src/HOL/Orderings.thy
author haftmann
Fri Aug 24 14:14:18 2007 +0200 (2007-08-24)
changeset 24422 c0b5ff9e9e4d
parent 24286 7619080e49f0
child 24641 448edc627ee4
permissions -rw-r--r--
moved class dense_linear_order to Orderings.thy
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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 {* 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)
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          else NONE
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      | dec _ = NONE;
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  in dec t end;
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in
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(* sorry - there is no preorder class
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structure Quasi_Tac = Quasi_Tac_Fun (
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struct
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  val le_trans = thm "order_trans";
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  val le_refl = thm "order_refl";
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  val eqD1 = thm "order_eq_refl";
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  val eqD2 = thm "sym" RS thm "order_eq_refl";
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  val less_reflE = thm "order_less_irrefl" RS thm "notE";
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  val less_imp_le = thm "order_less_imp_le";
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  val le_neq_trans = thm "order_le_neq_trans";
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  val neq_le_trans = thm "order_neq_le_trans";
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  val less_imp_neq = thm "less_imp_neq";
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  val decomp_trans = decomp_gen ["Orderings.preorder"];
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  val decomp_quasi = decomp_gen ["Orderings.preorder"];
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end);*)
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structure Order_Tac = Order_Tac_Fun (
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struct
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  val less_reflE = @{thm less_irrefl} RS @{thm notE};
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  val le_refl = @{thm order_refl};
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  val less_imp_le = @{thm less_imp_le};
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  val not_lessI = @{thm not_less} RS @{thm iffD2};
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  val not_leI = @{thm not_le} RS @{thm iffD2};
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  val not_lessD = @{thm not_less} RS @{thm iffD1};
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  val not_leD = @{thm not_le} RS @{thm iffD1};
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  val eqI = @{thm antisym};
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  val eqD1 = @{thm eq_refl};
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  val eqD2 = @{thm sym} RS @{thm eq_refl};
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  val less_trans = @{thm less_trans};
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  val less_le_trans = @{thm less_le_trans};
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  val le_less_trans = @{thm le_less_trans};
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  val le_trans = @{thm order_trans};
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  val le_neq_trans = @{thm le_neq_trans};
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  val neq_le_trans = @{thm neq_le_trans};
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  val less_imp_neq = @{thm less_imp_neq};
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  val eq_neq_eq_imp_neq = @{thm eq_neq_eq_imp_neq};
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  val not_sym = @{thm not_sym};
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  val decomp_part = decomp_gen ["Orderings.order"];
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  val decomp_lin = decomp_gen ["Orderings.linorder"];
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end);
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end;
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*}
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setup {*
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let
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fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   322
haftmann@21083
   323
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   324
  let val prems = prems_of_ss ss;
haftmann@22916
   325
      val less = Const (@{const_name less}, T);
haftmann@21083
   326
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   327
  in case find_first (prp t) prems of
haftmann@21083
   328
       NONE =>
haftmann@21083
   329
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   330
         in case find_first (prp t) prems of
haftmann@21083
   331
              NONE => NONE
haftmann@24422
   332
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
haftmann@21083
   333
         end
haftmann@24422
   334
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
haftmann@21083
   335
  end
haftmann@21083
   336
  handle THM _ => NONE;
nipkow@15524
   337
haftmann@21083
   338
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   339
  let val prems = prems_of_ss ss;
haftmann@22916
   340
      val le = Const (@{const_name less_eq}, T);
haftmann@21083
   341
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   342
  in case find_first (prp t) prems of
haftmann@21083
   343
       NONE =>
haftmann@21083
   344
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   345
         in case find_first (prp t) prems of
haftmann@21083
   346
              NONE => NONE
haftmann@24422
   347
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
haftmann@21083
   348
         end
haftmann@24422
   349
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
haftmann@21083
   350
  end
haftmann@21083
   351
  handle THM _ => NONE;
nipkow@15524
   352
haftmann@21248
   353
fun add_simprocs procs thy =
haftmann@21248
   354
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   355
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   356
