src/HOL/Orderings.thy
author wenzelm
Wed Dec 06 01:12:51 2006 +0100 (2006-12-06)
changeset 21673 a664ba87b3aa
parent 21656 43d709faa9dc
child 21737 f2be09171c9c
permissions -rw-r--r--
added basic ML bindings;
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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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section {* Abstract orders *}
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subsection {* Order syntax *}
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class ord =
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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)
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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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locale preorder = ord +
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  assumes 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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  and less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
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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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locale partial_order = preorder + 
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  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
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context partial_order
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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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axclass order < ord
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  order_refl [iff]: "x <= x"
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  order_trans: "x <= y ==> y <= z ==> x <= z"
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  order_antisym: "x <= y ==> y <= x ==> x = y"
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  order_less_le: "(x < y) = (x <= y & x ~= y)"
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interpretation order:
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  partial_order ["op \<le> \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool"]
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apply unfold_locales
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apply (rule order_refl)
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apply (erule (1) order_trans)
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apply (rule order_less_le)
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apply (erule (1) order_antisym)
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done
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subsection {* Linear (total) orders *}
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locale linorder = partial_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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  unfolding min_def le_less using less_linear by (auto intro: less_trans)
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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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  unfolding max_def le_less using less_linear by (auto intro: less_trans)
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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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  unfolding min_def le_less using less_linear by (auto intro: less_trans)
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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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  unfolding max_def le_less using less_linear by (auto intro: less_trans)
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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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axclass linorder < order
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  linorder_linear: "x <= y | y <= x"
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interpretation linorder:
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  linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"]
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  by unfold_locales (rule linorder_linear)
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subsection {* Name duplicates *}
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lemmas order_eq_refl [where 'b = "?'a::order"] = order.eq_refl
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lemmas order_less_irrefl [where 'b = "?'a::order"] = order.less_irrefl
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lemmas order_le_less [where 'b = "?'a::order"] = order.le_less
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lemmas order_le_imp_less_or_eq [where 'b = "?'a::order"] = order.le_imp_less_or_eq
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lemmas order_less_imp_le [where 'b = "?'a::order"] = order.less_imp_le
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lemmas order_less_not_sym [where 'b = "?'a::order"] = order.less_not_sym
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lemmas order_less_asym [where 'b = "?'a::order"] = order.less_asym
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lemmas order_eq_iff [where 'b = "?'a::order"] = order.eq_iff
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lemmas order_antisym_conv [where 'b = "?'a::order"] = order.antisym_conv
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lemmas less_imp_neq [where 'b = "?'a::order"] = order.less_imp_neq
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lemmas order_less_trans [where 'b = "?'a::order"] = order.less_trans
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lemmas order_le_less_trans [where 'b = "?'a::order"] = order.le_less_trans
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lemmas order_less_le_trans [where 'b = "?'a::order"] = order.less_le_trans
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lemmas order_less_imp_not_less [where 'b = "?'a::order"] = order.less_imp_not_less
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lemmas order_less_imp_triv [where 'b = "?'a::order"] = order.less_imp_triv
haftmann@21248
   320
lemmas order_less_imp_not_eq [where 'b = "?'a::order"] = order.less_imp_not_eq
haftmann@21248
   321
lemmas order_less_imp_not_eq2 [where 'b = "?'a::order"] = order.less_imp_not_eq2
haftmann@21248
   322
lemmas order_neq_le_trans [where 'b = "?'a::order"] = order.neq_le_trans
haftmann@21248
   323
lemmas order_le_neq_trans [where 'b = "?'a::order"] = order.le_neq_trans
haftmann@21248
   324
lemmas order_less_asym' [where 'b = "?'a::order"] = order.less_asym'
haftmann@21412
   325
lemmas linorder_less_linear [where 'b = "?'a::linorder"] = linorder.less_linear
haftmann@21248
   326
lemmas linorder_le_less_linear [where 'b = "?'a::linorder"] = linorder.le_less_linear
