src/HOL/Orderings.thy
author wenzelm
Wed Feb 28 22:05:43 2007 +0100 (2007-02-28)
changeset 22377 61610b1beedf
parent 22348 ab505d281015
child 22384 33a46e6c7f04
permissions -rw-r--r--
tuned ML setup;
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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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  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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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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  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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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
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lemmas order_neq_le_trans = preorder_class.neq_le_trans
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lemmas order_le_neq_trans = preorder_class.le_neq_trans
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lemmas order_antisym = order_class.antisym
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lemmas order_less_not_sym = order_class.less_not_sym
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lemmas order_less_asym = order_class.less_asym
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lemmas order_eq_iff = order_class.eq_iff
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lemmas order_antisym_conv = order_class.antisym_conv
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lemmas less_imp_neq = order_class.less_imp_neq
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lemmas order_less_trans = order_class.less_trans
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lemmas order_le_less_trans = order_class.le_less_trans
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lemmas order_less_le_trans = order_class.less_le_trans
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lemmas order_less_imp_not_less = order_class.less_imp_not_less
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lemmas order_less_imp_triv = order_class.less_imp_triv
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lemmas order_less_asym' = order_class.less_asym'
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lemmas linorder_linear = linorder_class.linear
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lemmas linorder_less_linear = linorder_class.less_linear
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lemmas linorder_le_less_linear = linorder_class.le_less_linear
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lemmas linorder_le_cases = linorder_class.le_cases
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lemmas linorder_cases = linorder_class.cases
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lemmas linorder_not_less = linorder_class.not_less
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lemmas linorder_not_le = linorder_class.not_le
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lemmas linorder_neq_iff = linorder_class.neq_iff
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lemmas linorder_neqE = linorder_class.neqE
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lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
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lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
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lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
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lemmas leI = linorder_class.leI
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lemmas leD = linorder_class.leD
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lemmas not_leE = linorder_class.not_leE
paulson@16796
   320
haftmann@21083
   321
haftmann@21083
   322
subsection {* Reasoning tools setup *}
haftmann@21083
   323
haftmann@21091
   324
ML {*
haftmann@21091
   325
local
haftmann@21091
   326
haftmann@21091
   327
fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248
   328
  let
haftmann@21248
   329
    fun of_sort t =
haftmann@21248
   330
      let
haftmann@21248
   331
        val T = type_of t
haftmann@21248
   332
      in
haftmann@21091
   333
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248
   334
        T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248
   335
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248
   336
      end;
haftmann@21248
   337
    fun dec (Const ("Not", _) $ t) = (case dec t
haftmann@21248
   338
          of NONE => NONE
haftmann@21248
   339
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21248
   340
      | dec (Const ("op =",  _) $ t1 $ t2) =
haftmann@21248
   341
          if of_sort t1
haftmann@21248
   342
          then SOME (t1, "=", t2)
haftmann@21248
   343
          else NONE
haftmann@21248
   344
      | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
haftmann@21248
   345
          if of_sort t1
haftmann@21248
   346
          then SOME (t1, "<=", t2)
haftmann@21248
   347
          else NONE
haftmann@21248
   348
      | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21248
   349
          if of_sort t1
haftmann@21248
   350
          then SOME (t1, "<", t2)
haftmann@21248
   351
          else NONE
haftmann@21248
   352
      | dec _ = NONE;
haftmann@21091
   353
  in dec t end;
haftmann@21091
   354
haftmann@21091
   355
in
haftmann@21091
   356
haftmann@21091
   357
(* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091
   358
   class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091
   359
   Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091
   360
haftmann@21248
   361
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   362
struct
haftmann@21248
   363
  val le_trans = thm "order_trans";
haftmann@21248
   364
  val le_refl = thm "order_refl";
haftmann@21248
   365
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   366
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   367
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   368
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   369
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   370
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   371
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   372
  val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21248
   373
  val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21248
   374
end);
haftmann@21091
   375
haftmann@21091
   376
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   377
struct
haftmann@21248
   378
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   379
  val le_refl = thm "order_refl";
haftmann@21248
   380
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   381
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   382
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   383
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   384
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   385
  val eqI = thm "order_antisym";
haftmann@21248
   386
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   387
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   388
  val less_trans = thm "order_less_trans";
haftmann@21248
   389
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   390
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   391
  val le_trans = thm "order_trans";
haftmann@21248
   392
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   393
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   394
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   395
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   396
  val not_sym = thm "not_sym";
haftmann@21248
   397
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   398
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   399
end);
haftmann@21091
   400
haftmann@21091
   401
end;
haftmann@21091
   402
*}
haftmann@21091
   403
haftmann@21083
   404
setup {*
haftmann@21083
   405
let
haftmann@21083
   406
haftmann@21083
   407
val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083
   408
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083
   409
