src/HOL/Orderings.thy
author haftmann
Wed Nov 15 17:05:42 2006 +0100 (2006-11-15)
changeset 21383 17e6275e13f5
parent 21329 7338206d75f1
child 21404 eb85850d3eb7
permissions -rw-r--r--
added transitivity rules, reworking of min/max lemmas
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(*  Title:      HOL/Orderings.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* Syntactic and abstract orders *}
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theory Orderings
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imports HOL
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begin
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section {* Abstract orders *}
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subsection {* Order syntax *}
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class ord =
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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begin
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notation
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  less_eq  ("op \<^loc><=")
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  less_eq  ("(_/ \<^loc><= _)" [51, 51] 50)
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  less  ("op \<^loc><")
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  less  ("(_/ \<^loc>< _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<^loc>\<le>")
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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>")
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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)
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  "x \<^loc>> y \<equiv> y \<^loc>< x"
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  greater_eq  (infix "\<^loc>>=" 50)
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  "x \<^loc>>= y \<equiv> y \<^loc><= x"
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notation (xsymbols)
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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 <=")
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  less_eq  ("(_/ <= _)" [51, 51] 50)
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  less  ("op <")
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  less  ("(_/ < _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<le>")
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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>")
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater  (infix ">" 50)
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  "x > y \<equiv> y < x"
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  greater_eq  (infix ">=" 50)
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  "x >= y \<equiv> y <= x"
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notation (xsymbols)
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  greater_eq  (infix "\<ge>" 50)
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subsection {* Quasiorders (preorders) *}
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locale preorder =
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
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  fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50)
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  assumes refl [iff]: "x \<sqsubseteq> x"
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  and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
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  and less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
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begin
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abbreviation (input)
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  greater_eq (infixl "\<sqsupseteq>" 50)
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  "x \<sqsupseteq> y \<equiv> y \<sqsubseteq> x"
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abbreviation (input)
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  greater (infixl "\<sqsupset>" 50)
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  "x \<sqsupset> y \<equiv> y \<sqsubset> x"
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text {* Reflexivity. *}
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lemma eq_refl: "x = y \<Longrightarrow> x \<sqsubseteq> y"
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    -- {* This form is useful with the classical reasoner. *}
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  by (erule ssubst) (rule refl)
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lemma less_irrefl [iff]: "\<not> x \<sqsubset> x"
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  by (simp add: less_le)
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lemma le_less: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubset> y \<or> x = y"
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    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
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  by (simp add: less_le) blast
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lemma le_imp_less_or_eq: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubset> y \<or> x = y"
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  unfolding less_le by blast
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lemma less_imp_le: "x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y"
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  unfolding less_le by blast
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lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
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  by (erule contrapos_pn, erule subst, rule less_irrefl)
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_eq: "x \<sqsubset> y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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  by auto
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lemma less_imp_not_eq2: "x \<sqsubset> y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
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  by auto
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text {* Transitivity rules for calculational reasoning *}
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lemma neq_le_trans: "\<lbrakk> a \<noteq> b; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
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  by (simp add: less_le)