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   357
fun add_solver name tac thy =
haftmann@21248
   358
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   359
    (mk_solver name (K tac))); thy);
haftmann@21083
   360
haftmann@21083
   361
in
haftmann@21248
   362
  add_simprocs [
haftmann@21248
   363
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   364
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   365
     ]
haftmann@21248
   366
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   367
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   368
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   369
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   370
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   371
     of 5 March 2004, was observed). *)
haftmann@21083
   372
end
haftmann@21083
   373
*}
nipkow@15524
   374
nipkow@15524
   375
haftmann@24422
   376
subsection {* Dense orders *}
haftmann@24422
   377
haftmann@24422
   378
class dense_linear_order = linorder + 
haftmann@24422
   379
  assumes gt_ex: "\<exists>y. x \<sqsubset> y" 
haftmann@24422
   380
  and lt_ex: "\<exists>y. y \<sqsubset> x"
haftmann@24422
   381
  and dense: "x \<sqsubset> y \<Longrightarrow> (\<exists>z. x \<sqsubset> z \<and> z \<sqsubset> y)"
haftmann@24422
   382
  (*see further theory Dense_Linear_Order*)
haftmann@24422
   383
haftmann@24422
   384
lemma interval_empty_iff:
haftmann@24422
   385
  fixes x y z :: "'a\<Colon>dense_linear_order"
haftmann@24422
   386
  shows "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
haftmann@24422
   387
  by (auto dest: dense)
haftmann@24422
   388
haftmann@24422
   389
subsection {* Name duplicates *}
haftmann@24422
   390
haftmann@24422
   391
lemmas order_less_le = less_le
haftmann@24422
   392
lemmas order_eq_refl = order_class.eq_refl
haftmann@24422
   393
lemmas order_less_irrefl = order_class.less_irrefl
haftmann@24422
   394
lemmas order_le_less = order_class.le_less
haftmann@24422
   395
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
haftmann@24422
   396
lemmas order_less_imp_le = order_class.less_imp_le
haftmann@24422
   397
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
haftmann@24422
   398
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
haftmann@24422
   399
lemmas order_neq_le_trans = order_class.neq_le_trans
haftmann@24422
   400
lemmas order_le_neq_trans = order_class.le_neq_trans
haftmann@24422
   401
haftmann@24422
   402
lemmas order_antisym = antisym
haftmann@24422
   403
lemmas order_less_not_sym = order_class.less_not_sym
haftmann@24422
   404
lemmas order_less_asym = order_class.less_asym
haftmann@24422
   405
lemmas order_eq_iff = order_class.eq_iff
haftmann@24422
   406
lemmas order_antisym_conv = order_class.antisym_conv
haftmann@24422
   407
lemmas order_less_trans = order_class.less_trans
haftmann@24422
   408
lemmas order_le_less_trans = order_class.le_less_trans
haftmann@24422
   409
lemmas order_less_le_trans = order_class.less_le_trans
haftmann@24422
   410
lemmas order_less_imp_not_less = order_class.less_imp_not_less
haftmann@24422
   411
lemmas order_less_imp_triv = order_class.less_imp_triv
haftmann@24422
   412
lemmas order_less_asym' = order_class.less_asym'
haftmann@24422
   413
haftmann@24422
   414
lemmas linorder_linear = linear
haftmann@24422
   415
lemmas linorder_less_linear = linorder_class.less_linear
haftmann@24422
   416
lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@24422
   417
lemmas linorder_le_cases = linorder_class.le_cases
haftmann@24422
   418
lemmas linorder_not_less = linorder_class.not_less
haftmann@24422
   419
lemmas linorder_not_le = linorder_class.not_le
haftmann@24422
   420
lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@24422
   421
lemmas linorder_neqE = linorder_class.neqE
haftmann@24422
   422
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@24422
   423
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@24422
   424
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
haftmann@24422
   425
haftmann@24422
   426
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj
haftmann@24422
   427
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj
haftmann@24422
   428
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj
haftmann@24422
   429
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj
haftmann@24422
   430
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj
haftmann@24422
   431
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj
haftmann@24422
   432
lemmas split_min = linorder_class.split_min
haftmann@24422
   433
lemmas split_max = linorder_class.split_max
haftmann@24422
   434
haftmann@24422
   435
haftmann@21083
   436
subsection {* Bounded quantifiers *}
haftmann@21083
   437
haftmann@21083
   438
syntax
wenzelm@21180
   439
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   440
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   441
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   442
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   443
wenzelm@21180
   444
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   445
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   446
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   447
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   448
haftmann@21083
   449
syntax (xsymbols)
wenzelm@21180
   450
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   451
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   452
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   453
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   454
wenzelm@21180
   455
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   456
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   457
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   458
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   459
haftmann@21083
   460
syntax (HOL)
wenzelm@21180
   461
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   462
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   463
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   464
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   465
haftmann@21083
   466
syntax (HTML output)
wenzelm@21180
   467
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   468
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   469
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   470
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   471
