src/HOL/Orderings.thy
author wenzelm
Tue Oct 09 00:20:13 2007 +0200 (2007-10-09)
changeset 24920 2a45e400fdad
parent 24867 e5b55d7be9bb
child 25062 af5ef0d4d655
permissions -rw-r--r--
generic Syntax.pretty/string_of operations;
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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 Set Fun
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uses
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  "~~/src/Provers/order.ML"
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begin
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subsection {* Partial orders *}
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class order = ord +
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  assumes less_le: "x \<^loc>< y \<longleftrightarrow> x \<^loc>\<le> y \<and> x \<noteq> y"
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  and order_refl [iff]: "x \<^loc>\<le> x"
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  and order_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> z"
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  assumes antisym: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y"
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begin
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notation (input)
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  less_eq (infix "\<sqsubseteq>" 50)
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and
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  less    (infix "\<sqsubset>" 50)
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text {* Reflexivity. *}
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lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
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    -- {* This form is useful with the classical reasoner. *}
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by (erule ssubst) (rule order_refl)
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lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
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by (simp add: less_le)
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lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< 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 \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
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unfolding less_le by blast
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lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
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unfolding less_le by blast
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lemma less_imp_neq: "x \<^loc>< 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 \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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by auto
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lemma less_imp_not_eq2: "x \<^loc>< 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: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
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by (simp add: less_le)
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lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
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by (simp add: less_le)
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text {* Asymmetry. *}
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lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
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by (simp add: less_le antisym)
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lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< 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 \<^loc>\<le> y \<and> y \<^loc>\<le> x"
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by (blast intro: antisym)
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lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym)
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lemma less_imp_neq: "x \<^loc>< 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: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_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 \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< 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': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
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by (rule less_asym)
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text {* Reverse order *}
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lemma order_reverse:
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  "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
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by unfold_locales
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   (simp add: less_le, auto intro: antisym order_trans)
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end
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subsection {* Linear (total) orders *}
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class linorder = order +
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  assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
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begin
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lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
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unfolding less_le using less_le linear by blast
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lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< 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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  "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
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using linear by blast
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lemma linorder_cases [case_names less equal greater]:
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  "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
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using less_linear by blast
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lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> 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_less_iff_gr_or_eq:
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 "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"
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apply(simp add:not_less le_less)
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apply blast
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done
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lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< 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 \<^loc>< y \<or> y \<^loc>< x"
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by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> 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 \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< 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 \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< 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 \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
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unfolding not_less .
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lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< 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 \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
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unfolding not_le .
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text {* Reverse order *}
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lemma linorder_reverse:
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  "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
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by unfold_locales
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  (simp add: less_le, auto intro: antisym order_trans simp add: linear)
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text {* min/max *}
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text {* for historic reasons, definitions are done in context ord *}
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definition (in ord)
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  min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  [code unfold, code inline del]: "min a b = (if a \<^loc>\<le> b then a else b)"
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definition (in ord)
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  max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  [code unfold, code inline del]: "max a b = (if a \<^loc>\<le> b then b else a)"
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lemma min_le_iff_disj:
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  "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
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unfolding min_def using linear by (auto intro: order_trans)
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lemma le_max_iff_disj:
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  "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
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unfolding max_def using linear by (auto intro: order_trans)
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lemma min_less_iff_disj:
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  "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< 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 \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< 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 \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< 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 \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< 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 [noatp]:
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  "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
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by (simp add: min_def)
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lemma split_max [noatp]:
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  "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
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by (simp add: max_def)
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end
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subsection {* Reasoning tools setup *}
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ML {*
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signature ORDERS =
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sig
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  val print_structures: Proof.context -> unit
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  val setup: theory -> theory
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  val order_tac: thm list -> Proof.context -> int -> tactic
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end;
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structure Orders: ORDERS =
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struct
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(** Theory and context data **)
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fun struct_eq ((s1: string, ts1), (s2, ts2)) =
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  (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
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structure Data = GenericDataFun
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(
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  type T = ((string * term list) * Order_Tac.less_arith) list;
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    (* Order structures:
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       identifier of the structure, list of operations and record of theorems
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       needed to set up the transitivity reasoner,
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       identifier and operations identify the structure uniquely. *)
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  val empty = [];
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  val extend = I;
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  fun merge _ = AList.join struct_eq (K fst);
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);
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fun print_structures ctxt =
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  let
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    val structs = Data.get (Context.Proof ctxt);
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    fun pretty_term t = Pretty.block
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      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
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        Pretty.str "::", Pretty.brk 1,
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        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
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    fun pretty_struct ((s, ts), _) = Pretty.block
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      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
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       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
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  in
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    Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
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  end;
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(** Method **)
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fun struct_tac ((s, [eq, le, less]), thms) prems =
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  let
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    fun decomp thy (Trueprop $ t) =
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      let
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        fun excluded t =
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          (* exclude numeric types: linear arithmetic subsumes transitivity *)
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          let val T = type_of t
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          in
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	    T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
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          end;
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	fun rel (bin_op $ t1 $ t2) =
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              if excluded t1 then NONE
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              else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
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              else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
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              else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
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              else NONE
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	  | rel _ = NONE;
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	fun dec (Const (@{const_name Not}, _) $ t) = (case rel t
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	      of NONE => NONE
