src/HOL/Orderings.thy
author paulson
Wed Aug 15 12:52:56 2007 +0200 (2007-08-15)
changeset 24286 7619080e49f0
parent 23948 261bd4678076
child 24422 c0b5ff9e9e4d
permissions -rw-r--r--
ATP blacklisting is now in theory data, attribute noatp
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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/quasi.ML"*)
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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 \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
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  and order_refl [iff]: "x \<sqsubseteq> x"
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  and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
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  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
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begin
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text {* 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 {* Name duplicates *}
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lemmas order_less_le = less_le
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lemmas order_eq_refl = order_class.eq_refl
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lemmas order_less_irrefl = order_class.less_irrefl
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lemmas order_le_less = order_class.le_less
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lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
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lemmas order_less_imp_le = order_class.less_imp_le
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lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
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lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
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lemmas order_neq_le_trans = order_class.neq_le_trans
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lemmas order_le_neq_trans = order_class.le_neq_trans
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lemmas order_antisym = antisym
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lemmas order_less_not_sym = order_class.less_not_sym
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lemmas order_less_asym = order_class.less_asym
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lemmas order_eq_iff = order_class.eq_iff
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lemmas order_antisym_conv = order_class.antisym_conv
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lemmas order_less_trans = order_class.less_trans
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lemmas order_le_less_trans = order_class.le_less_trans
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lemmas order_less_le_trans = order_class.less_le_trans
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lemmas order_less_imp_not_less = order_class.less_imp_not_less
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lemmas order_less_imp_triv = order_class.less_imp_triv
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lemmas order_less_asym' = order_class.less_asym'
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lemmas linorder_linear = linear
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lemmas linorder_less_linear = linorder_class.less_linear
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lemmas linorder_le_less_linear = linorder_class.le_less_linear
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lemmas linorder_le_cases = linorder_class.le_cases
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lemmas linorder_not_less = linorder_class.not_less
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lemmas linorder_not_le = linorder_class.not_le
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lemmas linorder_neq_iff = linorder_class.neq_iff
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lemmas linorder_neqE = linorder_class.neqE
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lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
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lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
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lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
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lemmas min_le_iff_disj = linorder_class.min_le_iff_disj
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lemmas le_max_iff_disj = linorder_class.le_max_iff_disj
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lemmas min_less_iff_disj = linorder_class.min_less_iff_disj
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lemmas less_max_iff_disj = linorder_class.less_max_iff_disj
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lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj
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lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj
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lemmas split_min = linorder_class.split_min
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lemmas split_max = linorder_class.split_max
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subsection {* Reasoning tools setup *}
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ML {*
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local
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fun decomp_gen sort thy (Trueprop $ t) =
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  let
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    fun of_sort t =
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      let
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        val T = type_of t
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      in
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        (* exclude numeric types: linear arithmetic subsumes transitivity *)
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        T <> HOLogic.natT andalso T <> HOLogic.intT
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          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
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      end;
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    fun dec (Const (@{const_name Not}, _) $ t) = (case dec t
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          of NONE => NONE
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           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
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      | dec (Const (@{const_name "op ="},  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "=", t2)
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          else NONE
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      | dec (Const (@{const_name HOL.less_eq},  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "<=", t2)