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lemma le_neq_trans: "\<lbrakk> a \<sqsubseteq> b; a \<noteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
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  by (simp add: less_le)
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end
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subsection {* Partial orderings *}
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locale partial_order = preorder + 
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  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
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context partial_order
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begin
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text {* Asymmetry. *}
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lemma less_not_sym: "x \<sqsubset> y \<Longrightarrow> \<not> (y \<sqsubset> x)"
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  by (simp add: less_le antisym)
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lemma less_asym: "x \<sqsubset> y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<sqsubset> x) \<Longrightarrow> P"
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  by (drule less_not_sym, erule contrapos_np) simp
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lemma eq_iff: "x = y \<longleftrightarrow> x \<sqsubseteq> y \<and> y \<sqsubseteq> x"
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  by (blast intro: antisym)
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lemma antisym_conv: "y \<sqsubseteq> x \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
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  by (blast intro: antisym)
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lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
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  by (erule contrapos_pn, erule subst, rule less_irrefl)
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text {* Transitivity. *}
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lemma less_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
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  by (simp add: less_le) (blast intro: trans antisym)
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lemma le_less_trans: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
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  by (simp add: less_le) (blast intro: trans antisym)
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lemma less_le_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
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  by (simp add: less_le) (blast intro: trans antisym)
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_less: "x \<sqsubset> y \<Longrightarrow> (\<not> y \<sqsubset> x) \<longleftrightarrow> True"
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  by (blast elim: less_asym)
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lemma less_imp_triv: "x \<sqsubset> y \<Longrightarrow> (y \<sqsubset> x \<longrightarrow> P) \<longleftrightarrow> True"
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  by (blast elim: less_asym)
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text {* Transitivity rules for calculational reasoning *}
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lemma less_asym': "\<lbrakk> a \<sqsubset> b; b \<sqsubset> a \<rbrakk> \<Longrightarrow> P"
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  by (rule less_asym)
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end
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axclass order < ord
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  order_refl [iff]: "x <= x"
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  order_trans: "x <= y ==> y <= z ==> x <= z"
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  order_antisym: "x <= y ==> y <= x ==> x = y"
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  order_less_le: "(x < y) = (x <= y & x ~= y)"
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interpretation order:
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  partial_order ["op \<le> \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool"]
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apply unfold_locales
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apply (rule order_refl)
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apply (erule (1) order_trans)
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apply (rule order_less_le)
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apply (erule (1) order_antisym)
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done
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subsection {* Linear (total) orders *}
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locale linorder = partial_order +
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  assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
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context linorder
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begin
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lemma trichotomy: "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 trichotomy)
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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 trichotomy 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 trichotomy, 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"
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  "min a b = (if a \<sqsubseteq> b then a else b)"
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  max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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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 trichotomy 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 trichotomy 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 trichotomy 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 trichotomy by (auto intro: less_trans)
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lemma split_min:
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  "P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)"
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  by (simp add: min_def)
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lemma split_max:
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  "P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)"
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  by (simp add: max_def)
