author  wenzelm 
Wed, 28 Feb 2007 22:05:43 +0100  
changeset 22377  61610b1beedf 
parent 22348  ab505d281015 
child 22384  33a46e6c7f04 
permissions  rwrr 
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(* Title: HOL/Orderings.thy 
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ID: $Id$ 

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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson 

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*) 

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header {* Syntactic and abstract orders *} 
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theory Orderings 

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imports HOL 
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begin 
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subsection {* Order syntax *} 
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class ord = 
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<sqsubset>" 50) 
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begin 
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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) 
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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) 
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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" 
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notation (input) 
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greater_eq (infix "\<^loc>\<ge>" 50) 
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end 

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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) 
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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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subsection {* Quasiorders (preorders) *} 
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class preorder = ord + 
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assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" 

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and refl [iff]: "x \<sqsubseteq> x" 

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and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z" 
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begin 
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text {* Reflexivity. *} 
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lemma eq_refl: "x = y \<Longrightarrow> x \<sqsubseteq> y" 
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 {* This form is useful with the classical reasoner. *} 
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by (erule ssubst) (rule refl) 
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lemma less_irrefl [iff]: "\<not> x \<sqsubset> x" 
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by (simp add: less_le) 

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lemma le_less: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubset> 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 \<sqsubseteq> y \<Longrightarrow> x \<sqsubset> y \<or> x = y" 
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unfolding less_le by blast 

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lemma less_imp_le: "x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y" 
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unfolding less_le by blast 

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lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y" 
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by (erule contrapos_pn, erule subst, rule less_irrefl) 

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

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lemma less_imp_not_eq: "x \<sqsubset> y \<Longrightarrow> (x = y) \<longleftrightarrow> False" 

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by auto 

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lemma less_imp_not_eq2: "x \<sqsubset> y \<Longrightarrow> (y = x) \<longleftrightarrow> False" 

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by auto 

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text {* Transitivity rules for calculational reasoning *} 

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lemma neq_le_trans: "\<lbrakk> a \<noteq> b; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b" 

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by (simp add: less_le) 

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lemma le_neq_trans: "\<lbrakk> a \<sqsubseteq> b; a \<noteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b" 

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by (simp add: less_le) 

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end 

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subsection {* Partial orderings *} 

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class order = preorder + 
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assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" 
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begin 

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text {* Asymmetry. *} 

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lemma less_not_sym: "x \<sqsubset> y \<Longrightarrow> \<not> (y \<sqsubset> x)" 
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by (simp add: less_le antisym) 

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lemma less_asym: "x \<sqsubset> y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<sqsubset> 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 \<sqsubseteq> y \<and> y \<sqsubseteq> x" 
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by (blast intro: antisym) 

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lemma antisym_conv: "y \<sqsubseteq> x \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym) 

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lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y" 
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by (erule contrapos_pn, erule subst, rule less_irrefl) 

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text {* Transitivity. *} 
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lemma less_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" 
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by (simp add: less_le) (blast intro: trans antisym) 

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lemma le_less_trans: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" 
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by (simp add: less_le) (blast intro: trans antisym) 

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lemma less_le_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" 
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by (simp add: less_le) (blast intro: trans antisym) 

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

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lemma less_imp_not_less: "x \<sqsubset> y \<Longrightarrow> (\<not> y \<sqsubset> x) \<longleftrightarrow> True" 
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by (blast elim: less_asym) 

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lemma less_imp_triv: "x \<sqsubset> y \<Longrightarrow> (y \<sqsubset> x \<longrightarrow> P) \<longleftrightarrow> True" 
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by (blast elim: less_asym) 

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text {* Transitivity rules for calculational reasoning *} 
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lemma less_asym': "\<lbrakk> a \<sqsubset> b; b \<sqsubset> a \<rbrakk> \<Longrightarrow> P" 
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by (rule less_asym) 

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end 

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subsection {* Linear (total) orders *} 

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class linorder = order + 
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assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" 
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begin 
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lemma less_linear: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x" 
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unfolding less_le using less_le linear by blast 
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lemma le_less_linear: "x \<sqsubseteq> y \<or> y \<sqsubset> x" 