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          else NONE
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      | dec (Const (@{const_name HOL.less},  _) $ t1 $ t2) =
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          if of_sort t1
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          then SOME (t1, "<", t2)
haftmann@21248
   315
          else NONE
haftmann@21248
   316
      | dec _ = NONE;
haftmann@21091
   317
  in dec t end;
haftmann@21091
   318
haftmann@21091
   319
in
haftmann@21091
   320
haftmann@22841
   321
(* sorry - there is no preorder class
haftmann@21248
   322
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   323
struct
haftmann@21248
   324
  val le_trans = thm "order_trans";
haftmann@21248
   325
  val le_refl = thm "order_refl";
haftmann@21248
   326
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   327
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   328
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   329
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   330
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   331
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   332
  val less_imp_neq = thm "less_imp_neq";
haftmann@22738
   333
  val decomp_trans = decomp_gen ["Orderings.preorder"];
haftmann@22738
   334
  val decomp_quasi = decomp_gen ["Orderings.preorder"];
haftmann@22841
   335
end);*)
haftmann@21091
   336
haftmann@21091
   337
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   338
struct
haftmann@21248
   339
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   340
  val le_refl = thm "order_refl";
haftmann@21248
   341
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   342
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   343
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   344
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   345
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   346
  val eqI = thm "order_antisym";
haftmann@21248
   347
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   348
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   349
  val less_trans = thm "order_less_trans";
haftmann@21248
   350
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   351
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   352
  val le_trans = thm "order_trans";
haftmann@21248
   353
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   354
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   355
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   356
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   357
  val not_sym = thm "not_sym";
haftmann@21248
   358
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   359
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   360
end);
haftmann@21091
   361
haftmann@21091
   362
end;
haftmann@21091
   363
*}
haftmann@21091
   364
haftmann@21083
   365
setup {*
haftmann@21083
   366
let
haftmann@21083
   367
haftmann@21083
   368
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   369
haftmann@21083
   370
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   371
  let val prems = prems_of_ss ss;
haftmann@22916
   372
      val less = Const (@{const_name less}, T);
haftmann@21083
   373
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   374
  in case find_first (prp t) prems of
haftmann@21083
   375
       NONE =>
haftmann@21083
   376
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   377
         in case find_first (prp t) prems of
haftmann@21083
   378
              NONE => NONE
haftmann@22738
   379
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
haftmann@21083
   380
         end
haftmann@22738
   381
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
haftmann@21083
   382
  end
haftmann@21083
   383
  handle THM _ => NONE;
nipkow@15524
   384
haftmann@21083
   385
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   386
  let val prems = prems_of_ss ss;
haftmann@22916
   387
      val le = Const (@{const_name less_eq}, T);
haftmann@21083
   388
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   389
  in case find_first (prp t) prems of
haftmann@21083
   390
       NONE =>
haftmann@21083
   391
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   392
         in case find_first (prp t) prems of
haftmann@21083
   393
              NONE => NONE
haftmann@22738
   394
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
haftmann@21083
   395
         end
haftmann@22738
   396
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2}))
haftmann@21083
   397
  end
haftmann@21083
   398
  handle THM _ => NONE;
nipkow@15524
   399
haftmann@21248
   400
fun add_simprocs procs thy =
haftmann@21248
   401
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   402
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   403
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   404
fun add_solver name tac thy =
haftmann@21248
   405
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   406
    (mk_solver name (K tac))); thy);
haftmann@21083
   407
haftmann@21083
   408
in
haftmann@21248
   409
  add_simprocs [
haftmann@21248
   410
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   411
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   412
     ]
haftmann@21248
   413
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   414
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   415
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   416
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   417
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   418
     of 5 March 2004, was observed). *)
haftmann@21083
   419
end
haftmann@21083
   420
*}
nipkow@15524
   421
nipkow@15524
   422
haftmann@21083
   423
subsection {* Bounded quantifiers *}
haftmann@21083
   424
haftmann@21083
   425
syntax
wenzelm@21180
   426
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   427
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   428
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   429
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   430
wenzelm@21180
   431
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   432
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   433
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   434
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   435
haftmann@21083
   436
syntax (xsymbols)
wenzelm@21180
   437
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   438
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   439
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   440
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   441
wenzelm@21180
   442
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   443
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   444
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   445
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   446
haftmann@21083
   447
syntax (HOL)
wenzelm@21180
   448