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end
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axclass linorder < order
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  linorder_linear: "x <= y | y <= x"
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interpretation linorder:
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  linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"]
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  by unfold_locales (rule linorder_linear)
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subsection {* Name duplicates *}
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lemmas order_eq_refl [where 'b = "?'a::order"] = order.eq_refl
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lemmas order_less_irrefl [where 'b = "?'a::order"] = order.less_irrefl
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lemmas order_le_less [where 'b = "?'a::order"] = order.le_less
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lemmas order_le_imp_less_or_eq [where 'b = "?'a::order"] = order.le_imp_less_or_eq
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lemmas order_less_imp_le [where 'b = "?'a::order"] = order.less_imp_le
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lemmas order_less_not_sym [where 'b = "?'a::order"] = order.less_not_sym
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lemmas order_less_asym [where 'b = "?'a::order"] = order.less_asym
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lemmas order_eq_iff [where 'b = "?'a::order"] = order.eq_iff
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lemmas order_antisym_conv [where 'b = "?'a::order"] = order.antisym_conv
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lemmas less_imp_neq [where 'b = "?'a::order"] = order.less_imp_neq
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lemmas order_less_trans [where 'b = "?'a::order"] = order.less_trans
haftmann@21248
   322
lemmas order_le_less_trans [where 'b = "?'a::order"] = order.le_less_trans
haftmann@21248
   323
lemmas order_less_le_trans [where 'b = "?'a::order"] = order.less_le_trans
haftmann@21248
   324
lemmas order_less_imp_not_less [where 'b = "?'a::order"] = order.less_imp_not_less
haftmann@21248
   325
lemmas order_less_imp_triv [where 'b = "?'a::order"] = order.less_imp_triv
haftmann@21248
   326
lemmas order_less_imp_not_eq [where 'b = "?'a::order"] = order.less_imp_not_eq
haftmann@21248
   327
lemmas order_less_imp_not_eq2 [where 'b = "?'a::order"] = order.less_imp_not_eq2
haftmann@21248
   328
lemmas order_neq_le_trans [where 'b = "?'a::order"] = order.neq_le_trans
haftmann@21248
   329
lemmas order_le_neq_trans [where 'b = "?'a::order"] = order.le_neq_trans
haftmann@21248
   330
lemmas order_less_asym' [where 'b = "?'a::order"] = order.less_asym'
haftmann@21248
   331
lemmas linorder_less_linear [where 'b = "?'a::linorder"] = linorder.trichotomy
haftmann@21248
   332
lemmas linorder_le_less_linear [where 'b = "?'a::linorder"] = linorder.le_less_linear
haftmann@21248
   333
lemmas linorder_le_cases [where 'b = "?'a::linorder"] = linorder.le_cases
haftmann@21248
   334
lemmas linorder_cases [where 'b = "?'a::linorder"] = linorder.cases
haftmann@21248
   335
lemmas linorder_not_less [where 'b = "?'a::linorder"] = linorder.not_less
haftmann@21248
   336
lemmas linorder_not_le [where 'b = "?'a::linorder"] = linorder.not_le
haftmann@21248
   337
lemmas linorder_neq_iff [where 'b = "?'a::linorder"] = linorder.neq_iff
haftmann@21248
   338
lemmas linorder_neqE [where 'b = "?'a::linorder"] = linorder.neqE
haftmann@21248
   339
lemmas linorder_antisym_conv1 [where 'b = "?'a::linorder"] = linorder.antisym_conv1
haftmann@21248
   340
lemmas linorder_antisym_conv2 [where 'b = "?'a::linorder"] = linorder.antisym_conv2
haftmann@21248
   341
lemmas linorder_antisym_conv3 [where 'b = "?'a::linorder"] = linorder.antisym_conv3
haftmann@21248
   342
lemmas leI [where 'b = "?'a::linorder"] = linorder.leI
haftmann@21248
   343
lemmas leD [where 'b = "?'a::linorder"] = linorder.leD
haftmann@21248
   344
lemmas not_leE [where 'b = "?'a::linorder"] = linorder.not_leE
paulson@16796
   345
haftmann@21083
   346
haftmann@21083
   347
subsection {* Reasoning tools setup *}
haftmann@21083
   348
haftmann@21091
   349
ML {*
haftmann@21091
   350
local
haftmann@21091
   351
haftmann@21091
   352
fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248
   353
  let
haftmann@21248
   354
    fun of_sort t =
haftmann@21248
   355
      let
haftmann@21248
   356
        val T = type_of t
haftmann@21248
   357
      in
haftmann@21091
   358
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248
   359
        T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248
   360
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248
   361
      end;
haftmann@21248
   362
    fun dec (Const ("Not", _) $ t) = (case dec t
haftmann@21248
   363
          of NONE => NONE
haftmann@21248
   364
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21248
   365
      | dec (Const ("op =",  _) $ t1 $ t2) =
haftmann@21248
   366
          if of_sort t1
haftmann@21248
   367
          then SOME (t1, "=", t2)
haftmann@21248
   368
          else NONE
haftmann@21248
   369
      | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
haftmann@21248
   370
          if of_sort t1
haftmann@21248
   371
          then SOME (t1, "<=", t2)
haftmann@21248
   372
          else NONE
haftmann@21248
   373
      | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21248
   374
          if of_sort t1
haftmann@21248
   375
          then SOME (t1, "<", t2)
haftmann@21248
   376
          else NONE
haftmann@21248
   377
      | dec _ = NONE;
haftmann@21091
   378
  in dec t end;
haftmann@21091
   379
haftmann@21091
   380
in
haftmann@21091
   381
haftmann@21091
   382
(* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091
   383
   class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091
   384
   Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091
   385
haftmann@21248
   386
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   387
struct
haftmann@21248
   388
  val le_trans = thm "order_trans";
haftmann@21248
   389
  val le_refl = thm "order_refl";
haftmann@21248
   390
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   391
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   392
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   393
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   394
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   395
