src/HOL/Orderings.thy
author haftmann
Mon Nov 27 13:42:33 2006 +0100 (2006-11-27)
changeset 21546 268b6bed0cc8
parent 21524 7843e2fd14a9
child 21620 3d4bfc7f6363
permissions -rw-r--r--
removed HOL structure
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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><=") 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 (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 <=") 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 (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) where
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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) where
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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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begin
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lemma less_linear: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
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  unfolding less_le using less_le linear by blast 
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lemma le_less_linear: "x \<sqsubseteq> y \<or> y \<sqsubset> x"
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  by (simp add: le_less less_linear)
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lemma le_cases [case_names le ge]:
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  "\<lbrakk> x \<sqsubseteq> y \<Longrightarrow> P; y \<sqsubseteq> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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  using linear by blast
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lemma cases [case_names less equal greater]:
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    "\<lbrakk> x \<sqsubset> y \<Longrightarrow> P; x = y \<Longrightarrow> P; y \<sqsubset> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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  using less_linear by blast
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lemma not_less: "\<not> x \<sqsubset> y \<longleftrightarrow> y \<sqsubseteq> x"
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  apply (simp add: less_le)
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  using linear apply (blast intro: antisym)
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  done
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lemma not_le: "\<not> x \<sqsubseteq> y \<longleftrightarrow> y \<sqsubset> x"
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  apply (simp add: less_le)
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  using linear apply (blast intro: antisym)
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  done
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<sqsubset> y \<or> y \<sqsubset> x"
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  by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "\<lbrakk> x \<noteq> y; x \<sqsubset> y \<Longrightarrow> R; y \<sqsubset> x \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
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  by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
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  by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv2: "x \<sqsubseteq> y \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
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  by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv3: "\<not> y \<sqsubset> x \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
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  by (blast intro: antisym dest: not_less [THEN iffD1])
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text{*Replacing the old Nat.leI*}
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lemma leI: "\<not> x \<sqsubset> y \<Longrightarrow> y \<sqsubseteq> x"
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  unfolding not_less .
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lemma leD: "y \<sqsubseteq> x \<Longrightarrow> \<not> x \<sqsubset> y"
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  unfolding not_less .
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(*FIXME inappropriate name (or delete altogether)*)
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lemma not_leE: "\<not> y \<sqsubseteq> x \<Longrightarrow> x \<sqsubset> y"
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  unfolding not_le .
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(* min/max *)
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definition
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  min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  "min a b = (if a \<sqsubseteq> b then a else b)"
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definition
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  max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  "max a b = (if a \<sqsubseteq> b then b else a)"
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lemma min_le_iff_disj:
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  "min x y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
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  unfolding min_def using linear by (auto intro: trans)
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lemma le_max_iff_disj:
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  "z \<sqsubseteq> max x y \<longleftrightarrow> z \<sqsubseteq> x \<or> z \<sqsubseteq> y"
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  unfolding max_def using linear by (auto intro: trans)
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lemma min_less_iff_disj:
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  "min x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<or> y \<sqsubset> z"
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  unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma less_max_iff_disj:
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  "z \<sqsubset> max x y \<longleftrightarrow> z \<sqsubset> x \<or> z \<sqsubset> y"
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  unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma min_less_iff_conj [simp]:
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  "z \<sqsubset> min x y \<longleftrightarrow> z \<sqsubset> x \<and> z \<sqsubset> y"
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  unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma max_less_iff_conj [simp]:
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  "max x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<and> y \<sqsubset> z"
