src/HOL/Orderings.thy
author wenzelm
Wed, 28 Feb 2007 22:05:43 +0100
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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
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subsection {* Reasoning tools setup *}
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ML {*
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local
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fun decomp_gen sort thy (Trueprop $ t) =
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  let
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    fun of_sort t =
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      let
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        val T = type_of t
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      in
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        (* exclude numeric types: linear arithmetic subsumes transitivity *)
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        T <> HOLogic.natT andalso T <> HOLogic.intT
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          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
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      end;
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    fun dec (Const ("Not", _) $ t) = (case dec t
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          of NONE => NONE
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           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
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      | dec (Const ("op =",  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "=", t2)
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          else NONE
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      | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "<=", t2)
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          else NONE
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      | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "<", t2)
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          else NONE
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      | dec _ = NONE;
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  in dec t end;
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in
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(* The setting up of Quasi_Tac serves as a demo.  Since there is no
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   class for quasi orders, the tactics Quasi_Tac.trans_tac and
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   Quasi_Tac.quasi_tac are not of much use. *)
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structure Quasi_Tac = Quasi_Tac_Fun (
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struct
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  val le_trans = thm "order_trans";
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  val le_refl = thm "order_refl";
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  val eqD1 = thm "order_eq_refl";
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  val eqD2 = thm "sym" RS thm "order_eq_refl";
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  val less_reflE = thm "order_less_irrefl" RS thm "notE";
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  val less_imp_le = thm "order_less_imp_le";
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  val le_neq_trans = thm "order_le_neq_trans";
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  val neq_le_trans = thm "order_neq_le_trans";
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  val less_imp_neq = thm "less_imp_neq";
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  val decomp_trans = decomp_gen ["Orderings.order"];
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  val decomp_quasi = decomp_gen ["Orderings.order"];
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end);
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structure Order_Tac = Order_Tac_Fun (
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struct
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  val less_reflE = thm "order_less_irrefl" RS thm "notE";
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  val le_refl = thm "order_refl";
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  val less_imp_le = thm "order_less_imp_le";
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  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
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  val not_leI = thm "linorder_not_le" RS thm "iffD2";
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  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
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  val not_leD = thm "linorder_not_le" RS thm "iffD1";
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  val eqI = thm "order_antisym";
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  val eqD1 = thm "order_eq_refl";
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  val eqD2 = thm "sym" RS thm "order_eq_refl";
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  val less_trans = thm "order_less_trans";
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  val less_le_trans = thm "order_less_le_trans";
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  val le_less_trans = thm "order_le_less_trans";
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  val le_trans = thm "order_trans";
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  val le_neq_trans = thm "order_le_neq_trans";
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  val neq_le_trans = thm "order_neq_le_trans";
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  val less_imp_neq = thm "less_imp_neq";
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  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
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  val not_sym = thm "not_sym";
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  val decomp_part = decomp_gen ["Orderings.order"];
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  val decomp_lin = decomp_gen ["Orderings.linorder"];
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   399
end);
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end;
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*}
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   403
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   404
setup {*
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   405
let
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   406
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val order_antisym_conv = thm "order_antisym_conv"
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   408
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
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   409
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
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   410
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
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   411
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fun prp t thm = (#prop (rep_thm thm) = t);
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   413
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fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
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  let val prems = prems_of_ss ss;
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      val less = Const("Orderings.less",T);
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      val t = HOLogic.mk_Trueprop(le $ s $ r);
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  in case find_first (prp t) prems of
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       NONE =>
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         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
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         in case find_first (prp t) prems of
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              NONE => NONE
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            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
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         end
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     | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
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  end
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  handle THM _ => NONE;
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   429
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
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   430
  let val prems = prems_of_ss ss;
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      val le = Const("Orderings.less_eq",T);
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      val t = HOLogic.mk_Trueprop(le $ r $ s);
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   433
  in case find_first (prp t) prems of
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       NONE =>
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         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
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   436
         in case find_first (prp t) prems of
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              NONE => NONE
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            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
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   439
         end
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     | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
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   441
  end
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   442
  handle THM _ => NONE;
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   443
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   444
fun add_simprocs procs thy =
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   445
  (Simplifier.change_simpset_of thy (fn ss => ss
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   446
    addsimprocs (map (fn (name, raw_ts, proc) =>
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   447
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
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   448
fun add_solver name tac thy =
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   449
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
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   450
    (mk_solver name (K tac))); thy);
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   451
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   452
in
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   453
  add_simprocs [
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   454
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
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   455
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
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   456
     ]
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   457
  #> add_solver "Trans_linear" Order_Tac.linear_tac
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   458
  #> add_solver "Trans_partial" Order_Tac.partial_tac
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   459
  (* Adding the transitivity reasoners also as safe solvers showed a slight
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   460
     speed up, but the reasoning strength appears to be not higher (at least
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   461
     no breaking of additional proofs in the entire HOL distribution, as
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   462
     of 5 March 2004, was observed). *)
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   463
end
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*}
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2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
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   466
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   467
subsection {* Bounded quantifiers *}
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   468
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   469
syntax
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   470
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
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  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
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  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
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  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