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   396
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   397
  val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21248
   398
  val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21248
   399
end);
haftmann@21091
   400
haftmann@21091
   401
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   402
struct
haftmann@21248
   403
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   404
  val le_refl = thm "order_refl";
haftmann@21248
   405
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   406
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   407
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   408
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   409
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   410
  val eqI = thm "order_antisym";
haftmann@21248
   411
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   412
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   413
  val less_trans = thm "order_less_trans";
haftmann@21248
   414
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   415
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   416
  val le_trans = thm "order_trans";
haftmann@21248
   417
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   418
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   419
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   420
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   421
  val not_sym = thm "not_sym";
haftmann@21248
   422
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   423
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   424
end);
haftmann@21091
   425
haftmann@21091
   426
end;
haftmann@21091
   427
*}
haftmann@21091
   428
haftmann@21083
   429
setup {*
haftmann@21083
   430
let
haftmann@21083
   431
haftmann@21083
   432
val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083
   433
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083
   434
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083
   435
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083
   436
haftmann@21083
   437
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   438
haftmann@21083
   439
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   440
  let val prems = prems_of_ss ss;
haftmann@21083
   441
      val less = Const("Orderings.less",T);
haftmann@21083
   442
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   443
  in case find_first (prp t) prems of
haftmann@21083
   444
       NONE =>
haftmann@21083
   445
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   446
         in case find_first (prp t) prems of
haftmann@21083
   447
              NONE => NONE
haftmann@21083
   448
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083
   449
         end
haftmann@21083
   450
     | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083
   451
  end
haftmann@21083
   452
  handle THM _ => NONE;
nipkow@15524
   453
haftmann@21083
   454
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   455
  let val prems = prems_of_ss ss;
haftmann@21083
   456
      val le = Const("Orderings.less_eq",T);
haftmann@21083
   457
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   458
  in case find_first (prp t) prems of
haftmann@21083
   459
       NONE =>
haftmann@21083
   460
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   461
         in case find_first (prp t) prems of
haftmann@21083
   462
              NONE => NONE
haftmann@21083
   463
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083
   464
         end
haftmann@21083
   465
     | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083
   466
  end
haftmann@21083
   467
  handle THM _ => NONE;
nipkow@15524
   468
haftmann@21248
   469
fun add_simprocs procs thy =
haftmann@21248
   470
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   471
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   472
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   473
fun add_solver name tac thy =
haftmann@21248
   474
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   475
    (mk_solver name (K tac))); thy);
haftmann@21083
   476
haftmann@21083
   477
in
haftmann@21248
   478
  add_simprocs [
haftmann@21248
   479
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   480
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   481
     ]
haftmann@21248
   482
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   483
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   484
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   485
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   486
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   487
     of 5 March 2004, was observed). *)
haftmann@21083
   488
end
haftmann@21083
   489
*}
nipkow@15524
   490
nipkow@15524
   491
haftmann@21083
   492
subsection {* Bounded quantifiers *}
haftmann@21083
   493
haftmann@21083
   494
syntax
wenzelm@21180
   495
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   496
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   497
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   498
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   499
wenzelm@21180
   500
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   501
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   502
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   503
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   504
haftmann@21083
   505
syntax (xsymbols)
wenzelm@21180
   506
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   507
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   508
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   509
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   510
wenzelm@21180
   511
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   512
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   513
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   514
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   515
haftmann@21083
   516
syntax (HOL)
wenzelm@21180
   517
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   518
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   519
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   520
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   521
haftmann@21083
   522
syntax (HTML output)
wenzelm@21180
   523
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   524
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   525
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   526