haftmann@21248
   327
lemmas linorder_le_cases [where 'b = "?'a::linorder"] = linorder.le_cases
haftmann@21248
   328
lemmas linorder_cases [where 'b = "?'a::linorder"] = linorder.cases
haftmann@21248
   329
lemmas linorder_not_less [where 'b = "?'a::linorder"] = linorder.not_less
haftmann@21248
   330
lemmas linorder_not_le [where 'b = "?'a::linorder"] = linorder.not_le
haftmann@21248
   331
lemmas linorder_neq_iff [where 'b = "?'a::linorder"] = linorder.neq_iff
haftmann@21248
   332
lemmas linorder_neqE [where 'b = "?'a::linorder"] = linorder.neqE
haftmann@21248
   333
lemmas linorder_antisym_conv1 [where 'b = "?'a::linorder"] = linorder.antisym_conv1
haftmann@21248
   334
lemmas linorder_antisym_conv2 [where 'b = "?'a::linorder"] = linorder.antisym_conv2
haftmann@21248
   335
lemmas linorder_antisym_conv3 [where 'b = "?'a::linorder"] = linorder.antisym_conv3
haftmann@21248
   336
lemmas leI [where 'b = "?'a::linorder"] = linorder.leI
haftmann@21248
   337
lemmas leD [where 'b = "?'a::linorder"] = linorder.leD
haftmann@21248
   338
lemmas not_leE [where 'b = "?'a::linorder"] = linorder.not_leE
paulson@16796
   339
haftmann@21083
   340
haftmann@21083
   341
subsection {* Reasoning tools setup *}
haftmann@21083
   342
haftmann@21091
   343
ML {*
haftmann@21091
   344
local
haftmann@21091
   345
haftmann@21091
   346
fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248
   347
  let
haftmann@21248
   348
    fun of_sort t =
haftmann@21248
   349
      let
haftmann@21248
   350
        val T = type_of t
haftmann@21248
   351
      in
haftmann@21091
   352
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248
   353
        T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248
   354
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248
   355
      end;
haftmann@21248
   356
    fun dec (Const ("Not", _) $ t) = (case dec t
haftmann@21248
   357
          of NONE => NONE
haftmann@21248
   358
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21248
   359
      | dec (Const ("op =",  _) $ t1 $ t2) =
haftmann@21248
   360
          if of_sort t1
haftmann@21248
   361
          then SOME (t1, "=", t2)
haftmann@21248
   362
          else NONE
haftmann@21248
   363
      | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
haftmann@21248
   364
          if of_sort t1
haftmann@21248
   365
          then SOME (t1, "<=", t2)
haftmann@21248
   366
          else NONE
haftmann@21248
   367
      | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21248
   368
          if of_sort t1
haftmann@21248
   369
          then SOME (t1, "<", t2)
haftmann@21248
   370
          else NONE
haftmann@21248
   371
      | dec _ = NONE;
haftmann@21091
   372
  in dec t end;
haftmann@21091
   373
haftmann@21091
   374
in
haftmann@21091
   375
haftmann@21091
   376
(* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091
   377
   class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091
   378
   Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091
   379
haftmann@21248
   380
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   381
struct
haftmann@21248
   382
  val le_trans = thm "order_trans";
haftmann@21248
   383
  val le_refl = thm "order_refl";
haftmann@21248
   384
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   385
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   386
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   387
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   388
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   389
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   390
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   391
  val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21248
   392
  val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21248
   393
end);
haftmann@21091
   394
haftmann@21091
   395
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   396
struct
haftmann@21248
   397
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   398
  val le_refl = thm "order_refl";
haftmann@21248
   399
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   400
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   401
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   402
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   403
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   404
  val eqI = thm "order_antisym";
haftmann@21248
   405
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   406
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   407
  val less_trans = thm "order_less_trans";
haftmann@21248
   408
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   409
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   410
  val le_trans = thm "order_trans";
haftmann@21248
   411
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   412
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   413
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   414
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   415
  val not_sym = thm "not_sym";
haftmann@21248
   416
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   417
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   418
end);
haftmann@21091
   419
haftmann@21091
   420
end;
haftmann@21091
   421
*}
haftmann@21091
   422
haftmann@21083
   423
setup {*
haftmann@21083
   424
let
haftmann@21083
   425
haftmann@21083
   426
val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083
   427
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083
   428
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083
   429
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083
   430
haftmann@21083
   431
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   432
haftmann@21083
   433
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   434
  let val prems = prems_of_ss ss;
haftmann@21083
   435
      val less = Const("Orderings.less",T);
haftmann@21083
   436
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   437
  in case find_first (prp t) prems of
haftmann@21083
   438
       NONE =>
haftmann@21083
   439
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   440
         in case find_first (prp t) prems of
haftmann@21083
   441
              NONE => NONE
haftmann@21083
   442
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083
   443
         end
haftmann@21083
   444
     | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083
   445
  end
haftmann@21083
   446
  handle THM _ => NONE;
nipkow@15524
   447
haftmann@21083
   448
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   449
  let val prems = prems_of_ss ss;
haftmann@21083
   450
      val le = Const("Orderings.less_eq",T);
haftmann@21083
   451
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   452
  in case find_first (prp t) prems of
haftmann@21083
   453
       NONE =>
haftmann@21083
   454