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  unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma split_min:
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  "P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)"
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  by (simp add: min_def)
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lemma split_max:
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  "P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)"
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  by (simp add: max_def)
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end
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axclass linorder < order
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  linorder_linear: "x <= y | y <= x"
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interpretation linorder:
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  linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"]
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  by unfold_locales (rule linorder_linear)
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subsection {* Name duplicates *}
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lemmas order_eq_refl [where 'b = "?'a::order"] = order.eq_refl
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lemmas order_less_irrefl [where 'b = "?'a::order"] = order.less_irrefl
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lemmas order_le_less [where 'b = "?'a::order"] = order.le_less
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lemmas order_le_imp_less_or_eq [where 'b = "?'a::order"] = order.le_imp_less_or_eq
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lemmas order_less_imp_le [where 'b = "?'a::order"] = order.less_imp_le
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lemmas order_less_not_sym [where 'b = "?'a::order"] = order.less_not_sym
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lemmas order_less_asym [where 'b = "?'a::order"] = order.less_asym
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lemmas order_eq_iff [where 'b = "?'a::order"] = order.eq_iff
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lemmas order_antisym_conv [where 'b = "?'a::order"] = order.antisym_conv
haftmann@21248
   324
lemmas less_imp_neq [where 'b = "?'a::order"] = order.less_imp_neq
haftmann@21248
   325
lemmas order_less_trans [where 'b = "?'a::order"] = order.less_trans
haftmann@21248
   326
lemmas order_le_less_trans [where 'b = "?'a::order"] = order.le_less_trans
haftmann@21248
   327
lemmas order_less_le_trans [where 'b = "?'a::order"] = order.less_le_trans
haftmann@21248
   328
lemmas order_less_imp_not_less [where 'b = "?'a::order"] = order.less_imp_not_less
haftmann@21248
   329
lemmas order_less_imp_triv [where 'b = "?'a::order"] = order.less_imp_triv
haftmann@21248
   330
lemmas order_less_imp_not_eq [where 'b = "?'a::order"] = order.less_imp_not_eq
haftmann@21248
   331
lemmas order_less_imp_not_eq2 [where 'b = "?'a::order"] = order.less_imp_not_eq2
haftmann@21248
   332
lemmas order_neq_le_trans [where 'b = "?'a::order"] = order.neq_le_trans
haftmann@21248
   333
lemmas order_le_neq_trans [where 'b = "?'a::order"] = order.le_neq_trans
haftmann@21248
   334
lemmas order_less_asym' [where 'b = "?'a::order"] = order.less_asym'
haftmann@21412
   335
lemmas linorder_less_linear [where 'b = "?'a::linorder"] = linorder.less_linear
haftmann@21248
   336
lemmas linorder_le_less_linear [where 'b = "?'a::linorder"] = linorder.le_less_linear
haftmann@21248
   337
lemmas linorder_le_cases [where 'b = "?'a::linorder"] = linorder.le_cases
haftmann@21248
   338
lemmas linorder_cases [where 'b = "?'a::linorder"] = linorder.cases
haftmann@21248
   339
lemmas linorder_not_less [where 'b = "?'a::linorder"] = linorder.not_less
haftmann@21248
   340
lemmas linorder_not_le [where 'b = "?'a::linorder"] = linorder.not_le
haftmann@21248
   341
lemmas linorder_neq_iff [where 'b = "?'a::linorder"] = linorder.neq_iff
haftmann@21248
   342
lemmas linorder_neqE [where 'b = "?'a::linorder"] = linorder.neqE
haftmann@21248
   343
lemmas linorder_antisym_conv1 [where 'b = "?'a::linorder"] = linorder.antisym_conv1
haftmann@21248
   344
lemmas linorder_antisym_conv2 [where 'b = "?'a::linorder"] = linorder.antisym_conv2
haftmann@21248
   345
lemmas linorder_antisym_conv3 [where 'b = "?'a::linorder"] = linorder.antisym_conv3
haftmann@21248
   346
lemmas leI [where 'b = "?'a::linorder"] = linorder.leI
haftmann@21248
   347
lemmas leD [where 'b = "?'a::linorder"] = linorder.leD
haftmann@21248
   348
lemmas not_leE [where 'b = "?'a::linorder"] = linorder.not_leE
paulson@16796
   349
haftmann@21083
   350
haftmann@21083
   351
subsection {* Reasoning tools setup *}
haftmann@21083
   352
haftmann@21091
   353
ML {*
haftmann@21091
   354
local
haftmann@21091
   355
haftmann@21091
   356
fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248
   357
  let
haftmann@21248
   358
    fun of_sort t =
haftmann@21248
   359
      let
haftmann@21248
   360
        val T = type_of t
haftmann@21248
   361
      in
haftmann@21091
   362
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248
   363
        T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248
   364
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248
   365
      end;
haftmann@21248
   366
    fun dec (Const ("Not", _) $ t) = (case dec t
haftmann@21248
   367
          of NONE => NONE
haftmann@21248
   368
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21248
   369
      | dec (Const ("op =",  _) $ 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_eq",  _) $ 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 (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21248
   378
          if of_sort t1
haftmann@21248
   379
          then SOME (t1, "<", t2)
haftmann@21248
   380
          else NONE
haftmann@21248
   381
      | dec _ = NONE;
haftmann@21091
   382
  in dec t end;
haftmann@21091
   383
haftmann@21091
   384
in
haftmann@21091
   385
haftmann@21091
   386
(* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091
   387
   class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091
   388
   Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091
   389
haftmann@21248
   390
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   391
struct
haftmann@21248
   392
  val le_trans = thm "order_trans";
haftmann@21248
   393
  val le_refl = thm "order_refl";
haftmann@21248
   394
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   395
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   396
