src/HOL/Orderings.thy
author paulson
Wed, 15 Aug 2007 12:52:56 +0200
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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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)
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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 "order_less_irrefl" RS thm "notE";
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  val le_refl = thm "order_refl";
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  val less_imp_le = thm "order_less_imp_le";
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  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
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  val not_leI = thm "linorder_not_le" RS thm "iffD2";
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  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
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  val not_leD = thm "linorder_not_le" RS thm "iffD1";
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  val eqI = thm "order_antisym";
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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_trans = thm "order_less_trans";
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  val less_le_trans = thm "order_less_le_trans";
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  val le_less_trans = thm "order_le_less_trans";
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  val le_trans = thm "order_trans";
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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 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);
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fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
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  let val prems = prems_of_ss ss;
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      val less = Const (@{const_name less}, T);
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      val t = HOLogic.mk_Trueprop(le $ s $ r);
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  in case find_first (prp t) prems of
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       NONE =>
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         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
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         in case find_first (prp t) prems of
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              NONE => NONE
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            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
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         end
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     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
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  end
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  handle THM _ => NONE;
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fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
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  let val prems = prems_of_ss ss;
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      val le = Const (@{const_name less_eq}, T);
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      val t = HOLogic.mk_Trueprop(le $ r $ s);
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  in case find_first (prp t) prems of
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       NONE =>
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         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
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         in case find_first (prp t) prems of
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              NONE => NONE
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            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
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         end
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     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2}))
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  end
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  handle THM _ => NONE;
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fun add_simprocs procs thy =
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  (Simplifier.change_simpset_of thy (fn ss => ss
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    addsimprocs (map (fn (name, raw_ts, proc) =>
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   403
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
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   404
fun add_solver name tac thy =
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   405
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
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    (mk_solver name (K tac))); thy);
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   407
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in
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   409
  add_simprocs [
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   410
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
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       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
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   412
     ]
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  #> add_solver "Trans_linear" Order_Tac.linear_tac
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   414
  #> add_solver "Trans_partial" Order_Tac.partial_tac
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   415
  (* Adding the transitivity reasoners also as safe solvers showed a slight
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   416
     speed up, but the reasoning strength appears to be not higher (at least
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   417
     no breaking of additional proofs in the entire HOL distribution, as
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     of 5 March 2004, was observed). *)
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   419
end
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*}
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subsection {* Bounded quantifiers *}
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   424
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syntax
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  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
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  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
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  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
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  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
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  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
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  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
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  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
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  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
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   435
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   436
syntax (xsymbols)
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  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
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  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
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  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
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  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
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  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
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  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
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  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
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  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
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syntax (HOL)
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  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
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  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
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  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
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  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
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syntax (HTML output)
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  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
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   455
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
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   456
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
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   457
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
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f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   459
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   460
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   461
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   462
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   463
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   464
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   465
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   466
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   467
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   468
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   469
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   470
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   471
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   472
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   473
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   474
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   475
let
22916
haftmann
parents: 22886
diff changeset
   476
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann
parents: 22886
diff changeset
   477
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   478
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   479
  val conj = @{const_syntax "op &"};
22916
haftmann
parents: 22886
diff changeset
   480
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   481
  val less_eq = @{const_syntax less_eq};
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   482
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   483
  val trans =
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   484
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   485
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   486
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   487
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   488
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   489
  fun matches_bound v t = 
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   490
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   491
              | _ => false
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   492
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   493
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   494
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   495
  fun tr' q = (q,
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   496
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   497
      (case AList.lookup (op =) trans (q, c, d) of
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   498
        NONE => raise Match
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   499
      | SOME (l, g) =>
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   500
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   501
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   502
          else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   503
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   504
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   505
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   506
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   507
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   508
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   509
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   510
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   511
by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   512
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   513
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   514
by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   515
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   516
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   517
by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   518
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   519
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   520
by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   521
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   522
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   523
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   524
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   525
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   526
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   527
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   528
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   529
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   530
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   531
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   532
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   533
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   534
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   535
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   536
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   537
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   538
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   539
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   540
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   541
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   542
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   543
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   544
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   545
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   546
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   547
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   548
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   549
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   550
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   551
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   552
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   553
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   554
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   555
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   556
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   557
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   558
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   559
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   560
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   561
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   562
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   563
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   564
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   565
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   566
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   567
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   568
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   569
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   570
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   571
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   572
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   573
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   574
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   575
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   576
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   577
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   578
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   579
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   580
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   581
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   582
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   583
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   584
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   585
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   586
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   587
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   588
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   589
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   590
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   591
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   592
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   593
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   594
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   595
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   596
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   597
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   598
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   599
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   600
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   601
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   602
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   603
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   604
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   605
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   606
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   607
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   608
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   609
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   610
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   611
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   612
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   613
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   614
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   615
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   616
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   617
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   618
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   619
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   620
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   621
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   622
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   623
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   624
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   625
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   626
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   627
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   628
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   629
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   630
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   631
  Note that this list of rules is in reverse order of priorities.