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	       | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
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          | dec x = rel x;
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      in dec t end;
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  in
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    case s of
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      "order" => Order_Tac.partial_tac decomp thms prems
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    | "linorder" => Order_Tac.linear_tac decomp thms prems
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    | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
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  end
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fun order_tac prems ctxt =
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  FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));
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(** Attribute **)
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fun add_struct_thm s tag =
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  Thm.declaration_attribute
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    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
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fun del_struct s =
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  Thm.declaration_attribute
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    (fn _ => Data.map (AList.delete struct_eq s));
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   332
ballarin@24641
   333
val attribute = Attrib.syntax
ballarin@24641
   334
     (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) ||
ballarin@24641
   335
          Args.del >> K NONE) --| Args.colon (* FIXME ||
ballarin@24641
   336
        Scan.succeed true *) ) -- Scan.lift Args.name --
ballarin@24641
   337
      Scan.repeat Args.term
ballarin@24641
   338
      >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
ballarin@24641
   339
           | ((NONE, n), ts) => del_struct (n, ts)));
ballarin@24641
   340
ballarin@24641
   341
ballarin@24641
   342
(** Diagnostic command **)
ballarin@24641
   343
ballarin@24641
   344
val print = Toplevel.unknown_context o
ballarin@24641
   345
  Toplevel.keep (Toplevel.node_case
ballarin@24641
   346
    (Context.cases (print_structures o ProofContext.init) print_structures)
ballarin@24641
   347
    (print_structures o Proof.context_of));
ballarin@24641
   348
wenzelm@24867
   349
val _ =
ballarin@24641
   350
  OuterSyntax.improper_command "print_orders"
ballarin@24641
   351
    "print order structures available to transitivity reasoner" OuterKeyword.diag
ballarin@24641
   352
    (Scan.succeed (Toplevel.no_timing o print));
ballarin@24641
   353
ballarin@24641
   354
ballarin@24641
   355
(** Setup **)
ballarin@24641
   356
wenzelm@24867
   357
val setup =
wenzelm@24867
   358
  Method.add_methods
wenzelm@24867
   359
    [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #>
wenzelm@24867
   360
  Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")];
haftmann@21091
   361
haftmann@21091
   362
end;
ballarin@24641
   363
haftmann@21091
   364
*}
haftmann@21091
   365
ballarin@24641
   366
setup Orders.setup
ballarin@24641
   367
ballarin@24641
   368
ballarin@24641
   369
text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
ballarin@24641
   370
ballarin@24641
   371
(* The type constraint on @{term op =} below is necessary since the operation
ballarin@24641
   372
   is not a parameter of the locale. *)
ballarin@24641
   373
lemmas (in order)
ballarin@24641
   374
  [order add less_reflE: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   375
  less_irrefl [THEN notE]
ballarin@24641
   376
lemmas (in order)
ballarin@24641
   377
  [order add le_refl: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   378
  order_refl
ballarin@24641
   379
lemmas (in order)
ballarin@24641
   380
  [order add less_imp_le: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   381
  less_imp_le
ballarin@24641
   382
lemmas (in order)
ballarin@24641
   383
  [order add eqI: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   384
  antisym
ballarin@24641
   385
lemmas (in order)
ballarin@24641
   386
  [order add eqD1: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   387
  eq_refl
ballarin@24641
   388
lemmas (in order)
ballarin@24641
   389
  [order add eqD2: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   390
  sym [THEN eq_refl]
ballarin@24641
   391
lemmas (in order)
ballarin@24641
   392
  [order add less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   393
  less_trans
ballarin@24641
   394
lemmas (in order)
ballarin@24641
   395
  [order add less_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   396
  less_le_trans
ballarin@24641
   397
lemmas (in order)
ballarin@24641
   398
  [order add le_less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   399
  le_less_trans
ballarin@24641
   400
lemmas (in order)
ballarin@24641
   401
  [order add le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   402
  order_trans
ballarin@24641
   403
lemmas (in order)
ballarin@24641
   404
  [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   405
  le_neq_trans
ballarin@24641
   406
lemmas (in order)
ballarin@24641
   407
  [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   408
  neq_le_trans
ballarin@24641
   409
lemmas (in order)
ballarin@24641
   410
  [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   411
  less_imp_neq
ballarin@24641
   412
lemmas (in order)
ballarin@24641
   413
  [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   414
   eq_neq_eq_imp_neq
ballarin@24641
   415
lemmas (in order)
ballarin@24641
   416
  [order add not_sym: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   417
  not_sym
ballarin@24641
   418
ballarin@24641
   419
lemmas (in linorder) [order del: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = _
ballarin@24641
   420
ballarin@24641
   421
lemmas (in linorder)
ballarin@24641
   422
  [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   423
  less_irrefl [THEN notE]
ballarin@24641
   424
lemmas (in linorder)
ballarin@24641
   425
  [order add le_refl: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   426
  order_refl
ballarin@24641
   427
lemmas (in linorder)
ballarin@24641
   428
  [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   429
  less_imp_le
ballarin@24641
   430
lemmas (in linorder)
ballarin@24641
   431
  [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   432
  not_less [THEN iffD2]
ballarin@24641
   433
lemmas (in linorder)
ballarin@24641
   434
  [order add not_leI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   435
  not_le [THEN iffD2]
ballarin@24641
   436
lemmas (in linorder)
ballarin@24641
   437
  [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   438
  not_less [THEN iffD1]
ballarin@24641
   439
lemmas (in linorder)
ballarin@24641
   440
  [order add not_leD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   441
  not_le [THEN iffD1]
ballarin@24641
   442
lemmas (in linorder)
ballarin@24641
   443
  [order add eqI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   444
  antisym
ballarin@24641
   445
lemmas (in linorder)
ballarin@24641
   446
  [order add eqD1: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   447
  eq_refl
ballarin@24641
   448
lemmas (in linorder)
ballarin@24641
   449
  [order add eqD2: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   450
  sym [THEN eq_refl]
ballarin@24641
   451
lemmas (in linorder)
ballarin@24641
   452
  [order add less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   453
  less_trans
ballarin@24641
   454
lemmas (in linorder)
ballarin@24641