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by (simp add: le_less less_linear) 
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lemma le_cases [case_names le ge]: 

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"\<lbrakk> x \<sqsubseteq> y \<Longrightarrow> P; y \<sqsubseteq> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" 

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using linear by blast 

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lemma cases [case_names less equal greater]: 

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"\<lbrakk> x \<sqsubset> y \<Longrightarrow> P; x = y \<Longrightarrow> P; y \<sqsubset> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" 

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using less_linear by blast 
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lemma not_less: "\<not> x \<sqsubset> y \<longleftrightarrow> y \<sqsubseteq> x" 

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apply (simp add: less_le) 

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using linear apply (blast intro: antisym) 

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done 
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lemma not_le: "\<not> x \<sqsubseteq> y \<longleftrightarrow> y \<sqsubset> x" 
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apply (simp add: less_le) 

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using linear apply (blast intro: antisym) 

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done 
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<sqsubset> y \<or> y \<sqsubset> x" 
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by (cut_tac x = x and y = y in less_linear, auto) 
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lemma neqE: "\<lbrakk> x \<noteq> y; x \<sqsubset> y \<Longrightarrow> R; y \<sqsubset> x \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" 
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by (simp add: neq_iff) blast 

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lemma antisym_conv1: "\<not> x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 

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lemma antisym_conv2: "x \<sqsubseteq> y \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 

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lemma antisym_conv3: "\<not> y \<sqsubset> x \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y" 
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by (blast intro: antisym dest: not_less [THEN iffD1]) 

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text{*Replacing the old Nat.leI*} 
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lemma leI: "\<not> x \<sqsubset> y \<Longrightarrow> y \<sqsubseteq> x" 
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unfolding not_less . 

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lemma leD: "y \<sqsubseteq> x \<Longrightarrow> \<not> x \<sqsubset> y" 
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unfolding not_less . 

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(*FIXME inappropriate name (or delete altogether)*) 

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lemma not_leE: "\<not> y \<sqsubseteq> x \<Longrightarrow> x \<sqsubset> y" 
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unfolding not_le . 

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(* min/max *) 
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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 \<sqsubseteq> b then a else b)" 
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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 \<sqsubseteq> b then b else a)" 
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lemma min_le_iff_disj: 
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"min x y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<or> y \<sqsubseteq> z" 
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unfolding min_def using linear by (auto intro: trans) 
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lemma le_max_iff_disj: 
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"z \<sqsubseteq> max x y \<longleftrightarrow> z \<sqsubseteq> x \<or> z \<sqsubseteq> y" 
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unfolding max_def using linear by (auto intro: trans) 
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lemma min_less_iff_disj: 
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"min x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<or> y \<sqsubset> z" 
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lemma less_max_iff_disj: 
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"z \<sqsubset> max x y \<longleftrightarrow> z \<sqsubset> x \<or> z \<sqsubset> y" 
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lemma min_less_iff_conj [simp]: 
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"z \<sqsubset> min x y \<longleftrightarrow> z \<sqsubset> x \<and> z \<sqsubset> y" 
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lemma max_less_iff_conj [simp]: 
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"max x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<and> y \<sqsubset> z" 
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lemma split_min: 
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"P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)" 
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by (simp add: min_def) 
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lemma split_max: 
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"P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)" 
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by (simp add: max_def) 
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end 
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subsection {* Name duplicates *} 

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lemmas order_refl [iff] = preorder_class.refl 
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lemmas order_trans = preorder_class.trans 

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lemmas order_less_le = preorder_class.less_le 

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lemmas order_eq_refl = preorder_class.eq_refl 

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lemmas order_less_irrefl = preorder_class.less_irrefl 

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lemmas order_le_less = preorder_class.le_less 

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lemmas order_le_imp_less_or_eq = preorder_class.le_imp_less_or_eq 

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lemmas order_less_imp_le = preorder_class.less_imp_le 

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lemmas order_less_imp_not_eq = preorder_class.less_imp_not_eq 

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lemmas order_less_imp_not_eq2 = preorder_class.less_imp_not_eq2 