wenzelm@21180
   472
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   473
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   474
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   475
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   476
haftmann@21083
   477
translations
haftmann@21083
   478
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   479
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   480
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   481
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   482
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   483
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   484
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   485
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   486
haftmann@21083
   487
print_translation {*
haftmann@21083
   488
let
haftmann@22916
   489
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann@22916
   490
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
wenzelm@22377
   491
  val impl = @{const_syntax "op -->"};
wenzelm@22377
   492
  val conj = @{const_syntax "op &"};
haftmann@22916
   493
  val less = @{const_syntax less};
haftmann@22916
   494
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   495
wenzelm@21180
   496
  val trans =
wenzelm@21524
   497
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   498
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   499
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   500
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   501
krauss@22344
   502
  fun matches_bound v t = 
krauss@22344
   503
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344
   504
              | _ => false
krauss@22344
   505
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344
   506
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180
   507
wenzelm@21180
   508
  fun tr' q = (q,
wenzelm@21180
   509
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   510
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   511
        NONE => raise Match
wenzelm@21180
   512
      | SOME (l, g) =>
krauss@22344
   513
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344
   514
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344
   515
          else raise Match)
wenzelm@21180
   516
     | _ => raise Match);
wenzelm@21524
   517
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   518
*}
haftmann@21083
   519
haftmann@21083
   520
haftmann@21383
   521
subsection {* Transitivity reasoning *}
haftmann@21383
   522
haftmann@21383
   523
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
nipkow@23212
   524
by (rule subst)
haftmann@21383
   525
haftmann@21383
   526
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
nipkow@23212
   527
by (rule ssubst)
haftmann@21383
   528
haftmann@21383
   529
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
nipkow@23212
   530
by (rule subst)
haftmann@21383
   531
haftmann@21383
   532
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
nipkow@23212
   533
by (rule ssubst)
haftmann@21383
   534
haftmann@21383
   535
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   536
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   537
proof -
haftmann@21383
   538
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   539
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   540
  also assume "f b < c"
haftmann@21383
   541
  finally (order_less_trans) show ?thesis .
haftmann@21383
   542
qed
haftmann@21383
   543
haftmann@21383
   544
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   545
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   546
proof -
haftmann@21383
   547
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   548
  assume "a < f b"
haftmann@21383
   549
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   550
  finally (order_less_trans) show ?thesis .
haftmann@21383
   551
qed
haftmann@21383
   552
haftmann@21383
   553
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   554
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   555
proof -
haftmann@21383
   556
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   557
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   558
  also assume "f b < c"
haftmann@21383
   559
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   560
qed
haftmann@21383
   561
haftmann@21383
   562
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   563
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   564
proof -
haftmann@21383
   565
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   566
  assume "a <= f b"
haftmann@21383
   567
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   568
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   569
qed
haftmann@21383
   570
haftmann@21383
   571
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   572
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   573
proof -
haftmann@21383
   574
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   575
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   576
  also assume "f b <= c"
haftmann@21383
   577
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   578
qed
haftmann@21383
   579
haftmann@21383
   580
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   581
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   582
proof -
haftmann@21383
   583
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   584
  assume "a < f b"
haftmann@21383
   585
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   586
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   587
qed
haftmann@21383
   588
haftmann@21383
   589
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   590
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   591
proof -
haftmann@21383
   592
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   593
  assume "a <= f b"
haftmann@21383
   594
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   595
  finally (order_trans) show ?thesis .
haftmann@21383
   596
qed
haftmann@21383
   597
haftmann@21383
   598
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   599
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   600
proof -
haftmann@21383
   601
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   602
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   603
  also assume "f b <= c"
haftmann@21383
   604
  finally (order_trans) show ?thesis .