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   527
wenzelm@21180
   528
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   529
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   530
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   531
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   532
haftmann@21083
   533
translations
haftmann@21083
   534
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   535
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   536
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   537
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   538
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   539
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   540
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   541
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   542
haftmann@21083
   543
print_translation {*
haftmann@21083
   544
let
wenzelm@21180
   545
  val syntax_name = Sign.const_syntax_name (the_context ());
wenzelm@21180
   546
  val impl = syntax_name "op -->";
wenzelm@21180
   547
  val conj = syntax_name "op &";
wenzelm@21180
   548
  val less = syntax_name "Orderings.less";
wenzelm@21180
   549
  val less_eq = syntax_name "Orderings.less_eq";
wenzelm@21180
   550
wenzelm@21180
   551
  val trans =
wenzelm@21180
   552
   [(("ALL ", impl, less), ("_All_less", "_All_greater")),
wenzelm@21180
   553
    (("ALL ", impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21180
   554
    (("EX ", conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21180
   555
    (("EX ", conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   556
haftmann@21083
   557
  fun mk v v' c n P =
wenzelm@21180
   558
    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
haftmann@21083
   559
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
wenzelm@21180
   560
wenzelm@21180
   561
  fun tr' q = (q,
wenzelm@21180
   562
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   563
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   564
        NONE => raise Match
wenzelm@21180
   565
      | SOME (l, g) =>
wenzelm@21180
   566
          (case (t, u) of
wenzelm@21180
   567
            (Const ("_bound", _) $ Free (v', _), n) => mk v v' l n P
wenzelm@21180
   568
          | (n, Const ("_bound", _) $ Free (v', _)) => mk v v' g n P
wenzelm@21180
   569
          | _ => raise Match))
wenzelm@21180
   570
     | _ => raise Match);
wenzelm@21180
   571
in [tr' "ALL ", tr' "EX "] end
haftmann@21083
   572
*}
haftmann@21083
   573
haftmann@21083
   574
haftmann@21383
   575
subsection {* Transitivity reasoning *}
haftmann@21383
   576
haftmann@21383
   577
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
haftmann@21383
   578
  by (rule subst)
haftmann@21383
   579
haftmann@21383
   580
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
haftmann@21383
   581
  by (rule ssubst)
haftmann@21383
   582
haftmann@21383
   583
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
haftmann@21383
   584
  by (rule subst)
haftmann@21383
   585
haftmann@21383
   586
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
haftmann@21383
   587
  by (rule ssubst)
haftmann@21383
   588
haftmann@21383
   589
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   590
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   591
proof -
haftmann@21383
   592
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   593
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   594
  also assume "f b < c"
haftmann@21383
   595
  finally (order_less_trans) show ?thesis .
haftmann@21383
   596
qed
haftmann@21383
   597
haftmann@21383
   598
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   599
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   600
proof -
haftmann@21383
   601
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   602
  assume "a < f b"
haftmann@21383
   603
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   604
  finally (order_less_trans) show ?thesis .
haftmann@21383
   605
qed
haftmann@21383
   606
haftmann@21383
   607
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   608
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   609
proof -
haftmann@21383
   610
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   611
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   612
  also assume "f b < c"
haftmann@21383
   613
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   614
qed
haftmann@21383
   615
haftmann@21383
   616
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   617
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   618
proof -
haftmann@21383
   619
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   620
  assume "a <= f b"
haftmann@21383
   621
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   622
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   623
qed
haftmann@21383
   624
haftmann@21383
   625
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   626
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   627
proof -
haftmann@21383
   628
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   629
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   630
  also assume "f b <= c"
haftmann@21383
   631
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   632
qed
haftmann@21383
   633
haftmann@21383
   634
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   635
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   636
proof -
haftmann@21383
   637
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   638
  assume "a < f b"
haftmann@21383
   639
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   640
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   641
qed
haftmann@21383
   642
haftmann@21383
   643
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   644
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   645
proof -
haftmann@21383
   646
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   647
  assume "a <= f b"
haftmann@21383
   648
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   649
  finally (order_trans) show ?thesis .
haftmann@21383
   650
qed
haftmann@21383
   651
haftmann@21383
   652
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   653
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   654
proof -
haftmann@21383
   655
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   656
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   657
  also assume "f b <= c"
haftmann@21383
   658
  finally (order_trans) show ?thesis .