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   449
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   450
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   451
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   452
haftmann@21083
   453
syntax (HTML output)
wenzelm@21180
   454
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   455
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   456
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   457
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   458
wenzelm@21180
   459
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   460
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   461
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   462
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   463
haftmann@21083
   464
translations
haftmann@21083
   465
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   466
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   467
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   468
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   469
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   470
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   471
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   472
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   473
haftmann@21083
   474
print_translation {*
haftmann@21083
   475
let
haftmann@22916
   476
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann@22916
   477
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
wenzelm@22377
   478
  val impl = @{const_syntax "op -->"};
wenzelm@22377
   479
  val conj = @{const_syntax "op &"};
haftmann@22916
   480
  val less = @{const_syntax less};
haftmann@22916
   481
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   482
wenzelm@21180
   483
  val trans =
wenzelm@21524
   484
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   485
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   486
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   487
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   488
krauss@22344
   489
  fun matches_bound v t = 
krauss@22344
   490
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344
   491
              | _ => false
krauss@22344
   492
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344
   493
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180
   494
wenzelm@21180
   495
  fun tr' q = (q,
wenzelm@21180
   496
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   497
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   498
        NONE => raise Match
wenzelm@21180
   499
      | SOME (l, g) =>
krauss@22344
   500
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344
   501
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344
   502
          else raise Match)
wenzelm@21180
   503
     | _ => raise Match);
wenzelm@21524
   504
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   505
*}
haftmann@21083
   506
haftmann@21083
   507
haftmann@21383
   508
subsection {* Transitivity reasoning *}
haftmann@21383
   509
haftmann@21383
   510
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
nipkow@23212
   511
by (rule subst)
haftmann@21383
   512
haftmann@21383
   513
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
nipkow@23212
   514
by (rule ssubst)
haftmann@21383
   515
haftmann@21383
   516
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
nipkow@23212
   517
by (rule subst)
haftmann@21383
   518
haftmann@21383
   519
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
nipkow@23212
   520
by (rule ssubst)
haftmann@21383
   521
haftmann@21383
   522
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   523
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   524
proof -
haftmann@21383
   525
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   526
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   527
  also assume "f b < c"
haftmann@21383
   528
  finally (order_less_trans) show ?thesis .
haftmann@21383
   529
qed
haftmann@21383
   530
haftmann@21383
   531
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   532
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   533
proof -
haftmann@21383
   534
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   535
  assume "a < f b"
haftmann@21383
   536
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   537
  finally (order_less_trans) show ?thesis .
haftmann@21383
   538
qed
haftmann@21383
   539
haftmann@21383
   540
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   541
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   542
proof -
haftmann@21383
   543
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   544
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   545
  also assume "f b < c"
haftmann@21383
   546
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   547
qed
haftmann@21383
   548
haftmann@21383
   549
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   550
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   551
proof -
haftmann@21383
   552
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   553
  assume "a <= f b"
haftmann@21383
   554
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   555
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   556
qed
haftmann@21383
   557
haftmann@21383
   558
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   559
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   560
proof -
haftmann@21383
   561
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   562
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   563
  also assume "f b <= c"
haftmann@21383
   564
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   565
qed
haftmann@21383
   566
haftmann@21383
   567
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   568
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   569
proof -
haftmann@21383
   570
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   571
  assume "a < f b"
haftmann@21383
   572
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   573
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   574
qed
haftmann@21383
   575
haftmann@21383
   576
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   577
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   578
proof -
haftmann@21383
   579
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   580
  assume "a <= f b"
haftmann@21383
   581
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   582
  finally (order_trans) show ?thesis .
haftmann@21383
   583
qed
haftmann@21383
   584
haftmann@21383
   585
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   586
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   587
proof -
haftmann@21383
   588
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   589
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   590
  also assume "f b <= c"
haftmann@21383
   591
  finally (order_trans) show ?thesis .