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   474
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   475
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
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  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
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   477
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   478
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   479
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   480
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   481
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   482
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   483
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   484
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   485
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   486
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   487
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   488
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   489
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   490
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   491
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   492
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   493
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   494
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   495
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   496
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   497
syntax (HTML output)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   498
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   499
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   500
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   501
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   502
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   503
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   504
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   505
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   506
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   507
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   508
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   509
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   510
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   511
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   512
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   513
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   514
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   515
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   516
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   517
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   518
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   519
let
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   520
  val All_binder = Syntax.binder_name @{const_syntax "All"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   521
  val Ex_binder = Syntax.binder_name @{const_syntax "Ex"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   522
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   523
  val conj = @{const_syntax "op &"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   524
  val less = @{const_syntax "less"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   525
  val less_eq = @{const_syntax "less_eq"};
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   526
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   527
  val trans =
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   528
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   529
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   530
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   531
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   532
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   533
  fun matches_bound v t = 
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   534
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   535
              | _ => false
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   536
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   537
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   538
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   539
  fun tr' q = (q,
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   540
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   541
      (case AList.lookup (op =) trans (q, c, d) of
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   542
        NONE => raise Match
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   543
      | SOME (l, g) =>
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   544
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   545
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   546
          else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   547
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   548
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   549
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   550
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   551
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   552
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   553
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   554
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   555
  by (rule subst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   556
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   557
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   558
  by (rule ssubst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   559
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   560
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   561
  by (rule subst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   562
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   563
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   564
  by (rule ssubst)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   565
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   566
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   567
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   568
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   569
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   570
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   571
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   572
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   573
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   574
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   575
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   576
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   577
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   578
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   579
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   580
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   581
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   582
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   583
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   584
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   585
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   586
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   587
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   588
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   589
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   590
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   591
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   592
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   593
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   594
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   595
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   596
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   597
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   598
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   599
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   600
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   601
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   602
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   603
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   604
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   605
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   606
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   607
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   608
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   609
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   610
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   611
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   612
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   613
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   614
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   615
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   616
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   617
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   618
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   619
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   620
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   621
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   622
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   623
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   624
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   625
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   626
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   627
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   628
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   629
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   630
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   631
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   632
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   633
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   634
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   635
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   636
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   637
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   638
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   639
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   640
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   641
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   642
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   643
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   644
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   645
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   646
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   647
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   648
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   649
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   650
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   651
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   652
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   653
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   654
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   655
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   656
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   657
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   658
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   659
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   660
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   661
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   662
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   663
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   664
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   665
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   666
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   667
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   668
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   669
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   670
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   671
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   672
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   673
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   674
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   675
  Note that this list of rules is in reverse order of priorities.