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083
   410
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083
   411
haftmann@21083
   412
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   413
haftmann@21083
   414
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   415
  let val prems = prems_of_ss ss;
haftmann@21083
   416
      val less = Const("Orderings.less",T);
haftmann@21083
   417
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   418
  in case find_first (prp t) prems of
haftmann@21083
   419
       NONE =>
haftmann@21083
   420
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   421
         in case find_first (prp t) prems of
haftmann@21083
   422
              NONE => NONE
haftmann@21083
   423
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083
   424
         end
haftmann@21083
   425
     | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083
   426
  end
haftmann@21083
   427
  handle THM _ => NONE;
nipkow@15524
   428
haftmann@21083
   429
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   430
  let val prems = prems_of_ss ss;
haftmann@21083
   431
      val le = Const("Orderings.less_eq",T);
haftmann@21083
   432
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   433
  in case find_first (prp t) prems of
haftmann@21083
   434
       NONE =>
haftmann@21083
   435
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   436
         in case find_first (prp t) prems of
haftmann@21083
   437
              NONE => NONE
haftmann@21083
   438
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083
   439
         end
haftmann@21083
   440
     | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083
   441
  end
haftmann@21083
   442
  handle THM _ => NONE;
nipkow@15524
   443
haftmann@21248
   444
fun add_simprocs procs thy =
haftmann@21248
   445
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   446
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   447
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   448
fun add_solver name tac thy =
haftmann@21248
   449
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   450
    (mk_solver name (K tac))); thy);
haftmann@21083
   451
haftmann@21083
   452
in
haftmann@21248
   453
  add_simprocs [
haftmann@21248
   454
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   455
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   456
     ]
haftmann@21248
   457
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   458
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   459
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   460
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   461
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   462
     of 5 March 2004, was observed). *)
haftmann@21083
   463
end
haftmann@21083
   464
*}
nipkow@15524
   465
nipkow@15524
   466
haftmann@21083
   467
subsection {* Bounded quantifiers *}
haftmann@21083
   468
haftmann@21083
   469
syntax
wenzelm@21180
   470
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   471
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   472
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   473
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   474
wenzelm@21180
   475
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   476
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   477
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   478
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   479
haftmann@21083
   480
syntax (xsymbols)
wenzelm@21180
   481
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   482
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   483
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   484
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   485
wenzelm@21180
   486
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   487
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   488
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   489
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   490
haftmann@21083
   491
syntax (HOL)
wenzelm@21180
   492
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   493
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   494
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   495
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   496
haftmann@21083
   497
syntax (HTML output)
wenzelm@21180
   498
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   499
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   500
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   501
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   502
wenzelm@21180
   503
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   504
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   505
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   506
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   507
haftmann@21083
   508
translations
haftmann@21083
   509
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   510
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   511
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   512
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   513
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   514
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   515
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   516
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   517
haftmann@21083
   518
print_translation {*
haftmann@21083
   519
let
wenzelm@22377
   520
  val All_binder = Syntax.binder_name @{const_syntax "All"};
wenzelm@22377
   521
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
wenzelm@22377
   522
  val impl = @{const_syntax "op -->"};
wenzelm@22377
   523
  val conj = @{const_syntax "op &"};
wenzelm@22377
   524
  val less = @{const_syntax "less"};
wenzelm@22377
   525
  val less_eq = @{const_syntax "less_eq"};
wenzelm@21180
   526
wenzelm@21180
   527
  val trans =
wenzelm@21524
   528
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   529
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   530
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   531
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   532
krauss@22344
   533
  fun matches_bound v t = 
krauss@22344
   534
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344
   535
              | _ => false
krauss@22344
   536
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344
   537
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180
   538
wenzelm@21180
   539
  fun tr' q = (q,
wenzelm@21180
   540
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   541
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   542
        NONE => raise Match
wenzelm@21180
   543
      | SOME (l, g) =>
krauss@22344
   544
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344
   545
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344
   546
          else raise Match)
wenzelm@21180
   547
     | _ => raise Match);
wenzelm@21524
   548
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   549
*}
haftmann@21083
   550
haftmann@21083
   551
haftmann@21383
   552
subsection {* Transitivity reasoning *}
haftmann@21383
   553
haftmann@21383
   554
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
haftmann@21383
   555
  by (rule subst)
haftmann@21383
   556
haftmann@21383
   557
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
haftmann@21383
   558
  by (rule ssubst)
haftmann@21383
   559
haftmann@21383
   560
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
haftmann@21383
   561
  by (rule subst)
haftmann@21383
   562
haftmann@21383
   563
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
haftmann@21383
   564
  by (rule ssubst)
haftmann@21383
   565
haftmann@21383
   566
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   567
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   568
proof -
haftmann@21383
   569
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   570
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   571
  also assume "f b < c"
haftmann@21383
   572
  finally (order_less_trans) show ?thesis .