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   455
         in case find_first (prp t) prems of
haftmann@21083
   456
              NONE => NONE
haftmann@21083
   457
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083
   458
         end
haftmann@21083
   459
     | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083
   460
  end
haftmann@21083
   461
  handle THM _ => NONE;
nipkow@15524
   462
haftmann@21248
   463
fun add_simprocs procs thy =
haftmann@21248
   464
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   465
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   466
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   467
fun add_solver name tac thy =
haftmann@21248
   468
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   469
    (mk_solver name (K tac))); thy);
haftmann@21083
   470
haftmann@21083
   471
in
haftmann@21248
   472
  add_simprocs [
haftmann@21248
   473
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   474
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   475
     ]
haftmann@21248
   476
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   477
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   478
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   479
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   480
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   481
     of 5 March 2004, was observed). *)
haftmann@21083
   482
end
haftmann@21083
   483
*}
nipkow@15524
   484
nipkow@15524
   485
haftmann@21083
   486
subsection {* Bounded quantifiers *}
haftmann@21083
   487
haftmann@21083
   488
syntax
wenzelm@21180
   489
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   490
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   491
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   492
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   493
wenzelm@21180
   494
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   495
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   496
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   497
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   498
haftmann@21083
   499
syntax (xsymbols)
wenzelm@21180
   500
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   501
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   502
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   503
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   504
wenzelm@21180
   505
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   506
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   507
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   508
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   509
haftmann@21083
   510
syntax (HOL)
wenzelm@21180
   511
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   512
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   513
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   514
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   515
haftmann@21083
   516
syntax (HTML output)
wenzelm@21180
   517
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   518
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   519
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   520
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   521
wenzelm@21180
   522
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   523
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   524
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   525
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   526
haftmann@21083
   527
translations
haftmann@21083
   528
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   529
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   530
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   531
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   532
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   533
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   534
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   535
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   536
haftmann@21083
   537
print_translation {*
haftmann@21083
   538
let
wenzelm@21180
   539
  val syntax_name = Sign.const_syntax_name (the_context ());
wenzelm@21524
   540
  val binder_name = Syntax.binder_name o syntax_name;
wenzelm@21524
   541
  val All_binder = binder_name "All";
wenzelm@21524
   542
  val Ex_binder = binder_name "Ex";
wenzelm@21180
   543
  val impl = syntax_name "op -->";
wenzelm@21180
   544
  val conj = syntax_name "op &";
wenzelm@21180
   545
  val less = syntax_name "Orderings.less";
wenzelm@21180
   546
  val less_eq = syntax_name "Orderings.less_eq";
wenzelm@21180
   547
wenzelm@21180
   548
  val trans =
wenzelm@21524
   549
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   550
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   551
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   552
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   553
haftmann@21083
   554
  fun mk v v' c n P =
wenzelm@21180
   555
    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
haftmann@21083
   556
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
wenzelm@21180
   557
wenzelm@21180
   558
  fun tr' q = (q,
wenzelm@21180
   559
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   560
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   561
        NONE => raise Match
wenzelm@21180
   562
      | SOME (l, g) =>
wenzelm@21180
   563
          (case (t, u) of
wenzelm@21180
   564
            (Const ("_bound", _) $ Free (v', _), n) => mk v v' l n P
wenzelm@21180
   565
          | (n, Const ("_bound", _) $ Free (v', _)) => mk v v' g n P
wenzelm@21180
   566
          | _ => raise Match))
wenzelm@21180
   567
     | _ => raise Match);
wenzelm@21524
   568
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   569
*}
haftmann@21083
   570
haftmann@21083
   571
haftmann@21383
   572
subsection {* Transitivity reasoning *}
haftmann@21383
   573
haftmann@21383
   574
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
haftmann@21383
   575
  by (rule subst)
haftmann@21383
   576
haftmann@21383
   577
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
haftmann@21383
   578
  by (rule ssubst)
haftmann@21383
   579
haftmann@21383
   580
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
haftmann@21383
   581
  by (rule subst)
haftmann@21383
   582
haftmann@21383
   583
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
haftmann@21383
   584
  by (rule ssubst)
haftmann@21383
   585
haftmann@21383
   586
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   587
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   588
proof -
haftmann@21383
   589
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   590
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   591
  also assume "f b < c"
haftmann@21383
   592
  finally (order_less_trans) show ?thesis .