haftmann@21383
   605
qed
haftmann@21383
   606
haftmann@21383
   607
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   608
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   609
proof -
haftmann@21383
   610
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   611
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   612
  also assume "f b = c"
haftmann@21383
   613
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   614
qed
haftmann@21383
   615
haftmann@21383
   616
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   617
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   618
proof -
haftmann@21383
   619
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   620
  assume "a = f b"
haftmann@21383
   621
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   622
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   623
qed
haftmann@21383
   624
haftmann@21383
   625
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   626
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   627
proof -
haftmann@21383
   628
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   629
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   630
  also assume "f b = c"
haftmann@21383
   631
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   632
qed
haftmann@21383
   633
haftmann@21383
   634
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   635
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   636
proof -
haftmann@21383
   637
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   638
  assume "a = f b"
haftmann@21383
   639
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   640
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   641
qed
haftmann@21383
   642
haftmann@21383
   643
text {*
haftmann@21383
   644
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   645
*}
haftmann@21383
   646
haftmann@21383
   647
lemmas order_trans_rules [trans] =
haftmann@21383
   648
  order_less_subst2
haftmann@21383
   649
  order_less_subst1
haftmann@21383
   650
  order_le_less_subst2
haftmann@21383
   651
  order_le_less_subst1
haftmann@21383
   652
  order_less_le_subst2
haftmann@21383
   653
  order_less_le_subst1
haftmann@21383
   654
  order_subst2
haftmann@21383
   655
  order_subst1
haftmann@21383
   656
  ord_le_eq_subst
haftmann@21383
   657
  ord_eq_le_subst
haftmann@21383
   658
  ord_less_eq_subst
haftmann@21383
   659
  ord_eq_less_subst
haftmann@21383
   660
  forw_subst
haftmann@21383
   661
  back_subst
haftmann@21383
   662
  rev_mp
haftmann@21383
   663
  mp
haftmann@21383
   664
  order_neq_le_trans
haftmann@21383
   665
  order_le_neq_trans
haftmann@21383
   666
  order_less_trans
haftmann@21383
   667
  order_less_asym'
haftmann@21383
   668
  order_le_less_trans
haftmann@21383
   669
  order_less_le_trans
haftmann@21383
   670
  order_trans
haftmann@21383
   671
  order_antisym
haftmann@21383
   672
  ord_le_eq_trans
haftmann@21383
   673
  ord_eq_le_trans
haftmann@21383
   674
  ord_less_eq_trans
haftmann@21383
   675
  ord_eq_less_trans
haftmann@21383
   676
  trans
haftmann@21383
   677
haftmann@21083
   678
wenzelm@21180
   679
(* FIXME cleanup *)
wenzelm@21180
   680
haftmann@21083
   681
text {* These support proving chains of decreasing inequalities
haftmann@21083
   682
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   683
haftmann@21083
   684
lemma xt1:
haftmann@21083
   685
  "a = b ==> b > c ==> a > c"
haftmann@21083
   686
  "a > b ==> b = c ==> a > c"
haftmann@21083
   687
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   688
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   689
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   690
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   691
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   692
  "(x::'a::order) >= y ==> y > z ==> x > z"
wenzelm@23417
   693
  "(a::'a::order) > b ==> b > a ==> P"
haftmann@21083
   694
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   695
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   696
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   697
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   698
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   699
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   700
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   701
by auto
haftmann@21083
   702
haftmann@21083
   703
lemma xt2:
haftmann@21083
   704
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   705
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   706
haftmann@21083
   707
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   708
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   709
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   710
haftmann@21083
   711
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   712
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   713
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   714
haftmann@21083
   715
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   716
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   717
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   718
haftmann@21083
   719
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   720
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   721
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   722
haftmann@21083
   723
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   724
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   725
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   726
haftmann@21083
   727
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   728
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   729
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   730
haftmann@21083
   731
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   732
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   733
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   734
haftmann@21083
   735
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   736
haftmann@21083
   737
(* 
haftmann@21083
   738
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   739
  for the wrong thing in an Isar proof.