haftmann@21383
   573
qed
haftmann@21383
   574
haftmann@21383
   575
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   576
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   577
proof -
haftmann@21383
   578
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   579
  assume "a < f b"
haftmann@21383
   580
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   581
  finally (order_less_trans) show ?thesis .
haftmann@21383
   582
qed
haftmann@21383
   583
haftmann@21383
   584
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   585
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   586
proof -
haftmann@21383
   587
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   588
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   589
  also assume "f b < c"
haftmann@21383
   590
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   591
qed
haftmann@21383
   592
haftmann@21383
   593
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   594
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   595
proof -
haftmann@21383
   596
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   597
  assume "a <= f b"
haftmann@21383
   598
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   599
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   600
qed
haftmann@21383
   601
haftmann@21383
   602
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   603
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   604
proof -
haftmann@21383
   605
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   606
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   607
  also assume "f b <= c"
haftmann@21383
   608
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   609
qed
haftmann@21383
   610
haftmann@21383
   611
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   612
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   613
proof -
haftmann@21383
   614
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   615
  assume "a < f b"
haftmann@21383
   616
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   617
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   618
qed
haftmann@21383
   619
haftmann@21383
   620
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   621
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   622
proof -
haftmann@21383
   623
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   624
  assume "a <= f b"
haftmann@21383
   625
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   626
  finally (order_trans) show ?thesis .
haftmann@21383
   627
qed
haftmann@21383
   628
haftmann@21383
   629
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   630
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   631
proof -
haftmann@21383
   632
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   633
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   634
  also assume "f b <= c"
haftmann@21383
   635
  finally (order_trans) show ?thesis .
haftmann@21383
   636
qed
haftmann@21383
   637
haftmann@21383
   638
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   639
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   640
proof -
haftmann@21383
   641
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   642
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   643
  also assume "f b = c"
haftmann@21383
   644
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   645
qed
haftmann@21383
   646
haftmann@21383
   647
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   648
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   649
proof -
haftmann@21383
   650
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   651
  assume "a = f b"
haftmann@21383
   652
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   653
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   654
qed
haftmann@21383
   655
haftmann@21383
   656
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   657
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   658
proof -
haftmann@21383
   659
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   660
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   661
  also assume "f b = c"
haftmann@21383
   662
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   663
qed
haftmann@21383
   664
haftmann@21383
   665
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   666
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   667
proof -
haftmann@21383
   668
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   669
  assume "a = f b"
haftmann@21383
   670
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   671
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   672
qed
haftmann@21383
   673
haftmann@21383
   674
text {*
haftmann@21383
   675
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   676
*}
haftmann@21383
   677
haftmann@21383
   678
lemmas order_trans_rules [trans] =
haftmann@21383
   679
  order_less_subst2
haftmann@21383
   680
  order_less_subst1
haftmann@21383
   681
  order_le_less_subst2
haftmann@21383
   682
  order_le_less_subst1
haftmann@21383
   683
  order_less_le_subst2
haftmann@21383
   684
  order_less_le_subst1
haftmann@21383
   685
  order_subst2
haftmann@21383
   686
  order_subst1
haftmann@21383
   687
  ord_le_eq_subst
haftmann@21383
   688
  ord_eq_le_subst
haftmann@21383
   689
  ord_less_eq_subst
haftmann@21383
   690
  ord_eq_less_subst
haftmann@21383
   691
  forw_subst
haftmann@21383
   692
  back_subst
haftmann@21383
   693
  rev_mp
haftmann@21383
   694
  mp
haftmann@21383
   695
  order_neq_le_trans