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   632
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   633
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   634
lemmas order_trans_rules [trans] =
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   635
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   636
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   637
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   638
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   639
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   640
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   641
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   642
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   643
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   644
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   645
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   646
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   647
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   648
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   649
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   650
  mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   651
  order_neq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   652
  order_le_neq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   653
  order_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   654
  order_less_asym'
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   655
  order_le_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   656
  order_less_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   657
  order_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   658
  order_antisym
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   659
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   660
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   661
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   662
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   663
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   664
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   665
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   666
(* FIXME cleanup *)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   667
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   668
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   669
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   670
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   671
lemma xt1:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   672
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   673
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   674
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   675
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   676
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   677
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   678
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   679
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   680
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   681
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   682
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   683
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   684
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   685
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   686
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   687
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   688
by auto
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   689
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   690
lemma xt2:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   691
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   692
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   693
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   694
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   695
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   696
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   697
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   698
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   699
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   700
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   701
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   702
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   703
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   704
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   705
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   706
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   707
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   708
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   709
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   710
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   711
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   712
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   713
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   714
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   715
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   716
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   717
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   718
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   719
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   720
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   721
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   722
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   723
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   724
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   725
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   726
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   727
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   728
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   729
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   730
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   731
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   732
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   733
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   734
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   735
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   736
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   737
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   738
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   739
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   740
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   741
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   742
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   743
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   744
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   745
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   746
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   747
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   748
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   749
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   750
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   751
subsection {* Order on bool *}
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   752
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   753
instance bool :: order 
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   754
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   755
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
22916
haftmann
parents: 22886
diff changeset
   756
  by intro_classes (auto simp add: le_bool_def less_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   757
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   758
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   759
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   760
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   761
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   762
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   763
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   764
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   765
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   766
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   767
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   768
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   769
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   770
lemma [code func]:
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   771
  "False \<le> b \<longleftrightarrow> True"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   772
  "True \<le> b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   773
  "False < b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   774
  "True < b \<longleftrightarrow> False"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   775
  unfolding le_bool_def less_bool_def by simp_all
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   776
22424
8a5412121687 *** empty log message ***
haftmann
parents: 22384
diff changeset
   777
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   778
subsection {* Order on sets *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   779
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   780
instance set :: (type) order
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   781
  by (intro_classes,
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   782
      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   783
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   784
lemmas basic_trans_rules [trans] =
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   785
  order_trans_rules set_rev_mp set_mp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   786
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   787
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   788
subsection {* Order on functions *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   789
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   790
instance "fun" :: (type, ord) ord
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   791
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   792
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   793
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   794
lemmas [code func del] = le_fun_def less_fun_def
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   795
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   796
instance "fun" :: (type, order) order
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   797
  by default
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   798
    (auto simp add: le_fun_def less_fun_def expand_fun_eq
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   799
       intro: order_trans order_antisym)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   800
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   801
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   802
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   803
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   804
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   805
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   806
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   807
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   808
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   809
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   810
text {*
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   811
  Handy introduction and elimination rules for @{text "\<le>"}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   812
  on unary and binary predicates
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   813
*}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   814
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   815
lemma predicate1I [Pure.intro!, intro!]:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   816
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   817
  shows "P \<le> Q"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   818
  apply (rule le_funI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   819
  apply (rule le_boolI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   820
  apply (rule PQ)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   821
  apply assumption
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   822
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   823
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   824
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   825
  apply (erule le_funE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   826
  apply (erule le_boolE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   827
  apply assumption+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   828
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   829
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   830
lemma predicate2I [Pure.intro!, intro!]:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   831
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   832
  shows "P \<le> Q"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   833
  apply (rule le_funI)+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   834
  apply (rule le_boolI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   835
  apply (rule PQ)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   836
  apply assumption
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   837
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   838
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   839
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   840
  apply (erule le_funE)+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   841
  apply (erule le_boolE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   842
  apply assumption+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   843
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   844
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   845
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   846
  by (rule predicate1D)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   847
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   848
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   849
  by (rule predicate2D)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   850
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   851
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   852
subsection {* Monotonicity, least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   853
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   854
locale mono =
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   855
  fixes f
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   856
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   857
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   858
lemmas monoI [intro?] = mono.intro
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
   859
  and monoD [dest?] = mono.mono
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   860
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   861
lemma LeastI2_order:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   862
  "[| P (x::'a::order);
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   863
      !!y. P y ==> x <= y;
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   864
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   865
   ==> Q (Least P)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   866
apply (unfold Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   867
apply (rule theI2)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   868
  apply (blast intro: order_antisym)+
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   869
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   870
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   871
lemma Least_mono:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   872
  "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   873
    ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   874
    -- {* Courtesy of Stephan Merz *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   875
  apply clarify
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   876
  apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   877
  apply (rule LeastI2_order)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   878
  apply (auto elim: monoD intro!: order_antisym)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   879
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   880
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   881
lemma Least_equality:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   882
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   883
apply (simp add: Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   884
apply (rule the_equality)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   885
apply (auto intro!: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   886
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   887
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   888
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   889
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   890
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   891
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   892
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   893
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   894
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   895
apply (simp add: min_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   896
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   897
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   898
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   899
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   900
apply (simp add: max_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   901
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   902
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   903
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   904
lemma min_of_mono:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   905
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   906
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   907
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   908
lemma max_of_mono:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   909
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   910
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   911
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   912
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   913
subsection {* legacy ML bindings *}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   914
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   915
ML {*
22548
6ce4bddf3bcb dropped legacy ML bindings
haftmann
parents: 22473
diff changeset
   916
val monoI = @{thm monoI};
22886
cdff6ef76009 moved recfun_codegen.ML to Code_Generator.thy
haftmann
parents: 22841
diff changeset
   917
*}
21673
a664ba87b3aa added basic ML bindings;
wenzelm
parents: 21656
diff changeset
   918
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   919
end