_class.split_min [unfolded min_linorder] 

864 
lemmas split_max = linorder_class.split_max [unfolded max_linorder] 

21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

865 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

866 
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

867 
by (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

868 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

869 
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

870 
by (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

871 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

872 
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

873 
apply (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

874 
apply (blast intro: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

875 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

876 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

877 
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

878 
apply (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

879 
apply (blast intro: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

880 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

881 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

882 
lemma min_of_mono: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

883 
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

884 
by (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

885 

17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

886 
lemma max_of_mono: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

887 
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

888 
by (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

889 

22548  890 

891 
subsection {* legacy ML bindings *} 

21673  892 

893 
ML {* 

22548  894 
val monoI = @{thm monoI}; 
22886  895 
*} 
21673  896 

15524  897 
end 