haftmann@21383
   592
qed
haftmann@21383
   593
haftmann@21383
   594
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   595
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   596
proof -
haftmann@21383
   597
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   598
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   599
  also assume "f b = c"
haftmann@21383
   600
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   601
qed
haftmann@21383
   602
haftmann@21383
   603
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   604
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   605
proof -
haftmann@21383
   606
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   607
  assume "a = f b"
haftmann@21383
   608
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   609
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   610
qed
haftmann@21383
   611
haftmann@21383
   612
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   613
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   614
proof -
haftmann@21383
   615
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   616
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   617
  also assume "f b = c"
haftmann@21383
   618
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   619
qed
haftmann@21383
   620
haftmann@21383
   621
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   622
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   623
proof -
haftmann@21383
   624
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   625
  assume "a = f b"
haftmann@21383
   626
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   627
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   628
qed
haftmann@21383
   629
haftmann@21383
   630
text {*
haftmann@21383
   631
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   632
*}
haftmann@21383
   633
haftmann@21383
   634
lemmas order_trans_rules [trans] =
haftmann@21383
   635
  order_less_subst2
haftmann@21383
   636
  order_less_subst1
haftmann@21383
   637
  order_le_less_subst2
haftmann@21383
   638
  order_le_less_subst1
haftmann@21383
   639
  order_less_le_subst2
haftmann@21383
   640
  order_less_le_subst1
haftmann@21383
   641
  order_subst2
haftmann@21383
   642
  order_subst1
haftmann@21383
   643
  ord_le_eq_subst
haftmann@21383
   644
  ord_eq_le_subst
haftmann@21383
   645
  ord_less_eq_subst
haftmann@21383
   646
  ord_eq_less_subst
haftmann@21383
   647
  forw_subst
haftmann@21383
   648
  back_subst
haftmann@21383
   649
  rev_mp
haftmann@21383
   650
  mp
haftmann@21383
   651
  order_neq_le_trans
haftmann@21383
   652
  order_le_neq_trans
haftmann@21383
   653
  order_less_trans
haftmann@21383
   654
  order_less_asym'
haftmann@21383
   655
  order_le_less_trans
haftmann@21383
   656
  order_less_le_trans
haftmann@21383
   657
  order_trans
haftmann@21383
   658
  order_antisym
haftmann@21383
   659
  ord_le_eq_trans
haftmann@21383
   660
  ord_eq_le_trans
haftmann@21383
   661
  ord_less_eq_trans
haftmann@21383
   662
  ord_eq_less_trans
haftmann@21383
   663
  trans
haftmann@21383
   664
haftmann@21083
   665
wenzelm@21180
   666
(* FIXME cleanup *)
wenzelm@21180
   667
haftmann@21083
   668
text {* These support proving chains of decreasing inequalities
haftmann@21083
   669
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   670
haftmann@21083
   671
lemma xt1:
haftmann@21083
   672
  "a = b ==> b > c ==> a > c"
haftmann@21083
   673
  "a > b ==> b = c ==> a > c"
haftmann@21083
   674
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   675
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   676
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   677
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   678
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   679
  "(x::'a::order) >= y ==> y > z ==> x > z"
wenzelm@23417
   680
  "(a::'a::order) > b ==> b > a ==> P"
haftmann@21083
   681
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   682
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   683
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   684
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   685
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   686
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   687
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   688
by auto
haftmann@21083
   689
haftmann@21083
   690
lemma xt2:
haftmann@21083
   691
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   692
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   693
haftmann@21083
   694
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   695
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   696
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   697
haftmann@21083
   698
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   699
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   700
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   701
haftmann@21083
   702
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   703
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   704
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   705
haftmann@21083
   706
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   707
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   708
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   709
haftmann@21083
   710
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   711
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   712
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   713
haftmann@21083
   714
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   715
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   716
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   717
haftmann@21083
   718
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   719
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   720
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   721
haftmann@21083
   722
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   723
haftmann@21083
   724
(* 
haftmann@21083
   725
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   726
  for the wrong thing in an Isar proof.
haftmann@21083
   727
haftmann@21083
   728
  The extra transitivity rules can be used as follows: 
haftmann@21083
   729
haftmann@21083
   730
lemma "(a::'a::order) > z"
haftmann@21083
   731
proof -
haftmann@21083
   732
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   733
    sorry
haftmann@21083
   734
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   735
    sorry
haftmann@21083
   736
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   737
    sorry
haftmann@21083
   738
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   739
    sorry
haftmann@21083
   740
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   741
    sorry
haftmann@21083
   742
  also (xtrans) have "?rhs > z"
haftmann@21083
   743
    sorry
haftmann@21083
   744
  finally (xtrans) show ?thesis .