   455
  [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   456
  less_le_trans
ballarin@24641
   457
lemmas (in linorder)
ballarin@24641
   458
  [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   459
  le_less_trans
ballarin@24641
   460
lemmas (in linorder)
ballarin@24641
   461
  [order add le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   462
  order_trans
ballarin@24641
   463
lemmas (in linorder)
ballarin@24641
   464
  [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   465
  le_neq_trans
ballarin@24641
   466
lemmas (in linorder)
ballarin@24641
   467
  [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   468
  neq_le_trans
ballarin@24641
   469
lemmas (in linorder)
ballarin@24641
   470
  [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   471
  less_imp_neq
ballarin@24641
   472
lemmas (in linorder)
ballarin@24641
   473
  [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   474
  eq_neq_eq_imp_neq
ballarin@24641
   475
lemmas (in linorder)
ballarin@24641
   476
  [order add not_sym: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
ballarin@24641
   477
  not_sym
ballarin@24641
   478
ballarin@24641
   479
haftmann@21083
   480
setup {*
haftmann@21083
   481
let
haftmann@21083
   482
haftmann@21083
   483
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   484
haftmann@21083
   485
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   486
  let val prems = prems_of_ss ss;
haftmann@22916
   487
      val less = Const (@{const_name less}, T);
haftmann@21083
   488
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   489
  in case find_first (prp t) prems of
haftmann@21083
   490
       NONE =>
haftmann@21083
   491
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   492
         in case find_first (prp t) prems of
haftmann@21083
   493
              NONE => NONE
haftmann@24422
   494
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
haftmann@21083
   495
         end
haftmann@24422
   496
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
haftmann@21083
   497
  end
haftmann@21083
   498
  handle THM _ => NONE;
nipkow@15524
   499
haftmann@21083
   500
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   501
  let val prems = prems_of_ss ss;
haftmann@22916
   502
      val le = Const (@{const_name less_eq}, T);
haftmann@21083
   503
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   504
  in case find_first (prp t) prems of
haftmann@21083
   505
       NONE =>
haftmann@21083
   506
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   507
         in case find_first (prp t) prems of
haftmann@21083
   508
              NONE => NONE
haftmann@24422
   509
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
haftmann@21083
   510
         end
haftmann@24422
   511
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
haftmann@21083
   512
  end
haftmann@21083
   513
  handle THM _ => NONE;
nipkow@15524
   514
haftmann@21248
   515
fun add_simprocs procs thy =
haftmann@21248
   516
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   517
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   518
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   519
fun add_solver name tac thy =
haftmann@21248
   520
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
ballarin@24704
   521
    (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy);
haftmann@21083
   522
haftmann@21083
   523
in
haftmann@21248
   524
  add_simprocs [
haftmann@21248
   525
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   526
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   527
     ]
ballarin@24641
   528
  #> add_solver "Transitivity" Orders.order_tac
haftmann@21248
   529
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   530
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   531
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   532
     of 5 March 2004, was observed). *)
haftmann@21083
   533
end
haftmann@21083
   534
*}
nipkow@15524
   535
nipkow@15524
   536
haftmann@24422
   537
subsection {* Dense orders *}
haftmann@24422
   538
haftmann@24422
   539
class dense_linear_order = linorder + 
haftmann@24422
   540
  assumes gt_ex: "\<exists>y. x \<sqsubset> y" 
haftmann@24422
   541
  and lt_ex: "\<exists>y. y \<sqsubset> x"
haftmann@24422
   542
  and dense: "x \<sqsubset> y \<Longrightarrow> (\<exists>z. x \<sqsubset> z \<and> z \<sqsubset> y)"
haftmann@24422
   543
  (*see further theory Dense_Linear_Order*)
haftmann@24422
   544
ballarin@24641
   545
haftmann@24422
   546
lemma interval_empty_iff:
haftmann@24422
   547
  fixes x y z :: "'a\<Colon>dense_linear_order"
haftmann@24422
   548
  shows "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
haftmann@24422
   549
  by (auto dest: dense)
haftmann@24422
   550
haftmann@24422
   551
subsection {* Name duplicates *}
haftmann@24422
   552
haftmann@24422
   553
lemmas order_less_le = less_le
haftmann@24422
   554
lemmas order_eq_refl = order_class.eq_refl
haftmann@24422
   555
lemmas order_less_irrefl = order_class.less_irrefl
haftmann@24422
   556
lemmas order_le_less = order_class.le_less
haftmann@24422
   557
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
haftmann@24422
   558
lemmas order_less_imp_le = order_class.less_imp_le
haftmann@24422
   559
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
haftmann@24422
   560
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
haftmann@24422
   561
lemmas order_neq_le_trans = order_class.neq_le_trans
haftmann@24422
   562
lemmas order_le_neq_trans = order_class.le_neq_trans
haftmann@24422
   563
haftmann@24422
   564
lemmas order_antisym = antisym
haftmann@24422
   565
lemmas order_less_not_sym = order_class.less_not_sym
haftmann@24422
   566
lemmas order_less_asym = order_class.less_asym
haftmann@24422
   567
lemmas order_eq_iff = order_class.eq_iff
haftmann@24422
   568
lemmas order_antisym_conv = order_class.antisym_conv
haftmann@24422
   569
lemmas order_less_trans = order_class.less_trans
haftmann@24422
   570
lemmas order_le_less_trans = order_class.le_less_trans
haftmann@24422
   571
lemmas order_less_le_trans = order_class.less_le_trans
haftmann@24422
   572
lemmas order_less_imp_not_less = order_class.less_imp_not_less
haftmann@24422
   573
lemmas order_less_imp_triv = order_class.less_imp_triv
haftmann@24422
   574
lemmas order_less_asym' = order_class.less_asym'
haftmann@24422
   575
haftmann@24422
   576
lemmas linorder_linear = linear
haftmann@24422
   577
lemmas linorder_less_linear = linorder_class.less_linear
haftmann@24422
   578
lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@24422
   579
lemmas linorder_le_cases = linorder_class.le_cases
haftmann@24422
   580
lemmas linorder_not_less = linorder_class.not_less
haftmann@24422
   581
lemmas linorder_not_le = linorder_class.not_le
haftmann@24422
   582
lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@24422
   583
lemmas linorder_neqE = linorder_class.neqE
haftmann@24422
   584
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@24422
   585
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@24422
   586
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
haftmann@24422
   587
haftmann@24422
   588
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj
haftmann@24422
   589
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj
haftmann@24422
   590
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj
haftmann@24422
   591
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj
haftmann@24422
   592
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj
haftmann@24422
   593
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj
haftmann@24422
   594
lemmas split_min = linorder_class.split_min
haftmann@24422
   595
lemmas split_max = linorder_class.split_max
haftmann@24422
   596