289 
lemmas order_neq_le_trans = preorder_class.neq_le_trans 

290 
lemmas order_le_neq_trans = preorder_class.le_neq_trans 

291 

292 
lemmas order_antisym = order_class.antisym 

293 
lemmas order_less_not_sym = order_class.less_not_sym 

294 
lemmas order_less_asym = order_class.less_asym 

295 
lemmas order_eq_iff = order_class.eq_iff 

296 
lemmas order_antisym_conv = order_class.antisym_conv 

297 
lemmas less_imp_neq = order_class.less_imp_neq 

298 
lemmas order_less_trans = order_class.less_trans 

299 
lemmas order_le_less_trans = order_class.le_less_trans 

300 
lemmas order_less_le_trans = order_class.less_le_trans 

301 
lemmas order_less_imp_not_less = order_class.less_imp_not_less 

302 
lemmas order_less_imp_triv = order_class.less_imp_triv 

303 
lemmas order_less_asym' = order_class.less_asym' 

304 

305 
lemmas linorder_linear = linorder_class.linear 

306 
lemmas linorder_less_linear = linorder_class.less_linear 

307 
lemmas linorder_le_less_linear = linorder_class.le_less_linear 

308 
lemmas linorder_le_cases = linorder_class.le_cases 

309 
lemmas linorder_cases = linorder_class.cases 

310 
lemmas linorder_not_less = linorder_class.not_less 

311 
lemmas linorder_not_le = linorder_class.not_le 

312 
lemmas linorder_neq_iff = linorder_class.neq_iff 

313 
lemmas linorder_neqE = linorder_class.neqE 

314 
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 

315 
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 

316 
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 

317 
lemmas leI = linorder_class.leI 

318 
lemmas leD = linorder_class.leD 

319 
lemmas not_leE = linorder_class.not_leE 

16796  320 

21083  321 

322 
subsection {* Reasoning tools setup *} 

323 

21091  324 
ML {* 
325 
local 

326 

327 
fun decomp_gen sort thy (Trueprop $ t) = 

21248  328 
let 
329 
fun of_sort t = 

330 
let 

331 
val T = type_of t 

332 
in 

21091  333 
(* exclude numeric types: linear arithmetic subsumes transitivity *) 
21248  334 
T <> HOLogic.natT andalso T <> HOLogic.intT 
335 
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) 

336 
end; 

337 
fun dec (Const ("Not", _) $ t) = (case dec t 

338 
of NONE => NONE 

339 
 SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) 

340 
 dec (Const ("op =", _) $ t1 $ t2) = 

341 
if of_sort t1 

342 
then SOME (t1, "=", t2) 

343 
else NONE 

344 
 dec (Const ("Orderings.less_eq", _) $ t1 $ t2) = 

345 
if of_sort t1 

346 
then SOME (t1, "<=", t2) 

347 
else NONE 

348 
 dec (Const ("Orderings.less", _) $ t1 $ t2) = 

349 
if of_sort t1 

350 
then SOME (t1, "<", t2) 

351 
else NONE 

352 
 dec _ = NONE; 

21091  353 
in dec t end; 
354 

355 
in 

356 

357 
(* The setting up of Quasi_Tac serves as a demo. Since there is no 

358 
class for quasi orders, the tactics Quasi_Tac.trans_tac and 

359 
Quasi_Tac.quasi_tac are not of much use. *) 

360 

21248  361 
structure Quasi_Tac = Quasi_Tac_Fun ( 
362 
struct 

363 
val le_trans = thm "order_trans"; 

364 
val le_refl = thm "order_refl"; 

365 
val eqD1 = thm "order_eq_refl"; 

366 
val eqD2 = thm "sym" RS thm "order_eq_refl"; 

367 
val less_reflE = thm "order_less_irrefl" RS thm "notE"; 

368 
val less_imp_le = thm "order_less_imp_le"; 

369 
val le_neq_trans = thm "order_le_neq_trans"; 

370 
val neq_le_trans = thm "order_neq_le_trans"; 

371 
val less_imp_neq = thm "less_imp_neq"; 

372 
val decomp_trans = decomp_gen ["Orderings.order"]; 

373 
val decomp_quasi = decomp_gen ["Orderings.order"]; 

374 
end); 