haftmann@21083
   740
haftmann@21083
   741
  The extra transitivity rules can be used as follows: 
haftmann@21083
   742
haftmann@21083
   743
lemma "(a::'a::order) > z"
haftmann@21083
   744
proof -
haftmann@21083
   745
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   746
    sorry
haftmann@21083
   747
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   748
    sorry
haftmann@21083
   749
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   750
    sorry
haftmann@21083
   751
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   752
    sorry
haftmann@21083
   753
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   754
    sorry
haftmann@21083
   755
  also (xtrans) have "?rhs > z"
haftmann@21083
   756
    sorry
haftmann@21083
   757
  finally (xtrans) show ?thesis .
haftmann@21083
   758
qed
haftmann@21083
   759
haftmann@21083
   760
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   761
  leave out the "(xtrans)" above.
haftmann@21083
   762
*)
haftmann@21083
   763
haftmann@21546
   764
subsection {* Order on bool *}
haftmann@21546
   765
haftmann@22886
   766
instance bool :: order 
haftmann@21546
   767
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   768
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@22916
   769
  by intro_classes (auto simp add: le_bool_def less_bool_def)
haftmann@24422
   770
lemmas [code func del] = le_bool_def less_bool_def
haftmann@21546
   771
haftmann@21546
   772
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
nipkow@23212
   773
by (simp add: le_bool_def)
haftmann@21546
   774
haftmann@21546
   775
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
nipkow@23212
   776
by (simp add: le_bool_def)
haftmann@21546
   777
haftmann@21546
   778
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23212
   779
by (simp add: le_bool_def)
haftmann@21546
   780
haftmann@21546
   781
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
nipkow@23212
   782
by (simp add: le_bool_def)
haftmann@21546
   783
haftmann@22348
   784
lemma [code func]:
haftmann@22348
   785
  "False \<le> b \<longleftrightarrow> True"
haftmann@22348
   786
  "True \<le> b \<longleftrightarrow> b"
haftmann@22348
   787
  "False < b \<longleftrightarrow> b"
haftmann@22348
   788
  "True < b \<longleftrightarrow> False"
haftmann@22348
   789
  unfolding le_bool_def less_bool_def by simp_all
haftmann@22348
   790
haftmann@22424
   791
haftmann@23881
   792
subsection {* Order on sets *}
haftmann@23881
   793
haftmann@23881
   794
instance set :: (type) order
haftmann@23881
   795
  by (intro_classes,
haftmann@23881
   796
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
haftmann@23881
   797
haftmann@23881
   798
lemmas basic_trans_rules [trans] =
haftmann@23881
   799
  order_trans_rules set_rev_mp set_mp
haftmann@23881
   800
haftmann@23881
   801
haftmann@23881
   802
subsection {* Order on functions *}
haftmann@23881
   803
haftmann@23881
   804
instance "fun" :: (type, ord) ord
haftmann@23881
   805
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
haftmann@23881
   806
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
haftmann@23881
   807
haftmann@23881
   808
lemmas [code func del] = le_fun_def less_fun_def
haftmann@23881
   809
haftmann@23881
   810
instance "fun" :: (type, order) order
haftmann@23881
   811
  by default
haftmann@23881
   812
    (auto simp add: le_fun_def less_fun_def expand_fun_eq
haftmann@23881
   813
       intro: order_trans order_antisym)
haftmann@23881
   814
haftmann@23881
   815
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
haftmann@23881
   816
  unfolding le_fun_def by simp
haftmann@23881
   817
haftmann@23881
   818
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@23881
   819
  unfolding le_fun_def by simp
haftmann@23881
   820
haftmann@23881
   821
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
haftmann@23881
   822
  unfolding le_fun_def by simp
haftmann@23881
   823
haftmann@23881
   824
text {*
haftmann@23881
   825
  Handy introduction and elimination rules for @{text "\<le>"}
haftmann@23881
   826
  on unary and binary predicates
haftmann@23881
   827
*}
haftmann@23881
   828
haftmann@23881
   829
lemma predicate1I [Pure.intro!, intro!]:
haftmann@23881
   830
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
haftmann@23881
   831
  shows "P \<le> Q"
haftmann@23881
   832
  apply (rule le_funI)
haftmann@23881