haftmann@24422
   597
haftmann@21083
   598
subsection {* Bounded quantifiers *}
haftmann@21083
   599
haftmann@21083
   600
syntax
wenzelm@21180
   601
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   602
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   603
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   604
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   605
wenzelm@21180
   606
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   607
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   608
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   609
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   610
haftmann@21083
   611
syntax (xsymbols)
wenzelm@21180
   612
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   613
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   614
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   615
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   616
wenzelm@21180
   617
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   618
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   619
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   620
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   621
haftmann@21083
   622
syntax (HOL)
wenzelm@21180
   623
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   624
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   625
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   626
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   627
haftmann@21083
   628
syntax (HTML output)
wenzelm@21180
   629
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   630
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   631
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   632
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   633
wenzelm@21180
   634
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   635
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   636
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   637
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   638
haftmann@21083
   639
translations
haftmann@21083
   640
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   641
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   642
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   643
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   644
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   645
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   646
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   647
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   648
haftmann@21083
   649
print_translation {*
haftmann@21083
   650
let
haftmann@22916
   651
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann@22916
   652
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
wenzelm@22377
   653
  val impl = @{const_syntax "op -->"};
wenzelm@22377
   654
  val conj = @{const_syntax "op &"};
haftmann@22916
   655
  val less = @{const_syntax less};
haftmann@22916
   656
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   657
wenzelm@21180
   658
  val trans =
wenzelm@21524
   659
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   660
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   661
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   662
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   663
krauss@22344
   664
  fun matches_bound v t = 
krauss@22344
   665
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344
   666
              | _ => false
krauss@22344
   667
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344
   668
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180
   669
wenzelm@21180
   670
  fun tr' q = (q,
wenzelm@21180
   671
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   672
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   673
        NONE => raise Match
wenzelm@21180
   674
      | SOME (l, g) =>
krauss@22344
   675
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344
   676
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344
   677
          else raise Match)
wenzelm@21180
   678
     | _ => raise Match);
wenzelm@21524
   679
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   680
*}
haftmann@21083
   681
haftmann@21083
   682
haftmann@21383
   683
subsection {* Transitivity reasoning *}
haftmann@21383
   684
haftmann@21383
   685
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
nipkow@23212
   686
by (rule subst)
haftmann@21383
   687
haftmann@21383
   688
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
nipkow@23212
   689
by (rule ssubst)
haftmann@21383
   690
haftmann@21383
   691
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
nipkow@23212
   692
by (rule subst)
haftmann@21383
   693
haftmann@21383
   694
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
nipkow@23212
   695
by (rule ssubst)
haftmann@21383
   696
haftmann@21383
   697
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   698
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   699
proof -
haftmann@21383
   700
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   701
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   702
  also assume "f b < c"
haftmann@21383
   703
  finally (order_less_trans) show ?thesis .
haftmann@21383
   704
qed
haftmann@21383
   705
haftmann@21383
   706
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   707
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   708
proof -
haftmann@21383
   709
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   710
  assume "a < f b"
haftmann@21383
   711
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   712
  finally (order_less_trans) show ?thesis .
haftmann@21383
   713
qed
haftmann@21383
   714
haftmann@21383
   715
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   716
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   717
proof -
haftmann@21383
   718
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   719
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   720
  also assume "f b < c"
haftmann@21383
   721
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   722
qed
haftmann@21383
   723
haftmann@21383
   724
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   725
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   726
proof -
haftmann@21383
   727
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   728
  assume "a <= f b"
haftmann@21383
   729
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   730
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   731
qed
haftmann@21383
   732
haftmann@21383
   733
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   734
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   735
proof -
haftmann@21383
   736
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   737
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   738
  also assume "f b <= c"
haftmann@21383
   739
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   740
qed
haftmann@21383
   741
haftmann@21383
   742
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   743
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   744
proof -
haftmann@21383
   745
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   746
  assume "a < f b"
haftmann@21383
   747
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   748
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   749
qed
haftmann@21383
   750
haftmann@21383
   751
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   752
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   753
proof -
haftmann@21383
   754
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   755
  assume "a <= f b"
haftmann@21383
   756
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   757
  finally (order_trans) show ?thesis .