21091  375 

376 
structure Order_Tac = Order_Tac_Fun ( 

21248  377 
struct 
378 
val less_reflE = thm "order_less_irrefl" RS thm "notE"; 

379 
val le_refl = thm "order_refl"; 

380 
val less_imp_le = thm "order_less_imp_le"; 

381 
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; 

382 
val not_leI = thm "linorder_not_le" RS thm "iffD2"; 

383 
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; 

384 
val not_leD = thm "linorder_not_le" RS thm "iffD1"; 

385 
val eqI = thm "order_antisym"; 

386 
val eqD1 = thm "order_eq_refl"; 

387 
val eqD2 = thm "sym" RS thm "order_eq_refl"; 

388 
val less_trans = thm "order_less_trans"; 

389 
val less_le_trans = thm "order_less_le_trans"; 

390 
val le_less_trans = thm "order_le_less_trans"; 

391 
val le_trans = thm "order_trans"; 

392 
val le_neq_trans = thm "order_le_neq_trans"; 

393 
val neq_le_trans = thm "order_neq_le_trans"; 

394 
val less_imp_neq = thm "less_imp_neq"; 

395 
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; 

396 
val not_sym = thm "not_sym"; 

397 
val decomp_part = decomp_gen ["Orderings.order"]; 

398 
val decomp_lin = decomp_gen ["Orderings.linorder"]; 

399 
end); 

21091  400 

401 
end; 

402 
*} 

403 

21083  404 
setup {* 
405 
let 

406 

407 
val order_antisym_conv = thm "order_antisym_conv" 

408 
val linorder_antisym_conv1 = thm "linorder_antisym_conv1" 

409 
val linorder_antisym_conv2 = thm "linorder_antisym_conv2" 

410 
val linorder_antisym_conv3 = thm "linorder_antisym_conv3" 

411 

412 
fun prp t thm = (#prop (rep_thm thm) = t); 

15524  413 

21083  414 
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = 
415 
let val prems = prems_of_ss ss; 

416 
val less = Const("Orderings.less",T); 

417 
val t = HOLogic.mk_Trueprop(le $ s $ r); 

418 
in case find_first (prp t) prems of 

419 
NONE => 

420 
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) 

421 
in case find_first (prp t) prems of 

422 
NONE => NONE 

423 
 SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1)) 

424 
end 

425 
 SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv)) 

426 
end 

427 
handle THM _ => NONE; 

15524  428 

21083  429 
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = 
430 
let val prems = prems_of_ss ss; 

431 
val le = Const("Orderings.less_eq",T); 

432 
val t = HOLogic.mk_Trueprop(le $ r $ s); 

433 
in case find_first (prp t) prems of 

434 
NONE => 

435 
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) 

436 
in case find_first (prp t) prems of 

437 
NONE => NONE 

438 
 SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3)) 

439 
end 

440 
 SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2)) 

441 
end 

442 
handle THM _ => NONE; 

15524  443 

21248  444 
fun add_simprocs procs thy = 
445 
(Simplifier.change_simpset_of thy (fn ss => ss 

446 
addsimprocs (map (fn (name, raw_ts, proc) => 

447 
Simplifier.simproc thy name raw_ts proc)) procs); thy); 

448 
fun add_solver name tac thy = 

449 
(Simplifier.change_simpset_of thy (fn ss => ss addSolver 

450 
(mk_solver name (K tac))); thy); 

21083  451 

452 
in 

21248  453 
add_simprocs [ 
454 
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), 

455 
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) 

456 
] 

457 
#> add_solver "Trans_linear" Order_Tac.linear_tac 

458 
#> add_solver "Trans_partial" Order_Tac.partial_tac 

459 
(* Adding the transitivity reasoners also as safe solvers showed a slight 

460 
speed up, but the reasoning strength appears to be not higher (at least 

461 
no breaking of additional proofs in the entire HOL distribution, as 

462 
of 5 March 2004, was observed). *) 

21083  463 
end 
464 
*} 

15524  465 

466 

21083  467 
subsection {* Bounded quantifiers *} 
468 

469 
syntax 

21180
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470 
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
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diff
changeset

471 
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
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diff
changeset