haftmann@21083
   745
qed
haftmann@21083
   746
haftmann@21083
   747
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   748
  leave out the "(xtrans)" above.
haftmann@21083
   749
*)
haftmann@21083
   750
haftmann@21546
   751
subsection {* Order on bool *}
haftmann@21546
   752
haftmann@22886
   753
instance bool :: order 
haftmann@21546
   754
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   755
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@22916
   756
  by intro_classes (auto simp add: le_bool_def less_bool_def)
haftmann@21546
   757
haftmann@21546
   758
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
nipkow@23212
   759
by (simp add: le_bool_def)
haftmann@21546
   760
haftmann@21546
   761
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
nipkow@23212
   762
by (simp add: le_bool_def)
haftmann@21546
   763
haftmann@21546
   764
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23212
   765
by (simp add: le_bool_def)
haftmann@21546
   766
haftmann@21546
   767
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
nipkow@23212
   768
by (simp add: le_bool_def)
haftmann@21546
   769
haftmann@22348
   770
lemma [code func]:
haftmann@22348
   771
  "False \<le> b \<longleftrightarrow> True"
haftmann@22348
   772
  "True \<le> b \<longleftrightarrow> b"
haftmann@22348
   773
  "False < b \<longleftrightarrow> b"
haftmann@22348
   774
  "True < b \<longleftrightarrow> False"
haftmann@22348
   775
  unfolding le_bool_def less_bool_def by simp_all
haftmann@22348
   776
haftmann@22424
   777
haftmann@23881
   778
subsection {* Order on sets *}
haftmann@23881
   779
haftmann@23881
   780
instance set :: (type) order
haftmann@23881
   781
  by (intro_classes,
haftmann@23881
   782
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
haftmann@23881
   783
haftmann@23881
   784
lemmas basic_trans_rules [trans] =
haftmann@23881
   785
  order_trans_rules set_rev_mp set_mp
haftmann@23881
   786
haftmann@23881
   787
haftmann@23881
   788
subsection {* Order on functions *}
haftmann@23881
   789
haftmann@23881
   790
instance "fun" :: (type, ord) ord
haftmann@23881
   791
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
haftmann@23881
   792
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
haftmann@23881
   793
haftmann@23881
   794
lemmas [code func del] = le_fun_def less_fun_def
haftmann@23881
   795
haftmann@23881
   796
instance "fun" :: (type, order) order
haftmann@23881
   797
  by default
haftmann@23881
   798
    (auto simp add: le_fun_def less_fun_def expand_fun_eq
haftmann@23881
   799
       intro: order_trans order_antisym)
haftmann@23881
   800
haftmann@23881
   801
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
haftmann@23881
   802
  unfolding le_fun_def by simp
haftmann@23881
   803
haftmann@23881
   804
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@23881
   805
  unfolding le_fun_def by simp
haftmann@23881
   806
haftmann@23881
   807
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
haftmann@23881
   808
  unfolding le_fun_def by simp
haftmann@23881
   809
haftmann@23881
   810
text {*
haftmann@23881
   811
  Handy introduction and elimination rules for @{text "\<le>"}
haftmann@23881
   812
  on unary and binary predicates
haftmann@23881
   813
*}
haftmann@23881
   814
haftmann@23881
   815
lemma predicate1I [Pure.intro!, intro!]:
haftmann@23881
   816
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
haftmann@23881
   817
  shows "P \<le> Q"
haftmann@23881
   818
  apply (rule le_funI)
haftmann@23881
   819
  apply (rule le_boolI)
haftmann@23881
   820
  apply (rule PQ)
haftmann@23881
   821
  apply assumption
haftmann@23881
   822
  done
haftmann@23881
   823
haftmann@23881
   824
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
haftmann@23881
   825
  apply (erule le_funE)
haftmann@23881
   826
  apply (erule le_boolE)
haftmann@23881
   827
  apply assumption+
haftmann@23881
   828
  done
haftmann@23881
   829
haftmann@23881