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   397
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   398
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   399
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   400
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   401
  val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21248
   402
  val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21248
   403
end);
haftmann@21091
   404
haftmann@21091
   405
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   406
struct
haftmann@21248
   407
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   408
  val le_refl = thm "order_refl";
haftmann@21248
   409
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   410
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   411
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   412
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   413
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   414
  val eqI = thm "order_antisym";
haftmann@21248
   415
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   416
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   417
  val less_trans = thm "order_less_trans";
haftmann@21248
   418
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   419
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   420
  val le_trans = thm "order_trans";
haftmann@21248
   421
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   422
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   423
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   424
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   425
  val not_sym = thm "not_sym";
haftmann@21248
   426
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   427
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   428
end);
haftmann@21091
   429
haftmann@21091
   430
end;
haftmann@21091
   431
*}
haftmann@21091
   432
haftmann@21083
   433
setup {*
haftmann@21083
   434
let
haftmann@21083
   435
haftmann@21083
   436
val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083
   437
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083
   438
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083
   439
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083
   440
haftmann@21083
   441
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   442
haftmann@21083
   443
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   444
  let val prems = prems_of_ss ss;
haftmann@21083
   445
      val less = Const("Orderings.less",T);
haftmann@21083
   446
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   447
  in case find_first (prp t) prems of
haftmann@21083
   448
       NONE =>
haftmann@21083
   449
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   450
         in case find_first (prp t) prems of
haftmann@21083
   451
              NONE => NONE
haftmann@21083
   452
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083
   453
         end
haftmann@21083
   454
     | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083
   455
  end
haftmann@21083
   456
  handle THM _ => NONE;
nipkow@15524
   457
haftmann@21083
   458
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   459
  let val prems = prems_of_ss ss;
haftmann@21083
   460
      val le = Const("Orderings.less_eq",T);
haftmann@21083
   461
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   462
  in case find_first (prp t) prems of
haftmann@21083
   463
       NONE =>
haftmann@21083
   464
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   465
         in case find_first (prp t) prems of
haftmann@21083
   466
              NONE => NONE
haftmann@21083
   467
            | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083
   468
         end
haftmann@21083
   469
     | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083
   470
  end
haftmann@21083
   471
  handle THM _ => NONE;
nipkow@15524
   472
haftmann@21248
   473
fun add_simprocs procs thy =
haftmann@21248
   474
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   475
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   476
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   477
fun add_solver name tac thy =
haftmann@21248
   478
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   479
    (mk_solver name (K tac))); thy);
haftmann@21083
   480
haftmann@21083
   481
in
haftmann@21248
   482
  add_simprocs [
haftmann@21248
   483
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   484
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   485
     ]
haftmann@21248
   486
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   487
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   488
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   489
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   490
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   491
     of 5 March 2004, was observed). *)
haftmann@21083
   492
end
haftmann@21083
   493
*}
nipkow@15524
   494
nipkow@15524
   495
haftmann@21083
   496
subsection {* Bounded quantifiers *}
haftmann@21083
   497
haftmann@21083
   498
syntax
wenzelm@21180
   499
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   500
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   501
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   502
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   503
wenzelm@21180
   504
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   505
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   506
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   507
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   508
haftmann@21083
   509
syntax (xsymbols)
wenzelm@21180
   510
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   511
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   512
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   513
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   514
wenzelm@21180
   515
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   516
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   517
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   518
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   519
haftmann@21083
   520
syntax (HOL)
wenzelm@21180
   521
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   522
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   523
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   524
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   525
haftmann@21083
   526
syntax (HTML output)
wenzelm@21180
   527