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   676
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   677
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   678
lemmas order_trans_rules [trans] =
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   679
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   680
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   681
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   682
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   683
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   684
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   685
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   686
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   687
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   688
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   689
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   690
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   691
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   692
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   693
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   694
  mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   695
  order_neq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   696
  order_le_neq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   697
  order_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   698
  order_less_asym'
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   699
  order_le_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   700
  order_less_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   701
  order_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   702
  order_antisym
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   703
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   704
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   705
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   706
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   707
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   708
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   709
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   710
(* FIXME cleanup *)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   711
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   712
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   713
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   714
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   715
lemma xt1:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   716
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   717
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   718
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   719
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   720
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   721
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   722
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   723
  "(x::'a::order) >= y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   724
  "(a::'a::order) > b ==> b > a ==> ?P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   725
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   726
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   727
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   728
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   729
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   730
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   731
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   732
by auto
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   733
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   734
lemma xt2:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   735
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   736
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   737
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   738
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   739
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   740
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   741
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   742
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   743
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   744
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   745
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   746
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   747
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   748
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   749
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   750
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   751
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   752
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   753
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   754
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   755
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   756
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   757
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   758
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   759
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   760
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   761
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   762
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   763
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   764
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   765
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   766
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   767
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   768
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   769
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   770
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   771
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   772
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   773
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   774
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   775
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   776
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   777
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   778
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   779
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   780
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   781
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   782
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   783
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   784
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   785
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   786
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   787
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   788
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   789
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   790
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   791
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   792
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   793
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   794
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   795
subsection {* Order on bool *}
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   796
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   797
instance bool :: linorder 
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   798
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   799
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   800
  by default (auto simp add: le_bool_def less_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   801
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   802
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   803
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   804
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   805
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   806
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   807
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   808
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   809
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   810
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   811
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   812
  by (simp add: le_bool_def)
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   813
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   814
lemma [code func]:
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   815
  "False \<le> b \<longleftrightarrow> True"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   816
  "True \<le> b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   817
  "False < b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   818
  "True < b \<longleftrightarrow> False"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   819
  unfolding le_bool_def less_bool_def by simp_all
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   820
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   821
subsection {* Monotonicity, syntactic least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   822
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   823
locale mono =
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   824
  fixes f
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   825
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   826
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   827
lemmas monoI [intro?] = mono.intro
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   828
  and monoD [dest?] = mono.mono
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   829
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   830
constdefs
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   831
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   832
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   833
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   834
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   835
lemma LeastI2_order:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   836
  "[| P (x::'a::order);
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   837
      !!y. P y ==> x <= y;
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   838
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   839
   ==> Q (Least P)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
  apply (unfold Least_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   841
  apply (rule theI2)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   842
    apply (blast intro: order_antisym)+
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   843
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   844
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   845
lemma Least_equality:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   846
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   847
  apply (simp add: Least_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   848
  apply (rule the_equality)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   849
  apply (auto intro!: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   850
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   851
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   852
constdefs
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   853
  min :: "['a::ord, 'a] => 'a"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   854
  "min a b == (if a <= b then a else b)"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   855
  max :: "['a::ord, 'a] => 'a"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   856
  "max a b == (if a <= b then b else a)"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   857
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   858
lemma min_linorder:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   859
  "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   860
  by rule+ (simp add: min_def linorder_class.min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   861
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   862
lemma max_linorder:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   863
  "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   864
  by rule+ (simp add: max_def linorder_class.max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   865
22316
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   866
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   867
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   868
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   869
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   870
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   871
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   872
lemmas split_min = linorder_class.split_min [unfolded min_linorder]
f662831459de added class "preorder"
haftmann
parents: 22295
diff changeset
   873
lemmas split_max = linorder_class.split_max [unfolded max_linorder]
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   874
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   875
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   876
  by (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   877
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   878
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   879
  by (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   880
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   881
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   882
  apply (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   883
  apply (blast intro: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   884
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   885
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   886
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   887
  apply (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   888
  apply (blast intro: order_antisym)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   889
  done
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   890
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   891
lemma min_of_mono:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   892
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   893
  by (simp add: min_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   894
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   895
lemma max_of_mono:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   896
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   897
  by (simp add: max_def)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   898
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   899
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   900
subsection {* Basic ML bindings *}
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   901
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   902
ML {*
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   903
val leD = thm "leD";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   904
val leI = thm "leI";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   905
val linorder_neqE = thm "linorder_neqE";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   906
val linorder_neq_iff = thm "linorder_neq_iff";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   907
val linorder_not_le = thm "linorder_not_le";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   908
val linorder_not_less = thm "linorder_not_less";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   909
val monoD = thm "monoD";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   910
val monoI = thm "monoI";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   911
val order_antisym = thm "order_antisym";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   912
val order_less_irrefl = thm "order_less_irrefl";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   913
val order_refl = thm "order_refl";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   914
val order_trans = thm "order_trans";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   915
val split_max = thm "split_max";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   916
val split_min = thm "split_min";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   917
*}
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   918
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   919
ML {*
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   920
structure HOL =
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   921
struct
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   922
  val thy = theory "HOL";
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   923
end;
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   924
*}  -- "belongs to theory HOL"
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   925
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   926
end