472 
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) 
f27f12bcafb8
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wenzelm
parents:
21091
diff
changeset

473 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) 
21083  474 

21180
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475 
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) 
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changeset

476 
"_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

477 
"_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

478 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) 
21083  479 

480 
syntax (xsymbols) 

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

481 
"_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

482 
"_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

483 
"_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

484 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  485 

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

486 
"_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

487 
"_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

488 
"_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

489 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  490 

491 
syntax (HOL) 

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

492 
"_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

493 
"_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

494 
"_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

495 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) 
21083  496 

497 
syntax (HTML output) 

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

498 
"_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

499 
"_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

500 
"_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

501 
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) 
21083  502 

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

503 
"_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

504 
"_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

505 
"_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

506 
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) 
21083  507 

508 
translations 

509 
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" 

510 
"EX x<y. P" => "EX x. x < y \<and> P" 

511 
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" 

512 
"EX x<=y. P" => "EX x. x <= y \<and> P" 

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 

518 
print_translation {* 

519 
let 

22377  520 
val All_binder = Syntax.binder_name @{const_syntax "All"}; 
521 
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; 

522 
val impl = @{const_syntax "op >"}; 

523 
val conj = @{const_syntax "op &"}; 

524 
val less = @{const_syntax "less"}; 

525 
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

526 

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

527 
val trans = 
21524  528 
[((All_binder, impl, less), ("_All_less", "_All_greater")), 
529 
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), 

530 
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), 

531 
((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

532 

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

533 
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

534 
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

535 
 _ => false 
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

536 
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

537 
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

538 

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

539 
fun tr' q = (q, 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

540 
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

541 
(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

542 
NONE => raise Match 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

543 
 SOME (l, g) => 
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset

544 
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

545 
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

546 
else raise Match) 
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

547 
 _ => raise Match); 
21524  548 
in [tr' All_binder, tr' Ex_binder] end 
21083  549 
*} 
550 

551 

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

552 
subsection {* Transitivity reasoning *} 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

553 

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

554 
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

555 
by (rule subst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

556 

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

557 
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

558 
by (rule ssubst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

559 

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

560 
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

561 
by (rule subst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

562 

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

563 
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

564 
by (rule ssubst) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

565 

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

566 
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

567 
(!!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

568 
proof  
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

569 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

570 
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

571 
also assume "f b < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

572 
finally (order_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

573 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

574 

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

575 
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

576 
(!!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

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

578 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

579 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

580 
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

581 
finally (order_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

582 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

583 

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

584 
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

585 
(!!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

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

587 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

588 
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

589 
also assume "f b < c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

590 
finally (order_le_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

591 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

592 

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

593 
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

594 
(!!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

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

596 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

597 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

598 
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

599 
finally (order_le_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

600 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

601 

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

602 
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

603 
(!!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

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

605 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

606 
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

607 
also assume "f b <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

608 
finally (order_less_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

609 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

610 

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

611 
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

612 
(!!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

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

614 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

615 
assume "a < f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

616 
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

617 
finally (order_less_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

618 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

619 

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

620 
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

621 
(!!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

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

623 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

624 
assume "a <= f b" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

625 
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

626 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

627 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

628 

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

629 
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

630 
(!!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

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

632 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

633 
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

634 
also assume "f b <= c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

635 
finally (order_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

636 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

637 

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

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

639 
(!!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

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

641 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

642 
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

643 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

644 
finally (ord_le_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

645 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

646 

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

647 
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

648 
(!!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

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

650 
assume r: "!!x y. x <= y ==> f x <= f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

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

652 
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

653 
finally (ord_eq_le_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

654 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

655 

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

656 
lemma ord_less_eq_subst: "a < b ==> f b = c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

657 
(!!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

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

659 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

660 
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

661 
also assume "f b = c" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

662 
finally (ord_less_eq_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

663 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

664 

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

665 
lemma ord_eq_less_subst: "a = f b ==> b < c ==> 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

666 
(!!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

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

668 
assume r: "!!x y. x < y ==> f x < f y" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

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

670 
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

671 
finally (ord_eq_less_trans) show ?thesis . 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

672 
qed 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

673 

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

674 
text {* 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

675 
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

676 
*} 
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 
lemmas order_trans_rules [trans] = 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