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   528
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   529
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   530
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   531
wenzelm@21180
   532
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   533
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   534
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   535
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   536
haftmann@21083
   537
translations
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
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   543
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   544
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   545
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   546
haftmann@21083
   547
print_translation {*
haftmann@21083
   548
let
wenzelm@21180
   549
  val syntax_name = Sign.const_syntax_name (the_context ());
wenzelm@21524
   550
  val binder_name = Syntax.binder_name o syntax_name;
wenzelm@21524
   551
  val All_binder = binder_name "All";
wenzelm@21524
   552
  val Ex_binder = binder_name "Ex";
wenzelm@21180
   553
  val impl = syntax_name "op -->";
wenzelm@21180
   554
  val conj = syntax_name "op &";
wenzelm@21180
   555
  val less = syntax_name "Orderings.less";
wenzelm@21180
   556
  val less_eq = syntax_name "Orderings.less_eq";
wenzelm@21180
   557
wenzelm@21180
   558
  val trans =
wenzelm@21524
   559
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   560
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   561
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   562
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   563
haftmann@21083
   564
  fun mk v v' c n P =
wenzelm@21180
   565
    if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
haftmann@21083
   566
    then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
wenzelm@21180
   567
wenzelm@21180
   568
  fun tr' q = (q,
wenzelm@21180
   569
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   570
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   571
        NONE => raise Match
wenzelm@21180
   572
      | SOME (l, g) =>
wenzelm@21180
   573
          (case (t, u) of
wenzelm@21180
   574
            (Const ("_bound", _) $ Free (v', _), n) => mk v v' l n P
wenzelm@21180
   575
          | (n, Const ("_bound", _) $ Free (v', _)) => mk v v' g n P
wenzelm@21180
   576
          | _ => raise Match))
wenzelm@21180
   577
     | _ => raise Match);
wenzelm@21524
   578
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   579
*}
haftmann@21083
   580
haftmann@21083
   581
haftmann@21383
   582
subsection {* Transitivity reasoning *}
haftmann@21383
   583
haftmann@21383
   584
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
haftmann@21383
   585
  by (rule subst)
haftmann@21383
   586
haftmann@21383
   587
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
haftmann@21383
   588
  by (rule ssubst)
haftmann@21383
   589
haftmann@21383
   590
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
haftmann@21383
   591
  by (rule subst)
haftmann@21383
   592
haftmann@21383
   593
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
haftmann@21383
   594
  by (rule ssubst)
haftmann@21383
   595
haftmann@21383
   596
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   597
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   598
proof -
haftmann@21383
   599
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   600
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   601
  also assume "f b < c"
haftmann@21383
   602
  finally (order_less_trans) show ?thesis .
haftmann@21383
   603
qed
haftmann@21383
   604
haftmann@21383
   605
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   606
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   607
proof -
haftmann@21383
   608
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   609
  assume "a < f b"
haftmann@21383
   610
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   611
  finally (order_less_trans) show ?thesis .
haftmann@21383
   612
qed
haftmann@21383
   613
haftmann@21383
   614
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   615
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   616
proof -
haftmann@21383
   617
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   618
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   619
  also assume "f b < c"
haftmann@21383
   620
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   621
qed
haftmann@21383
   622
haftmann@21383
   623
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   624
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   625
proof -
haftmann@21383
   626
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   627
  assume "a <= f b"
haftmann@21383
   628
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   629
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   630
qed
haftmann@21383
   631
haftmann@21383
   632
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   633
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   634
proof -
haftmann@21383
   635
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   636
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   637
  also assume "f b <= c"
haftmann@21383
   638
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   639
qed
haftmann@21383
   640
haftmann@21383
   641
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   642
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   643
proof -
haftmann@21383
   644
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   645
  assume "a < f b"
haftmann@21383
   646
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   647
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   648
qed
haftmann@21383
   649
haftmann@21383
   650
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   651
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   652
proof -
haftmann@21383
   653
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   654
  assume "a <= f b"
haftmann@21383
   655
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   656
  finally (order_trans) show ?thesis .
haftmann@21383
   657
qed
haftmann@21383
   658
haftmann@21383
   659
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   660
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   661
proof -
haftmann@21383
   662
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   663
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   664
  also assume "f b <= c"
haftmann@21383
   665
  finally (order_trans) show ?thesis .