   830
lemma predicate2I [Pure.intro!, intro!]:
haftmann@23881
   831
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
haftmann@23881
   832
  shows "P \<le> Q"
haftmann@23881
   833
  apply (rule le_funI)+
haftmann@23881
   834
  apply (rule le_boolI)
haftmann@23881
   835
  apply (rule PQ)
haftmann@23881
   836
  apply assumption
haftmann@23881
   837
  done
haftmann@23881
   838
haftmann@23881
   839
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
haftmann@23881
   840
  apply (erule le_funE)+
haftmann@23881
   841
  apply (erule le_boolE)
haftmann@23881
   842
  apply assumption+
haftmann@23881
   843
  done
haftmann@23881
   844
haftmann@23881
   845
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
haftmann@23881
   846
  by (rule predicate1D)
haftmann@23881
   847
haftmann@23881
   848
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
haftmann@23881
   849
  by (rule predicate2D)
haftmann@23881
   850
haftmann@23881
   851
haftmann@23881
   852
subsection {* Monotonicity, least value operator and min/max *}
haftmann@21083
   853
haftmann@21216
   854
locale mono =
haftmann@21216
   855
  fixes f
haftmann@21216
   856
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   857
haftmann@21216
   858
lemmas monoI [intro?] = mono.intro
haftmann@21216
   859
  and monoD [dest?] = mono.mono
haftmann@21083
   860
haftmann@21383
   861
lemma LeastI2_order:
haftmann@21383
   862
  "[| P (x::'a::order);
haftmann@21383
   863
      !!y. P y ==> x <= y;
haftmann@21383
   864
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   865
   ==> Q (Least P)"
nipkow@23212
   866
apply (unfold Least_def)
nipkow@23212
   867
apply (rule theI2)
nipkow@23212
   868
  apply (blast intro: order_antisym)+
nipkow@23212
   869
done
haftmann@21383
   870
haftmann@23881
   871
lemma Least_mono:
haftmann@23881
   872
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
haftmann@23881
   873
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
haftmann@23881
   874
    -- {* Courtesy of Stephan Merz *}
haftmann@23881
   875
  apply clarify
haftmann@23881
   876
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
haftmann@23881
   877
  apply (rule LeastI2_order)
haftmann@23881
   878
  apply (auto elim: monoD intro!: order_antisym)
haftmann@23881
   879
  done
haftmann@23881
   880
haftmann@21383
   881
lemma Least_equality:
nipkow@23212
   882
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
nipkow@23212
   883
apply (simp add: Least_def)
nipkow@23212
   884
apply (rule the_equality)
nipkow@23212
   885
apply (auto intro!: order_antisym)
nipkow@23212
   886
done
haftmann@21383
   887
haftmann@21383
   888
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
nipkow@23212
   889
by (simp add: min_def)
haftmann@21383
   890
haftmann@21383
   891
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
nipkow@23212
   892
by (simp add: max_def)
haftmann@21383
   893
haftmann@21383
   894
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
nipkow@23212
   895
apply (simp add: min_def)
nipkow@23212
   896
apply (blast intro: order_antisym)
nipkow@23212
   897
done
haftmann@21383
   898
haftmann@21383
   899
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
nipkow@23212
   900
apply (simp add: max_def)
nipkow@23212
   901
apply (blast intro: order_antisym)
nipkow@23212
   902
done
haftmann@21383
   903
haftmann@21383
   904
lemma min_of_mono:
nipkow@23212
   905
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
nipkow@23212
   906
by (simp add: min_def)
haftmann@21383
   907
haftmann@21383
   908
lemma max_of_mono:
nipkow@23212
   909
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
nipkow@23212
   910
by (simp add: max_def)
haftmann@21383
   911
haftmann@22548
   912
haftmann@22548
   913
subsection {* legacy ML bindings *}
wenzelm@21673
   914
wenzelm@21673
   915
ML {*
haftmann@22548
   916
val monoI = @{thm monoI};
haftmann@22886
   917
*}
wenzelm@21673
   918
nipkow@15524
   919
end