haftmann@21383
   659
qed
haftmann@21383
   660
haftmann@21383
   661
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   662
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   663
proof -
haftmann@21383
   664
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   665
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   666
  also assume "f b = c"
haftmann@21383
   667
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   668
qed
haftmann@21383
   669
haftmann@21383
   670
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   671
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   672
proof -
haftmann@21383
   673
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   674
  assume "a = f b"
haftmann@21383
   675
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   676
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   677
qed
haftmann@21383
   678
haftmann@21383
   679
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   680
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   681
proof -
haftmann@21383
   682
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   683
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   684
  also assume "f b = c"
haftmann@21383
   685
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   686
qed
haftmann@21383
   687
haftmann@21383
   688
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   689
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   690
proof -
haftmann@21383
   691
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   692
  assume "a = f b"
haftmann@21383
   693
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   694
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   695
qed
haftmann@21383
   696
haftmann@21383
   697
text {*
haftmann@21383
   698
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   699
*}
haftmann@21383
   700
haftmann@21383
   701
lemmas order_trans_rules [trans] =
haftmann@21383
   702
  order_less_subst2
haftmann@21383
   703
  order_less_subst1
haftmann@21383
   704
  order_le_less_subst2
haftmann@21383
   705
  order_le_less_subst1
haftmann@21383
   706
  order_less_le_subst2
haftmann@21383
   707
  order_less_le_subst1
haftmann@21383
   708
  order_subst2
haftmann@21383
   709
  order_subst1
haftmann@21383
   710
  ord_le_eq_subst
haftmann@21383
   711
  ord_eq_le_subst
haftmann@21383
   712
  ord_less_eq_subst
haftmann@21383
   713
  ord_eq_less_subst
haftmann@21383
   714
  forw_subst
haftmann@21383
   715
  back_subst
haftmann@21383
   716
  rev_mp
haftmann@21383
   717
  mp
haftmann@21383
   718
  order_neq_le_trans
haftmann@21383
   719
  order_le_neq_trans
haftmann@21383
   720
  order_less_trans
haftmann@21383
   721
  order_less_asym'
haftmann@21383
   722
  order_le_less_trans
haftmann@21383
   723
  order_less_le_trans
haftmann@21383
   724
  order_trans
haftmann@21383
   725
  order_antisym
haftmann@21383
   726
  ord_le_eq_trans
haftmann@21383
   727
  ord_eq_le_trans
haftmann@21383
   728
  ord_less_eq_trans
haftmann@21383
   729
  ord_eq_less_trans
haftmann@21383
   730
  trans
haftmann@21383
   731
haftmann@21083
   732
wenzelm@21180
   733
(* FIXME cleanup *)
wenzelm@21180
   734
haftmann@21083
   735
text {* These support proving chains of decreasing inequalities
haftmann@21083
   736
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   737
haftmann@21083
   738
lemma xt1:
haftmann@21083
   739
  "a = b ==> b > c ==> a > c"
haftmann@21083
   740
  "a > b ==> b = c ==> a > c"
haftmann@21083
   741
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   742
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   743
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   744
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   745
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   746
  "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083
   747
  "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083
   748
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   749
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   750
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   751
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   752
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   753
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   754
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   755
by auto
haftmann@21083
   756
haftmann@21083
   757
lemma xt2:
haftmann@21083
   758
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   759
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   760
haftmann@21083
   761
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   762
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   763
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   764
haftmann@21083
   765
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   766
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   767
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   768
haftmann@21083
   769
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   770
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   771
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   772
haftmann@21083
   773
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   774
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   775
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   776
haftmann@21083
   777
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   778
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   779
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   780
haftmann@21083
   781
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   782
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   783
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   784
haftmann@21083
   785
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   786
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   787
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   788
haftmann@21083
   789
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   790
haftmann@21083
   791
(* 
haftmann@21083
   792
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   793
  for the wrong thing in an Isar proof.
haftmann@21083
   794
haftmann@21083
   795
  The extra transitivity rules can be used as follows: 
haftmann@21083
   796
haftmann@21083
   797
lemma "(a::'a::order) > z"
haftmann@21083
   798
proof -
haftmann@21083
   799
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   800
    sorry
haftmann@21083
   801
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   802
    sorry
haftmann@21083
   803
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   804
    sorry
haftmann@21083
   805
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   806
    sorry
haftmann@21083
   807
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   808
    sorry
haftmann@21083
   809
  also (xtrans) have "?rhs > z"
haftmann@21083
   810
    sorry
haftmann@21083
   811
  finally (xtrans) show ?thesis .