   833
  apply (rule le_boolI)
haftmann@23881
   834
  apply (rule PQ)
haftmann@23881
   835
  apply assumption
haftmann@23881
   836
  done
haftmann@23881
   837
haftmann@23881
   838
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
haftmann@23881
   839
  apply (erule le_funE)
haftmann@23881
   840
  apply (erule le_boolE)
haftmann@23881
   841
  apply assumption+
haftmann@23881
   842
  done
haftmann@23881
   843
haftmann@23881
   844
lemma predicate2I [Pure.intro!, intro!]:
haftmann@23881
   845
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
haftmann@23881
   846
  shows "P \<le> Q"
haftmann@23881
   847
  apply (rule le_funI)+
haftmann@23881
   848
  apply (rule le_boolI)
haftmann@23881
   849
  apply (rule PQ)
haftmann@23881
   850
  apply assumption
haftmann@23881
   851
  done
haftmann@23881
   852
haftmann@23881
   853
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
haftmann@23881
   854
  apply (erule le_funE)+
haftmann@23881
   855
  apply (erule le_boolE)
haftmann@23881
   856
  apply assumption+
haftmann@23881
   857
  done
haftmann@23881
   858
haftmann@23881
   859
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
haftmann@23881
   860
  by (rule predicate1D)
haftmann@23881
   861
haftmann@23881
   862
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
haftmann@23881
   863
  by (rule predicate2D)
haftmann@23881
   864
haftmann@23881
   865
haftmann@23881
   866
subsection {* Monotonicity, least value operator and min/max *}
haftmann@21083
   867
haftmann@21216
   868
locale mono =
haftmann@21216
   869
  fixes f
haftmann@21216
   870
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   871
haftmann@21216
   872
lemmas monoI [intro?] = mono.intro
haftmann@21216
   873
  and monoD [dest?] = mono.mono
haftmann@21083
   874
haftmann@21383
   875
lemma LeastI2_order:
haftmann@21383
   876
  "[| P (x::'a::order);
haftmann@21383
   877
      !!y. P y ==> x <= y;
haftmann@21383
   878
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   879
   ==> Q (Least P)"
nipkow@23212
   880
apply (unfold Least_def)
nipkow@23212
   881
apply (rule theI2)
nipkow@23212
   882
  apply (blast intro: order_antisym)+
nipkow@23212
   883
done
haftmann@21383
   884
haftmann@23881
   885
lemma Least_mono:
haftmann@23881
   886
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
haftmann@23881
   887
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
haftmann@23881
   888
    -- {* Courtesy of Stephan Merz *}
haftmann@23881
   889
  apply clarify
haftmann@23881
   890
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
haftmann@23881
   891
  apply (rule LeastI2_order)
haftmann@23881
   892
  apply (auto elim: monoD intro!: order_antisym)
haftmann@23881
   893
  done
haftmann@23881
   894
haftmann@21383
   895
lemma Least_equality:
nipkow@23212
   896
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
nipkow@23212
   897
apply (simp add: Least_def)
nipkow@23212
   898
apply (rule the_equality)
nipkow@23212
   899
apply (auto intro!: order_antisym)
nipkow@23212
   900
done
haftmann@21383
   901
haftmann@21383
   902
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
nipkow@23212
   903
by (simp add: min_def)
haftmann@21383
   904
haftmann@21383
   905
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
nipkow@23212
   906
by (simp add: max_def)
haftmann@21383
   907
haftmann@21383
   908
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
nipkow@23212
   909
apply (simp add: min_def)
nipkow@23212
   910
apply (blast intro: order_antisym)
nipkow@23212
   911
done
haftmann@21383
   912
haftmann@21383
   913
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
nipkow@23212
   914
apply (simp add: max_def)
nipkow@23212
   915
apply (blast intro: order_antisym)
nipkow@23212
   916
done
haftmann@21383
   917
haftmann@21383
   918
lemma min_of_mono:
nipkow@23212
   919
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
nipkow@23212
   920
by (simp add: min_def)
haftmann@21383
   921
haftmann@21383
   922
lemma max_of_mono:
nipkow@23212
   923
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
nipkow@23212
   924
by (simp add: max_def)
haftmann@21383
   925
haftmann@22548
   926
haftmann@22548
   927
subsection {* legacy ML bindings *}
wenzelm@21673
   928
wenzelm@21673
   929
ML {*
haftmann@22548
   930
val monoI = @{thm monoI};
haftmann@22886
   931
*}
wenzelm@21673
   932
nipkow@15524
   933
end