haftmann@21383
   666
qed
haftmann@21383
   667
haftmann@21383
   668
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   669
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   670
proof -
haftmann@21383
   671
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   672
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   673
  also assume "f b = c"
haftmann@21383
   674
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   675
qed
haftmann@21383
   676
haftmann@21383
   677
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   678
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   679
proof -
haftmann@21383
   680
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   681
  assume "a = f b"
haftmann@21383
   682
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   683
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   684
qed
haftmann@21383
   685
haftmann@21383
   686
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   687
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   688
proof -
haftmann@21383
   689
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   690
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   691
  also assume "f b = c"
haftmann@21383
   692
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   693
qed
haftmann@21383
   694
haftmann@21383
   695
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   696
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   697
proof -
haftmann@21383
   698
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   699
  assume "a = f b"
haftmann@21383
   700
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   701
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   702
qed
haftmann@21383
   703
haftmann@21383
   704
text {*
haftmann@21383
   705
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   706
*}
haftmann@21383
   707
haftmann@21383
   708
lemmas order_trans_rules [trans] =
haftmann@21383
   709
  order_less_subst2
haftmann@21383
   710
  order_less_subst1
haftmann@21383
   711
  order_le_less_subst2
haftmann@21383
   712
  order_le_less_subst1
haftmann@21383
   713
  order_less_le_subst2
haftmann@21383
   714
  order_less_le_subst1
haftmann@21383
   715
  order_subst2
haftmann@21383
   716
  order_subst1
haftmann@21383
   717
  ord_le_eq_subst
haftmann@21383
   718
  ord_eq_le_subst
haftmann@21383
   719
  ord_less_eq_subst
haftmann@21383
   720
  ord_eq_less_subst
haftmann@21383
   721
  forw_subst
haftmann@21383
   722
  back_subst
haftmann@21383
   723
  rev_mp
haftmann@21383
   724
  mp
haftmann@21383
   725
  order_neq_le_trans
haftmann@21383
   726
  order_le_neq_trans
haftmann@21383
   727
  order_less_trans
haftmann@21383
   728
  order_less_asym'
haftmann@21383
   729
  order_le_less_trans
haftmann@21383
   730
  order_less_le_trans
haftmann@21383
   731
  order_trans
haftmann@21383
   732
  order_antisym
haftmann@21383
   733
  ord_le_eq_trans
haftmann@21383
   734
  ord_eq_le_trans
haftmann@21383
   735
  ord_less_eq_trans
haftmann@21383
   736
  ord_eq_less_trans
haftmann@21383
   737
  trans
haftmann@21383
   738
haftmann@21083
   739
wenzelm@21180
   740
(* FIXME cleanup *)
wenzelm@21180
   741
haftmann@21083
   742
text {* These support proving chains of decreasing inequalities
haftmann@21083
   743
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   744
haftmann@21083
   745
lemma xt1:
haftmann@21083
   746
  "a = b ==> b > c ==> a > c"
haftmann@21083
   747
  "a > b ==> b = c ==> a > c"
haftmann@21083
   748
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   749
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   750
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   751
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   752
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   753
  "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083
   754
  "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083
   755
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   756
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   757
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   758
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   759
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   760
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   761
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   762
by auto
haftmann@21083
   763
haftmann@21083
   764
lemma xt2:
haftmann@21083
   765
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   766
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   767
haftmann@21083
   768
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   769
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   770
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   771
haftmann@21083
   772
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   773
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   774
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   775
haftmann@21083
   776
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   777
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   778
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   779
haftmann@21083
   780
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   781
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   782
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   783
haftmann@21083
   784
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   785
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   786
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   787
haftmann@21083
   788
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   789
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   790
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   791
haftmann@21083
   792
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   793
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   794
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   795
haftmann@21083
   796
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   797
haftmann@21083
   798
(* 
haftmann@21083
   799
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   800
  for the wrong thing in an Isar proof.
haftmann@21083
   801
haftmann@21083
   802
  The extra transitivity rules can be used as follows: 
haftmann@21083
   803
haftmann@21083
   804
lemma "(a::'a::order) > z"
haftmann@21083
   805
proof -
haftmann@21083
   806
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   807
    sorry
haftmann@21083
   808
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   809
    sorry
haftmann@21083
   810
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   811
    sorry
haftmann@21083
   812
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   813
    sorry
haftmann@21083
   814
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   815
    sorry
haftmann@21083
   816
  also (xtrans) have "?rhs > z"
haftmann@21083
   817
    sorry
haftmann@21083
   818
  finally (xtrans) show ?thesis .
haftmann@21083
   819
qed
haftmann@21083
   820
haftmann@21083
   821
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   822
  leave out the "(xtrans)" above.