haftmann@21383
   758
qed
haftmann@21383
   759
haftmann@21383
   760
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   761
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   762
proof -
haftmann@21383
   763
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   764
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   765
  also assume "f b <= c"
haftmann@21383
   766
  finally (order_trans) show ?thesis .
haftmann@21383
   767
qed
haftmann@21383
   768
haftmann@21383
   769
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   770
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   771
proof -
haftmann@21383
   772
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   773
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   774
  also assume "f b = c"
haftmann@21383
   775
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   776
qed
haftmann@21383
   777
haftmann@21383
   778
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   779
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   780
proof -
haftmann@21383
   781
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   782
  assume "a = f b"
haftmann@21383
   783
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   784
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   785
qed
haftmann@21383
   786
haftmann@21383
   787
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   788
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   789
proof -
haftmann@21383
   790
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   791
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   792
  also assume "f b = c"
haftmann@21383
   793
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   794
qed
haftmann@21383
   795
haftmann@21383
   796
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   797
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   798
proof -
haftmann@21383
   799
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   800
  assume "a = f b"
haftmann@21383
   801
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   802
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   803
qed
haftmann@21383
   804
haftmann@21383
   805
text {*
haftmann@21383
   806
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   807
*}
haftmann@21383
   808
haftmann@21383
   809
lemmas order_trans_rules [trans] =
haftmann@21383
   810
  order_less_subst2
haftmann@21383
   811
  order_less_subst1
haftmann@21383
   812
  order_le_less_subst2
haftmann@21383
   813
  order_le_less_subst1
haftmann@21383
   814
  order_less_le_subst2
haftmann@21383
   815
  order_less_le_subst1
haftmann@21383
   816
  order_subst2
haftmann@21383
   817
  order_subst1
haftmann@21383
   818
  ord_le_eq_subst
haftmann@21383
   819
  ord_eq_le_subst
haftmann@21383
   820
  ord_less_eq_subst
haftmann@21383
   821
  ord_eq_less_subst
haftmann@21383
   822
  forw_subst
haftmann@21383
   823
  back_subst
haftmann@21383
   824
  rev_mp
haftmann@21383
   825
  mp
haftmann@21383
   826
  order_neq_le_trans
haftmann@21383
   827
  order_le_neq_trans
haftmann@21383
   828
  order_less_trans
haftmann@21383
   829
  order_less_asym'
haftmann@21383
   830
  order_le_less_trans
haftmann@21383
   831
  order_less_le_trans
haftmann@21383
   832
  order_trans
haftmann@21383
   833
  order_antisym
haftmann@21383
   834
  ord_le_eq_trans
haftmann@21383
   835
  ord_eq_le_trans
haftmann@21383
   836
  ord_less_eq_trans
haftmann@21383
   837
  ord_eq_less_trans
haftmann@21383
   838
  trans
haftmann@21383
   839
haftmann@21083
   840
wenzelm@21180
   841
(* FIXME cleanup *)
wenzelm@21180
   842
haftmann@21083
   843
text {* These support proving chains of decreasing inequalities
haftmann@21083
   844
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   845
haftmann@21083
   846
lemma xt1:
haftmann@21083
   847
  "a = b ==> b > c ==> a > c"
haftmann@21083
   848
  "a > b ==> b = c ==> a > c"
haftmann@21083
   849
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   850
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   851
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   852
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   853
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   854
  "(x::'a::order) >= y ==> y > z ==> x > z"
wenzelm@23417
   855
  "(a::'a::order) > b ==> b > a ==> P"
haftmann@21083
   856
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   857
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   858
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   859
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   860
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   861
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   862
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   863
by auto
haftmann@21083
   864
haftmann@21083
   865
lemma xt2:
haftmann@21083
   866
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   867
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   868
haftmann@21083
   869
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   870
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   871
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   872
haftmann@21083
   873
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   874
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   875
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   876
haftmann@21083
   877
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   878
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   879
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   880
haftmann@21083
   881
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   882
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   883
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   884
haftmann@21083
   885
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   886
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   887
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   888
haftmann@21083
   889
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   890
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   891
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   892
haftmann@21083
   893
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   894
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   895
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   896
haftmann@21083
   897
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   898
haftmann@21083
   899
(* 
haftmann@21083
   900
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   901
  for the wrong thing in an Isar proof.