679 
order_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

680 
order_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

681 
order_le_less_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

682 
order_le_less_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

683 
order_less_le_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

684 
order_less_le_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

685 
order_subst2 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

686 
order_subst1 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

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

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

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

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

691 
forw_subst 
17e6275e13f5
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haftmann
parents:
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diff
changeset

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

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

694 
mp 
17e6275e13f5
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haftmann
parents:
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diff
changeset

695 
order_neq_le_trans 
17e6275e13f5
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haftmann
parents:
21329
diff
changeset

696 
order_le_neq_trans 
17e6275e13f5
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haftmann
parents:
21329
diff
changeset

697 
order_less_trans 
17e6275e13f5
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haftmann
parents:
21329
diff
changeset

698 
order_less_asym' 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

699 
order_le_less_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

700 
order_less_le_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

701 
order_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

702 
order_antisym 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

703 
ord_le_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

704 
ord_eq_le_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

705 
ord_less_eq_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

706 
ord_eq_less_trans 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

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

708 

21083  709 

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

710 
(* FIXME cleanup *) 
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset

711 

21083  712 
text {* These support proving chains of decreasing inequalities 
713 
a >= b >= c ... in Isar proofs. *} 

714 

715 
lemma xt1: 

716 
"a = b ==> b > c ==> a > c" 

717 
"a > b ==> b = c ==> a > c" 

718 
"a = b ==> b >= c ==> a >= c" 

719 
"a >= b ==> b = c ==> a >= c" 

720 
"(x::'a::order) >= y ==> y >= x ==> x = y" 

721 
"(x::'a::order) >= y ==> y >= z ==> x >= z" 

722 
"(x::'a::order) > y ==> y >= z ==> x > z" 

723 
"(x::'a::order) >= y ==> y > z ==> x > z" 

724 
"(a::'a::order) > b ==> b > a ==> ?P" 

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

726 
"(a::'a::order) >= b ==> a ~= b ==> a > b" 

727 
"(a::'a::order) ~= b ==> a >= b ==> a > b" 

728 
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 

729 
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" 

730 
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" 

731 
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" 

732 
by auto 

733 

734 
lemma xt2: 

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

736 
by (subgoal_tac "f b >= f c", force, force) 

737 

738 
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 

739 
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" 

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

741 

742 
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> 

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

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

745 

746 
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> 

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

748 
by (subgoal_tac "f a > f b", force, force) 

749 

750 
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> 

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

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

753 

754 
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> 

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

756 
by (subgoal_tac "f a >= f b", force, force) 

757 

758 
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> 

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

760 
by (subgoal_tac "f b > f c", force, force) 

761 

762 
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> 

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

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

765 

766 
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 

767 

768 
(* 

769 
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands 

770 
for the wrong thing in an Isar proof. 

771 

772 
The extra transitivity rules can be used as follows: 

773 

774 
lemma "(a::'a::order) > z" 

775 
proof  

776 
have "a >= b" (is "_ >= ?rhs") 

777 
sorry 

778 
also have "?rhs >= c" (is "_ >= ?rhs") 

779 
sorry 

780 
also (xtrans) have "?rhs = d" (is "_ = ?rhs") 

781 
sorry 

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

783 
sorry 

784 
also (xtrans) have "?rhs > f" (is "_ > ?rhs") 

785 
sorry 

786 
also (xtrans) have "?rhs > z" 

787 
sorry 

788 
finally (xtrans) show ?thesis . 

789 
qed 

790 

791 
Alternatively, one can use "declare xtrans [trans]" and then 

792 
leave out the "(xtrans)" above. 