haftmann@21383
   696
  order_le_neq_trans
haftmann@21383
   697
  order_less_trans
haftmann@21383
   698
  order_less_asym'
haftmann@21383
   699
  order_le_less_trans
haftmann@21383
   700
  order_less_le_trans
haftmann@21383
   701
  order_trans
haftmann@21383
   702
  order_antisym
haftmann@21383
   703
  ord_le_eq_trans
haftmann@21383
   704
  ord_eq_le_trans
haftmann@21383
   705
  ord_less_eq_trans
haftmann@21383
   706
  ord_eq_less_trans
haftmann@21383
   707
  trans
haftmann@21383
   708
haftmann@21083
   709
wenzelm@21180
   710
(* FIXME cleanup *)
wenzelm@21180
   711
haftmann@21083
   712
text {* These support proving chains of decreasing inequalities
haftmann@21083
   713
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   714
haftmann@21083
   715
lemma xt1:
haftmann@21083
   716
  "a = b ==> b > c ==> a > c"
haftmann@21083
   717
  "a > b ==> b = c ==> a > c"
haftmann@21083
   718
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   719
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   720
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   721
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   722
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   723
  "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083
   724
  "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083
   725
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   726
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   727
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   728
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   729
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   730
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   731
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   732
by auto
haftmann@21083
   733
haftmann@21083
   734
lemma xt2:
haftmann@21083
   735
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   736
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   737
haftmann@21083
   738
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   739
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   740
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   741
haftmann@21083
   742
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   743
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   744
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   745
haftmann@21083
   746
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   747
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   748
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   749
haftmann@21083
   750
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   751
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   752
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   753
haftmann@21083
   754
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   755
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   756
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   757
haftmann@21083
   758
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   759
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   760
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   761
haftmann@21083
   762
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   763
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   764
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   765
haftmann@21083
   766
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   767
haftmann@21083
   768
(* 
haftmann@21083
   769
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   770
  for the wrong thing in an Isar proof.
haftmann@21083
   771
haftmann@21083
   772
  The extra transitivity rules can be used as follows: 
haftmann@21083
   773
haftmann@21083
   774
lemma "(a::'a::order) > z"
haftmann@21083
   775
proof -
haftmann@21083
   776
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   777
    sorry
haftmann@21083
   778
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   779
    sorry
haftmann@21083
   780
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   781
    sorry
haftmann@21083
   782
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   783
    sorry
haftmann@21083
   784
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   785
    sorry
haftmann@21083
   786
  also (xtrans) have "?rhs > z"
haftmann@21083
   787
    sorry
haftmann@21083
   788
  finally (xtrans) show ?thesis .
haftmann@21083
   789
qed
haftmann@21083
   790
haftmann@21083
   791
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   792
  leave out the "(xtrans)" above.
haftmann@21083
   793
*)
haftmann@21083
   794
haftmann@21546
   795
subsection {* Order on bool *}
haftmann@21546
   796
haftmann@21546
   797
instance bool :: linorder 
haftmann@21546
   798
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   799
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@21546
   800
  by default (auto simp add: le_bool_def less_bool_def)
haftmann@21546
   801
haftmann@21546
   802
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@21546
   803
  by (simp add: le_bool_def)
haftmann@21546
   804
haftmann@21546
   805
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@21546
   806
  by (simp add: le_bool_def)
haftmann@21546
   807
haftmann@21546
   808
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@21546
   809
  by (simp add: le_bool_def)
haftmann@21546
   810
haftmann@21546
   811
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@21546
   812
  by (simp add: le_bool_def)
haftmann@21546
   813