haftmann@21083
   823
*)
haftmann@21083
   824
haftmann@21546
   825
subsection {* Order on bool *}
haftmann@21546
   826
haftmann@21546
   827
instance bool :: linorder 
haftmann@21546
   828
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   829
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@21546
   830
  by default (auto simp add: le_bool_def less_bool_def)
haftmann@21546
   831
haftmann@21546
   832
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@21546
   833
  by (simp add: le_bool_def)
haftmann@21546
   834
haftmann@21546
   835
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@21546
   836
  by (simp add: le_bool_def)
haftmann@21546
   837
haftmann@21546
   838
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@21546
   839
  by (simp add: le_bool_def)
haftmann@21546
   840
haftmann@21546
   841
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@21546
   842
  by (simp add: le_bool_def)
haftmann@21546
   843
haftmann@21383
   844
subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083
   845
haftmann@21216
   846
locale mono =
haftmann@21216
   847
  fixes f
haftmann@21216
   848
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   849
haftmann@21216
   850
lemmas monoI [intro?] = mono.intro
haftmann@21216
   851
  and monoD [dest?] = mono.mono
haftmann@21083
   852
haftmann@21083
   853
constdefs
haftmann@21083
   854
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083
   855
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083
   856
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083
   857
haftmann@21383
   858
lemma LeastI2_order:
haftmann@21383
   859
  "[| P (x::'a::order);
haftmann@21383
   860
      !!y. P y ==> x <= y;
haftmann@21383
   861
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   862
   ==> Q (Least P)"
haftmann@21383
   863
  apply (unfold Least_def)
haftmann@21383
   864
  apply (rule theI2)
haftmann@21383
   865
    apply (blast intro: order_antisym)+
haftmann@21383
   866
  done
haftmann@21383
   867
haftmann@21383
   868
lemma Least_equality:
haftmann@21383
   869
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
haftmann@21383
   870
  apply (simp add: Least_def)
haftmann@21383
   871
  apply (rule the_equality)
haftmann@21383
   872
  apply (auto intro!: order_antisym)
haftmann@21383
   873
  done
haftmann@21383
   874
haftmann@21083
   875
constdefs
haftmann@21083
   876
  min :: "['a::ord, 'a] => 'a"
haftmann@21083
   877
  "min a b == (if a <= b then a else b)"
haftmann@21083
   878
  max :: "['a::ord, 'a] => 'a"
haftmann@21083
   879
  "max a b == (if a <= b then b else a)"
haftmann@21083
   880
haftmann@21383
   881
lemma min_linorder:
haftmann@21383
   882
  "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
haftmann@21383
   883
  by (rule+) (simp add: min_def linorder.min_def)
haftmann@21383
   884
haftmann@21383
   885
lemma max_linorder:
haftmann@21383
   886
  "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
haftmann@21383
   887
  by (rule+) (simp add: max_def linorder.max_def)
haftmann@21383
   888
haftmann@21383
   889
lemmas min_le_iff_disj = linorder.min_le_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   890
lemmas le_max_iff_disj = linorder.le_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   891
lemmas min_less_iff_disj = linorder.min_less_iff_disj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   892
lemmas less_max_iff_disj = linorder.less_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   893
lemmas min_less_iff_conj [simp] = linorder.min_less_iff_conj [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   894
lemmas max_less_iff_conj [simp] = linorder.max_less_iff_conj [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   895
lemmas split_min = linorder.split_min [where 'b = "?'a::linorder", simplified min_linorder]
haftmann@21383
   896
lemmas split_max = linorder.split_max [where 'b = "?'a::linorder", simplified max_linorder]
haftmann@21383
   897
haftmann@21383
   898
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
haftmann@21383
   899
  by (simp add: min_def)
haftmann@21383
   900
haftmann@21383
   901
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
haftmann@21383
   902
  by (simp add: max_def)
haftmann@21383
   903
haftmann@21383
   904
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
haftmann@21383
   905
  apply (simp add: min_def)
haftmann@21383
   906
  apply (blast intro: order_antisym)
haftmann@21383
   907
  done
haftmann@21383
   908
haftmann@21383
   909
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
haftmann@21383
   910
  apply (simp add: max_def)
haftmann@21383
   911
  apply (blast intro: order_antisym)
haftmann@21383
   912
  done
haftmann@21383
   913
haftmann@21383
   914
lemma min_of_mono:
haftmann@21383
   915
    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
haftmann@21383
   916
  by (simp add: min_def)
haftmann@21383
   917
haftmann@21383
   918
lemma max_of_mono:
haftmann@21383
   919
    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
haftmann@21383
   920
  by (simp add: max_def)
haftmann@21383
   921
nipkow@15524
   922
end