haftmann@21383
   593
qed
haftmann@21383
   594
haftmann@21383
   595
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   596
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   597
proof -
haftmann@21383
   598
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   599
  assume "a < f b"
haftmann@21383
   600
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   601
  finally (order_less_trans) show ?thesis .
haftmann@21383
   602
qed
haftmann@21383
   603
haftmann@21383
   604
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   605
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   606
proof -
haftmann@21383
   607
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   608
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   609
  also assume "f b < c"
haftmann@21383
   610
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   611
qed
haftmann@21383
   612
haftmann@21383
   613
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   614
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   615
proof -
haftmann@21383
   616
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   617
  assume "a <= f b"
haftmann@21383
   618
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   619
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   620
qed
haftmann@21383
   621
haftmann@21383
   622
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   623
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   624
proof -
haftmann@21383
   625
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   626
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   627
  also assume "f b <= c"
haftmann@21383
   628
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   629
qed
haftmann@21383
   630
haftmann@21383
   631
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   632
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   633
proof -
haftmann@21383
   634
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   635
  assume "a < f b"
haftmann@21383
   636
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   637
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   638
qed
haftmann@21383
   639
haftmann@21383
   640
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   641
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   642
proof -
haftmann@21383
   643
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   644
  assume "a <= f b"
haftmann@21383
   645
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   646
  finally (order_trans) show ?thesis .
haftmann@21383
   647
qed
haftmann@21383
   648
haftmann@21383
   649
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   650
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   651
proof -
haftmann@21383
   652
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   653
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   654
  also assume "f b <= c"
haftmann@21383
   655
  finally (order_trans) show ?thesis .
haftmann@21383
   656
qed
haftmann@21383
   657
haftmann@21383
   658
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   659
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   660
proof -
haftmann@21383
   661
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   662
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   663
  also assume "f b = c"
haftmann@21383
   664
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   665
qed
haftmann@21383
   666
haftmann@21383
   667
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   668
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   669
proof -
haftmann@21383
   670
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   671
  assume "a = f b"
haftmann@21383
   672
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   673
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   674
qed
haftmann@21383
   675
haftmann@21383
   676
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   677
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   678
proof -
haftmann@21383
   679
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   680
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   681
  also assume "f b = c"
haftmann@21383
   682
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   683
qed
haftmann@21383
   684
haftmann@21383
   685
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   686
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   687
proof -
haftmann@21383
   688
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   689
  assume "a = f b"
haftmann@21383
   690
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   691
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   692
qed
haftmann@21383
   693
haftmann@21383
   694
text {*
haftmann@21383
   695
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   696
*}
haftmann@21383
   697
haftmann@21383
   698
lemmas order_trans_rules [trans] =
haftmann@21383
   699
  order_less_subst2
haftmann@21383
   700
  order_less_subst1
haftmann@21383
   701
  order_le_less_subst2
haftmann@21383
   702
  order_le_less_subst1
haftmann@21383
   703
  order_less_le_subst2
haftmann@21383
   704
  order_less_le_subst1
haftmann@21383
   705
  order_subst2
haftmann@21383
   706
  order_subst1
haftmann@21383
   707
  ord_le_eq_subst
haftmann@21383
   708
  ord_eq_le_subst
haftmann@21383
   709
  ord_less_eq_subst
haftmann@21383
   710
  ord_eq_less_subst
haftmann@21383
   711
  forw_subst
haftmann@21383
   712
  back_subst
haftmann@21383
   713