haftmann@22348
   814
lemma [code func]:
haftmann@22348
   815
  "False \<le> b \<longleftrightarrow> True"
haftmann@22348
   816
  "True \<le> b \<longleftrightarrow> b"
haftmann@22348
   817
  "False < b \<longleftrightarrow> b"
haftmann@22348
   818
  "True < b \<longleftrightarrow> False"
haftmann@22348
   819
  unfolding le_bool_def less_bool_def by simp_all
haftmann@22348
   820
haftmann@21383
   821
subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083
   822
haftmann@21216
   823
locale mono =
haftmann@21216
   824
  fixes f
haftmann@21216
   825
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   826
haftmann@21216
   827
lemmas monoI [intro?] = mono.intro
haftmann@21216
   828
  and monoD [dest?] = mono.mono
haftmann@21083
   829
haftmann@21083
   830
constdefs
haftmann@21083
   831
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083
   832
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083
   833
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083
   834
haftmann@21383
   835
lemma LeastI2_order:
haftmann@21383
   836
  "[| P (x::'a::order);
haftmann@21383
   837
      !!y. P y ==> x <= y;
haftmann@21383
   838
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   839
   ==> Q (Least P)"
haftmann@21383
   840
  apply (unfold Least_def)
haftmann@21383
   841
  apply (rule theI2)
haftmann@21383
   842
    apply (blast intro: order_antisym)+
haftmann@21383
   843
  done
haftmann@21383
   844
haftmann@21383
   845
lemma Least_equality:
haftmann@21383
   846
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
haftmann@21383
   847
  apply (simp add: Least_def)
haftmann@21383
   848
  apply (rule the_equality)
haftmann@21383
   849
  apply (auto intro!: order_antisym)
haftmann@21383
   850
  done
haftmann@21383
   851
haftmann@21083
   852
constdefs
haftmann@21083
   853
  min :: "['a::ord, 'a] => 'a"
haftmann@21083
   854
  "min a b == (if a <= b then a else b)"
haftmann@21083
   855
  max :: "['a::ord, 'a] => 'a"
haftmann@21083
   856
  "max a b == (if a <= b then b else a)"
haftmann@21083
   857
haftmann@21383
   858
lemma min_linorder:
haftmann@21383
   859
  "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
haftmann@22316
   860
  by rule+ (simp add: min_def linorder_class.min_def)
haftmann@21383
   861
haftmann@21383
   862
lemma max_linorder:
haftmann@21383
   863
  "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
haftmann@22316
   864
  by rule+ (simp add: max_def linorder_class.max_def)
haftmann@21383
   865
haftmann@22316
   866
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder]
haftmann@22316
   867
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder]
haftmann@22316
   868
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder]
haftmann@22316
   869
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder]
haftmann@22316
   870
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder]
haftmann@22316
   871
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder]
haftmann@22316
   872
lemmas split_min = linorder_class.split_min [unfolded min_linorder]
haftmann@22316
   873
lemmas split_max = linorder_class.split_max [unfolded max_linorder]
haftmann@21383
   874
haftmann@21383
   875
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
haftmann@21383
   876
  by (simp add: min_def)
haftmann@21383
   877
haftmann@21383
   878
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
haftmann@21383
   879
  by (simp add: max_def)
haftmann@21383
   880
haftmann@21383
   881
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
haftmann@21383
   882
  apply (simp add: min_def)
haftmann@21383
   883
  apply (blast intro: order_antisym)
haftmann@21383
   884
  done
haftmann@21383
   885
haftmann@21383
   886
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
haftmann@21383
   887
  apply (simp add: max_def)
haftmann@21383
   888
  apply (blast intro: order_antisym)
haftmann@21383
   889
  done
haftmann@21383
   890
haftmann@21383
   891
lemma min_of_mono:
haftmann@21383
   892
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
haftmann@21383
   893
  by (simp add: min_def)
haftmann@21383
   894
haftmann@21383
   895
lemma max_of_mono:
haftmann@21383
   896
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
haftmann@21383
   897
  by (simp add: max_def)
haftmann@21383
   898
wenzelm@21673
   899
wenzelm@21673
   900
subsection {* Basic ML bindings *}
wenzelm@21673
   901
wenzelm@21673
   902
ML {*
wenzelm@21673
   903
val leD = thm "leD";
wenzelm@21673
   904
val leI = thm "leI";
wenzelm@21673
   905
val linorder_neqE = thm "linorder_neqE";
wenzelm@21673
   906
val linorder_neq_iff = thm "linorder_neq_iff";
wenzelm@21673
   907
val linorder_not_le = thm "linorder_not_le";
wenzelm@21673
   908
val linorder_not_less = thm "linorder_not_less";
wenzelm@21673
   909
val monoD = thm "monoD";
wenzelm@21673
   910
val monoI = thm "monoI";
wenzelm@21673
   911
val order_antisym = thm "order_antisym";
wenzelm@21673
   912
val order_less_irrefl = thm "order_less_irrefl";
wenzelm@21673
   913
val order_refl = thm "order_refl";
wenzelm@21673
   914
val order_trans = thm "order_trans";
wenzelm@21673
   915
val split_max = thm "split_max";
wenzelm@21673
   916
val split_min = thm "split_min";
wenzelm@21673
   917
*}
wenzelm@21673
   918
wenzelm@21673
   919
ML {*
wenzelm@21673
   920
structure HOL =
wenzelm@21673
   921
struct
wenzelm@21673
   922
  val thy = theory "HOL";
wenzelm@21673
   923
end;
wenzelm@21673
   924
*}  -- "belongs to theory HOL"
wenzelm@21673
   925
nipkow@15524
   926
end