haftmann@21083
   812
qed
haftmann@21083
   813
haftmann@21083
   814
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   815
  leave out the "(xtrans)" above.
haftmann@21083
   816
*)
haftmann@21083
   817
haftmann@21383
   818
subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083
   819
haftmann@21216
   820
locale mono =
haftmann@21216
   821
  fixes f
haftmann@21216
   822
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   823
haftmann@21216
   824
lemmas monoI [intro?] = mono.intro
haftmann@21216
   825
  and monoD [dest?] = mono.mono
haftmann@21083
   826
haftmann@21083
   827
constdefs
haftmann@21083
   828
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083
   829
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083
   830
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083
   831
haftmann@21383
   832
lemma LeastI2_order:
haftmann@21383
   833
  "[| P (x::'a::order);
haftmann@21383
   834
      !!y. P y ==> x <= y;
haftmann@21383
   835
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   836
   ==> Q (Least P)"
haftmann@21383
   837
  apply (unfold Least_def)
haftmann@21383
   838
  apply (rule theI2)
haftmann@21383
   839
    apply (blast intro: order_antisym)+
haftmann@21383
   840
  done
haftmann@21383
   841
haftmann@21383
   842
lemma Least_equality:
haftmann@21383
   843
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
haftmann@21383
   844
  apply (simp add: Least_def)
haftmann@21383
   845
  apply (rule the_equality)
haftmann@21383
   846
  apply (auto intro!: order_antisym)
haftmann@21383
   847
  done
haftmann@21383
   848
haftmann@21083
   849
constdefs
haftmann@21083
   850
  min :: "['a::ord, 'a] => 'a"
haftmann@21083
   851
  "min a b == (if a <= b then a else b)"
haftmann@21083
   852
  max :: "['a::ord, 'a] => 'a"
haftmann@21083
   853
  "max a b == (if a <= b then b else a)"
haftmann@21083
   854
haftmann@21383
   855
lemma min_linorder:
haftmann@21383
   856
  "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
haftmann@21383
   857
  by (rule+) (simp add: min_def linorder.min_def)
haftmann@21383
   858
haftmann@21383
   859
lemma max_linorder:
haftmann@21383
   860
  "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
haftmann@21383
   861
  by (rule+) (simp add: max_def linorder.max_def)
haftmann@21383
   862
haftmann@21383
   863
lemmas min_le_iff_disj = linorder.min_le_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   864
lemmas le_max_iff_disj = linorder.le_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   865
lemmas min_less_iff_disj = linorder.min_less_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   866
lemmas less_max_iff_disj = linorder.less_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   867
lemmas min_less_iff_conj [simp] = linorder.min_less_iff_conj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   868
lemmas max_less_iff_conj [simp] = linorder.max_less_iff_conj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   869
lemmas split_min = linorder.split_min [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   870
lemmas split_max = linorder.split_max [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   871
haftmann@21383
   872
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
haftmann@21383
   873
  by (simp add: min_def)
haftmann@21383
   874
haftmann@21383
   875
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
haftmann@21383
   876
  by (simp add: max_def)
haftmann@21383
   877
haftmann@21383
   878
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
haftmann@21383
   879
  apply (simp add: min_def)
haftmann@21383
   880
  apply (blast intro: order_antisym)
haftmann@21383
   881
  done
haftmann@21383
   882
haftmann@21383
   883
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
haftmann@21383
   884
  apply (simp add: max_def)
haftmann@21383
   885
  apply (blast intro: order_antisym)
haftmann@21383
   886
  done
haftmann@21383
   887
haftmann@21383
   888
lemma min_of_mono:
haftmann@21383
   889
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
haftmann@21383
   890
  by (simp add: min_def)
haftmann@21383
   891
haftmann@21383
   892
lemma max_of_mono:
haftmann@21383
   893
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
haftmann@21383
   894
  by (simp add: max_def)
haftmann@21383
   895
nipkow@15524
   896
end