  rev_mp
haftmann@21383
   714
  mp
haftmann@21383
   715
  order_neq_le_trans
haftmann@21383
   716
  order_le_neq_trans
haftmann@21383
   717
  order_less_trans
haftmann@21383
   718
  order_less_asym'
haftmann@21383
   719
  order_le_less_trans
haftmann@21383
   720
  order_less_le_trans
haftmann@21383
   721
  order_trans
haftmann@21383
   722
  order_antisym
haftmann@21383
   723
  ord_le_eq_trans
haftmann@21383
   724
  ord_eq_le_trans
haftmann@21383
   725
  ord_less_eq_trans
haftmann@21383
   726
  ord_eq_less_trans
haftmann@21383
   727
  trans
haftmann@21383
   728
haftmann@21083
   729
wenzelm@21180
   730
(* FIXME cleanup *)
wenzelm@21180
   731
haftmann@21083
   732
text {* These support proving chains of decreasing inequalities
haftmann@21083
   733
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   734
haftmann@21083
   735
lemma xt1:
haftmann@21083
   736
  "a = b ==> b > c ==> a > c"
haftmann@21083
   737
  "a > b ==> b = c ==> a > c"
haftmann@21083
   738
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   739
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   740
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   741
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   742
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   743
  "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083
   744
  "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083
   745
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   746
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   747
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   748
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   749
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   750
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   751
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   752
by auto
haftmann@21083
   753
haftmann@21083
   754
lemma xt2:
haftmann@21083
   755
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   756
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   757
haftmann@21083
   758
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   759
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   760
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   761
haftmann@21083
   762
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   763
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   764
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   765
haftmann@21083
   766
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   767
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   768
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   769
haftmann@21083
   770
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   771
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   772
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   773
haftmann@21083
   774
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   775
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   776
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   777
haftmann@21083
   778
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   779
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   780
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   781
haftmann@21083
   782
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   783
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   784
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   785
haftmann@21083
   786
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   787
haftmann@21083
   788
(* 
haftmann@21083
   789
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   790
  for the wrong thing in an Isar proof.
haftmann@21083
   791
haftmann@21083
   792
  The extra transitivity rules can be used as follows: 
haftmann@21083
   793
haftmann@21083
   794
lemma "(a::'a::order) > z"
haftmann@21083
   795
proof -
haftmann@21083
   796
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   797
    sorry
haftmann@21083
   798
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   799
    sorry
haftmann@21083
   800
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   801
    sorry
haftmann@21083
   802
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   803
    sorry
haftmann@21083
   804
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   805
    sorry
haftmann@21083
   806
  also (xtrans) have "?rhs > z"
haftmann@21083
   807
    sorry
haftmann@21083
   808
  finally (xtrans) show ?thesis .
haftmann@21083
   809
qed
haftmann@21083
   810
haftmann@21083
   811
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   812
  leave out the "(xtrans)" above.
haftmann@21083
   813
*)
haftmann@21083
   814
haftmann@21546
   815
subsection {* Order on bool *}
haftmann@21546
   816
haftmann@21546
   817
instance bool :: linorder 
haftmann@21546
   818
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   819
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@21546
   820
  by default (auto simp add: le_bool_def less_bool_def)
haftmann@21546
   821
haftmann@21546
   822
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@21546
   823
  by (simp add: le_bool_def)
haftmann@21546
   824
haftmann@21546
   825
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@21546
   826
  by (simp add: le_bool_def)
haftmann@21546
   827
haftmann@21546
   828
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@21546
   829
  by (simp add: le_bool_def)
haftmann@21546