haftmann@21083
   902
haftmann@21083
   903
  The extra transitivity rules can be used as follows: 
haftmann@21083
   904
haftmann@21083
   905
lemma "(a::'a::order) > z"
haftmann@21083
   906
proof -
haftmann@21083
   907
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   908
    sorry
haftmann@21083
   909
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   910
    sorry
haftmann@21083
   911
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   912
    sorry
haftmann@21083
   913
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   914
    sorry
haftmann@21083
   915
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   916
    sorry
haftmann@21083
   917
  also (xtrans) have "?rhs > z"
haftmann@21083
   918
    sorry
haftmann@21083
   919
  finally (xtrans) show ?thesis .
haftmann@21083
   920
qed
haftmann@21083
   921
haftmann@21083
   922
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   923
  leave out the "(xtrans)" above.
haftmann@21083
   924
*)
haftmann@21083
   925
haftmann@21546
   926
subsection {* Order on bool *}
haftmann@21546
   927
haftmann@22886
   928
instance bool :: order 
haftmann@21546
   929
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   930
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@22916
   931
  by intro_classes (auto simp add: le_bool_def less_bool_def)
haftmann@24422
   932
lemmas [code func del] = le_bool_def less_bool_def
haftmann@21546
   933
haftmann@21546
   934
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
nipkow@23212
   935
by (simp add: le_bool_def)
haftmann@21546
   936
haftmann@21546
   937
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
nipkow@23212
   938
by (simp add: le_bool_def)
haftmann@21546
   939
haftmann@21546
   940
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23212
   941
by (simp add: le_bool_def)
haftmann@21546
   942
haftmann@21546
   943
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
nipkow@23212
   944
by (simp add: le_bool_def)
haftmann@21546
   945
haftmann@22348
   946
lemma [code func]:
haftmann@22348
   947
  "False \<le> b \<longleftrightarrow> True"
haftmann@22348
   948
  "True \<le> b \<longleftrightarrow> b"
haftmann@22348
   949
  "False < b \<longleftrightarrow> b"
haftmann@22348
   950
  "True < b \<longleftrightarrow> False"
haftmann@22348
   951
  unfolding le_bool_def less_bool_def by simp_all
haftmann@22348
   952
haftmann@22424
   953
haftmann@23881
   954
subsection {* Order on sets *}
haftmann@23881
   955
haftmann@23881
   956
instance set :: (type) order
haftmann@23881
   957
  by (intro_classes,
haftmann@23881
   958
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
haftmann@23881
   959
haftmann@23881
   960
lemmas basic_trans_rules [trans] =
haftmann@23881
   961
  order_trans_rules set_rev_mp set_mp
haftmann@23881
   962
haftmann@23881
   963
haftmann@23881
   964
subsection {* Order on functions *}
haftmann@23881
   965
haftmann@23881
   966
instance "fun" :: (type, ord) ord
haftmann@23881
   967
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
haftmann@23881
   968
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
haftmann@23881
   969
haftmann@23881
   970
lemmas [code func del] = le_fun_def less_fun_def
haftmann@23881
   971
haftmann@23881
   972
instance "fun" :: (type, order) order
haftmann@23881
   973
  by default
haftmann@23881
   974
    (auto simp add: le_fun_def less_fun_def expand_fun_eq
haftmann@23881
   975
       intro: order_trans order_antisym)
haftmann@23881
   976
haftmann@23881
   977
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
haftmann@23881
   978
  unfolding le_fun_def by simp
haftmann@23881
   979
haftmann@23881
   980
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@23881
   981
  unfolding le_fun_def by simp
haftmann@23881
   982
haftmann@23881
   983
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
haftmann@23881
   984
  unfolding le_fun_def by simp
haftmann@23881
   985
haftmann@23881
   986
text {*
haftmann@23881
   987
  Handy introduction and elimination rules for @{text "\<le>"}
haftmann@23881
   988
  on unary and binary predicates
haftmann@23881
   989
*}
haftmann@23881
   990
haftmann@23881
   991
lemma predicate1I [Pure.intro!, intro!]:
haftmann@23881
   992
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
haftmann@23881
   993
  shows "P \<le> Q"
haftmann@23881
   994
  apply (rule le_funI)
haftmann@23881