793 
*) 

794 

21546  795 
subsection {* Order on bool *} 
796 

797 
instance bool :: linorder 

798 
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" 

799 
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" 

800 
by default (auto simp add: le_bool_def less_bool_def) 

801 

802 
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" 

803 
by (simp add: le_bool_def) 

804 

805 
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" 

806 
by (simp add: le_bool_def) 

807 

808 
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" 

809 
by (simp add: le_bool_def) 

810 

811 
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" 

812 
by (simp add: le_bool_def) 

813 

22348  814 
lemma [code func]: 
815 
"False \<le> b \<longleftrightarrow> True" 

816 
"True \<le> b \<longleftrightarrow> b" 

817 
"False < b \<longleftrightarrow> b" 

818 
"True < b \<longleftrightarrow> False" 

819 
unfolding le_bool_def less_bool_def by simp_all 

820 

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

821 
subsection {* Monotonicity, syntactic least value operator and min/max *} 
21083  822 

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

823 
locale mono = 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

824 
fixes f 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

825 
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

826 

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

827 
lemmas monoI [intro?] = mono.intro 
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset

828 
and monoD [dest?] = mono.mono 
21083  829 

830 
constdefs 

831 
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) 

832 
"Least P == THE x. P x & (ALL y. P y > x <= y)" 

833 
 {* We can no longer use LeastM because the latter requires HilbertAC. *} 

834 

21383
17e6275e13f5
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diff
changeset

835 
lemma LeastI2_order: 
17e6275e13f5
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haftmann
parents:
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diff
changeset

836 
"[ P (x::'a::order); 
17e6275e13f5
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haftmann
parents:
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diff
changeset

837 
!!y. P y ==> x <= y; 
17e6275e13f5
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parents:
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diff
changeset

838 
!!x. [ P x; ALL y. P y > x \<le> y ] ==> Q x ] 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

839 
==> Q (Least P)" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
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diff
changeset

840 
apply (unfold Least_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

841 
apply (rule theI2) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

842 
apply (blast intro: order_antisym)+ 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

843 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

844 

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

845 
lemma Least_equality: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

846 
"[ 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:
21329
diff
changeset

847 
apply (simp add: Least_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

848 
apply (rule the_equality) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

849 
apply (auto intro!: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

850 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

851 

21083  852 
constdefs 
853 
min :: "['a::ord, 'a] => 'a" 

854 
"min a b == (if a <= b then a else b)" 

855 
max :: "['a::ord, 'a] => 'a" 

856 
"max a b == (if a <= b then b else a)" 

857 

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

858 
lemma min_linorder: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

859 
"linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min" 
22316  860 
by rule+ (simp add: min_def linorder_class.min_def) 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

861 

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

862 
lemma max_linorder: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

863 
"linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max" 
22316  864 
by rule+ (simp add: max_def linorder_class.max_def) 
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

865 

22316  866 
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder] 
867 
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder] 

868 
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder] 

869 
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder] 

870 
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder] 

871 
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder] 

872 
lemmas split_min = linorder_class.split_min [unfolded min_linorder] 

873 
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

874 

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

875 
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

876 
by (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

877 

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

878 
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

879 
by (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

880 

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

881 
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

882 
apply (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

883 
apply (blast intro: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

884 
done 
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_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

887 
apply (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

888 
apply (blast intro: order_antisym) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

889 
done 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

890 

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

891 
lemma min_of_mono: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

892 
"(!!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

893 
by (simp add: min_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

894 

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

895 
lemma max_of_mono: 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

896 
"(!!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

897 
by (simp add: max_def) 
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset

898 

21673  899 

900 
subsection {* Basic ML bindings *} 

901 

902 
ML {* 

903 
val leD = thm "leD"; 

904 
val leI = thm "leI"; 

905 
val linorder_neqE = thm "linorder_neqE"; 

906 
val linorder_neq_iff = thm "linorder_neq_iff"; 

907 
val linorder_not_le = thm "linorder_not_le"; 

908 
val linorder_not_less = thm "linorder_not_less"; 

909 
val monoD = thm "monoD"; 

910 
val monoI = thm "monoI"; 

911 
val order_antisym = thm "order_antisym"; 

912 
val order_less_irrefl = thm "order_less_irrefl"; 

913 
val order_refl = thm "order_refl"; 

914 
val order_trans = thm "order_trans"; 

915 
val split_max = thm "split_max"; 

916 
val split_min = thm "split_min"; 

917 
*} 

918 

919 
ML {* 

920 
structure HOL = 

921 
struct 

922 
val thy = theory "HOL"; 

923 
end; 

924 
*}  "belongs to theory HOL" 

925 

15524  926 
end 