   830
haftmann@21546
   831
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@21546
   832
  by (simp add: le_bool_def)
haftmann@21546
   833
haftmann@21383
   834
subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083
   835
haftmann@21216
   836
locale mono =
haftmann@21216
   837
  fixes f
haftmann@21216
   838
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   839
haftmann@21216
   840
lemmas monoI [intro?] = mono.intro
haftmann@21216
   841
  and monoD [dest?] = mono.mono
haftmann@21083
   842
haftmann@21083
   843
constdefs
haftmann@21083
   844
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083
   845
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083
   846
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083
   847
haftmann@21383
   848
lemma LeastI2_order:
haftmann@21383
   849
  "[| P (x::'a::order);
haftmann@21383
   850
      !!y. P y ==> x <= y;
haftmann@21383
   851
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   852
   ==> Q (Least P)"
haftmann@21383
   853
  apply (unfold Least_def)
haftmann@21383
   854
  apply (rule theI2)
haftmann@21383
   855
    apply (blast intro: order_antisym)+
haftmann@21383
   856
  done
haftmann@21383
   857
haftmann@21383
   858
lemma Least_equality:
haftmann@21383
   859
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
haftmann@21383
   860
  apply (simp add: Least_def)
haftmann@21383
   861
  apply (rule the_equality)
haftmann@21383
   862
  apply (auto intro!: order_antisym)
haftmann@21383
   863
  done
haftmann@21383
   864
haftmann@21083
   865
constdefs
haftmann@21083
   866
  min :: "['a::ord, 'a] => 'a"
haftmann@21083
   867
  "min a b == (if a <= b then a else b)"
haftmann@21083
   868
  max :: "['a::ord, 'a] => 'a"
haftmann@21083
   869
  "max a b == (if a <= b then b else a)"
haftmann@21083
   870
haftmann@21383
   871
lemma min_linorder:
haftmann@21383
   872
  "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
haftmann@21383
   873
  by (rule+) (simp add: min_def linorder.min_def)
haftmann@21383
   874
haftmann@21383
   875
lemma max_linorder:
haftmann@21383
   876
  "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
haftmann@21383
   877
  by (rule+) (simp add: max_def linorder.max_def)
haftmann@21383
   878
haftmann@21383
   879
lemmas min_le_iff_disj = linorder.min_le_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   880
lemmas le_max_iff_disj = linorder.le_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   881
lemmas min_less_iff_disj = linorder.min_less_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   882
lemmas less_max_iff_disj = linorder.less_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   883
lemmas min_less_iff_conj [simp] = linorder.min_less_iff_conj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   884
lemmas max_less_iff_conj [simp] = linorder.max_less_iff_conj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   885
lemmas split_min = linorder.split_min [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   886
lemmas split_max = linorder.split_max [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   887
haftmann@21383
   888
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
haftmann@21383
   889
  by (simp add: min_def)
haftmann@21383
   890
haftmann@21383
   891
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
haftmann@21383
   892
  by (simp add: max_def)
haftmann@21383
   893
haftmann@21383
   894
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
haftmann@21383
   895
  apply (simp add: min_def)
haftmann@21383
   896
  apply (blast intro: order_antisym)
haftmann@21383
   897
  done
haftmann@21383
   898
haftmann@21383
   899
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
haftmann@21383
   900
  apply (simp add: max_def)
haftmann@21383
   901
  apply (blast intro: order_antisym)
haftmann@21383
   902
  done
haftmann@21383
   903
haftmann@21383
   904
lemma min_of_mono:
haftmann@21383
   905
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
haftmann@21383
   906
  by (simp add: min_def)
haftmann@21383
   907
haftmann@21383
   908
lemma max_of_mono:
haftmann@21383
   909
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
haftmann@21383
   910
  by (simp add: max_def)
haftmann@21383
   911
wenzelm@21673
   912
wenzelm@21673
   913
subsection {* Basic ML bindings *}
wenzelm@21673
   914
wenzelm@21673
   915
ML {*
wenzelm@21673
   916
val leD = thm "leD";
wenzelm@21673
   917
val leI = thm "leI";
wenzelm@21673
   918
val linorder_neqE = thm "linorder_neqE";
wenzelm@21673
   919
val linorder_neq_iff = thm "linorder_neq_iff";
wenzelm@21673
   920
val linorder_not_le = thm "linorder_not_le";
wenzelm@21673
   921
val linorder_not_less = thm "linorder_not_less";
wenzelm@21673
   922
val monoD = thm "monoD";
wenzelm@21673
   923
val monoI = thm "monoI";
wenzelm@21673
   924
val order_antisym = thm "order_antisym";
wenzelm@21673
   925
val order_less_irrefl = thm "order_less_irrefl";
wenzelm@21673
   926
val order_refl = thm "order_refl";
wenzelm@21673
   927
val order_trans = thm "order_trans";
wenzelm@21673
   928
val split_max = thm "split_max";
wenzelm@21673
   929
val split_min = thm "split_min";
wenzelm@21673
   930
*}
wenzelm@21673
   931
wenzelm@21673
   932
ML {*
wenzelm@21673
   933
structure HOL =
wenzelm@21673
   934
struct
wenzelm@21673
   935
  val thy = theory "HOL";
wenzelm@21673
   936
end;
wenzelm@21673
   937
*}  -- "belongs to theory HOL"
wenzelm@21673
   938
nipkow@15524
   939
end