   995
  apply (rule le_boolI)
haftmann@23881
   996
  apply (rule PQ)
haftmann@23881
   997
  apply assumption
haftmann@23881
   998
  done
haftmann@23881
   999
haftmann@23881
  1000
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
haftmann@23881
  1001
  apply (erule le_funE)
haftmann@23881
  1002
  apply (erule le_boolE)
haftmann@23881
  1003
  apply assumption+
haftmann@23881
  1004
  done
haftmann@23881
  1005
haftmann@23881
  1006
lemma predicate2I [Pure.intro!, intro!]:
haftmann@23881
  1007
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
haftmann@23881
  1008
  shows "P \<le> Q"
haftmann@23881
  1009
  apply (rule le_funI)+
haftmann@23881
  1010
  apply (rule le_boolI)
haftmann@23881
  1011
  apply (rule PQ)
haftmann@23881
  1012
  apply assumption
haftmann@23881
  1013
  done
haftmann@23881
  1014
haftmann@23881
  1015
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
haftmann@23881
  1016
  apply (erule le_funE)+
haftmann@23881
  1017
  apply (erule le_boolE)
haftmann@23881
  1018
  apply assumption+
haftmann@23881
  1019
  done
haftmann@23881
  1020
haftmann@23881
  1021
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
haftmann@23881
  1022
  by (rule predicate1D)
haftmann@23881
  1023
haftmann@23881
  1024
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
haftmann@23881
  1025
  by (rule predicate2D)
haftmann@23881
  1026
haftmann@23881
  1027
haftmann@23881
  1028
subsection {* Monotonicity, least value operator and min/max *}
haftmann@21083
  1029
haftmann@21216
  1030
locale mono =
haftmann@21216
  1031
  fixes f
haftmann@21216
  1032
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
  1033
haftmann@21216
  1034
lemmas monoI [intro?] = mono.intro
haftmann@21216
  1035
  and monoD [dest?] = mono.mono
haftmann@21083
  1036
haftmann@21383
  1037
lemma LeastI2_order:
haftmann@21383
  1038
  "[| P (x::'a::order);
haftmann@21383
  1039
      !!y. P y ==> x <= y;
haftmann@21383
  1040
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
  1041
   ==> Q (Least P)"
nipkow@23212
  1042
apply (unfold Least_def)
nipkow@23212
  1043
apply (rule theI2)
nipkow@23212
  1044
  apply (blast intro: order_antisym)+
nipkow@23212
  1045
done
haftmann@21383
  1046
haftmann@23881
  1047
lemma Least_mono:
haftmann@23881
  1048
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
haftmann@23881
  1049
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
haftmann@23881
  1050
    -- {* Courtesy of Stephan Merz *}
haftmann@23881
  1051
  apply clarify
haftmann@23881
  1052
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
haftmann@23881
  1053
  apply (rule LeastI2_order)
haftmann@23881
  1054
  apply (auto elim: monoD intro!: order_antisym)
haftmann@23881
  1055
  done
haftmann@23881
  1056
haftmann@21383
  1057
lemma Least_equality:
nipkow@23212
  1058
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
nipkow@23212
  1059
apply (simp add: Least_def)
nipkow@23212
  1060
apply (rule the_equality)
nipkow@23212
  1061
apply (auto intro!: order_antisym)
nipkow@23212
  1062
done
haftmann@21383
  1063
haftmann@21383
  1064
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
nipkow@23212
  1065
by (simp add: min_def)
haftmann@21383
  1066
haftmann@21383
  1067
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
nipkow@23212
  1068
by (simp add: max_def)
haftmann@21383
  1069
haftmann@21383
  1070
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
nipkow@23212
  1071
apply (simp add: min_def)
nipkow@23212
  1072
apply (blast intro: order_antisym)
nipkow@23212
  1073
done
haftmann@21383
  1074
haftmann@21383
  1075
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
nipkow@23212
  1076
apply (simp add: max_def)
nipkow@23212
  1077
apply (blast intro: order_antisym)
nipkow@23212
  1078
done
haftmann@21383
  1079
haftmann@21383
  1080
lemma min_of_mono:
nipkow@23212
  1081
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
nipkow@23212
  1082
by (simp add: min_def)
haftmann@21383
  1083
haftmann@21383
  1084
lemma max_of_mono:
nipkow@23212
  1085
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
nipkow@23212
  1086
by (simp add: max_def)
haftmann@21383
  1087
haftmann@22548
  1088
haftmann@22548
  1089
subsection {* legacy ML bindings *}
wenzelm@21673
  1090
wenzelm@21673
  1091
ML {*
haftmann@22548
  1092
val monoI = @{thm monoI};
haftmann@22886
  1093
*}
wenzelm@21673
  1094
nipkow@15524
  1095
end