src/HOL/Orderings.thy
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(*  Title:      HOL/Orderings.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* Abstract orderings *}
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theory Orderings
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imports Set Fun
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uses
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  "~~/src/Provers/order.ML"
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begin
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subsection {* Partial orders *}
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class order = ord +
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  assumes less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
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  and order_refl [iff]: "x \<le> x"
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  and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
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  assumes antisym: "x \<le> y \<Longrightarrow> y \<le> 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 \<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 < x"
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by (simp add: less_le)
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lemma le_less: "x \<le> y \<longleftrightarrow> x < 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 \<le> y \<Longrightarrow> x < y \<or> x = y"
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unfolding less_le by blast
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
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unfolding less_le by blast
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lemma less_imp_neq: "x < 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 < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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by auto
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lemma less_imp_not_eq2: "x < 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 \<le> b \<Longrightarrow> a < b"
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by (simp add: less_le)
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < 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 < y \<Longrightarrow> \<not> (y < x)"
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by (simp add: less_le antisym)
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lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < 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 \<le> y \<and> y \<le> x"
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by (blast intro: antisym)
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lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym)
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lemma less_imp_neq: "x < 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 < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < 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 < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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lemma less_imp_triv: "x < y \<Longrightarrow> (y < 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 < b \<Longrightarrow> b < 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 (op \<ge>) (op >)"
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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 \<le> y \<or> y \<le> x"
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begin
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lemma less_linear: "x < y \<or> x = y \<or> y < x"
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lemma le_less_linear: "x \<le> y \<or> y < 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 \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<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 < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
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using less_linear by blast
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lemma not_less: "\<not> x < y \<longleftrightarrow> y \<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 < y) \<longleftrightarrow> (x > 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 \<le> y \<longleftrightarrow> y < 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 < y \<or> y < 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 < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<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 \<le> y \<Longrightarrow> \<not> x < 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 < x \<Longrightarrow> \<not> x < 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 < y \<Longrightarrow> y \<le> x"
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unfolding not_less .
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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < 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 \<le> x \<Longrightarrow> x < 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 (op \<ge>) (op >)"
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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 \<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 \<le> b then b else a)"
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lemma min_le_iff_disj:
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  "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<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 \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<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 < z \<longleftrightarrow> x < z \<or> y < 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 < max x y \<longleftrightarrow> z < x \<or> z < 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 < min x y \<longleftrightarrow> z < x \<and> z < 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 < z \<longleftrightarrow> x < z \<and> y < 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 \<le> j \<longrightarrow> P i) \<and> (\<not> i \<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 \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
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by (simp add: max_def)
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end
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subsection {* Reasoning tools setup *}
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ML {*
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signature ORDERS =
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sig
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  val print_structures: Proof.context -> unit
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  val setup: theory -> theory
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  val order_tac: thm list -> Proof.context -> int -> tactic
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end;
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structure Orders: ORDERS =
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struct
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(** Theory and context data **)
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fun struct_eq ((s1: string, ts1), (s2, ts2)) =
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  (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
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structure Data = GenericDataFun
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(
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  type T = ((string * term list) * Order_Tac.less_arith) list;
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    (* Order structures:
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       identifier of the structure, list of operations and record of theorems
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       needed to set up the transitivity reasoner,
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       identifier and operations identify the structure uniquely. *)
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  val empty = [];
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  val extend = I;
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  fun merge _ = AList.join struct_eq (K fst);
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);
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fun print_structures ctxt =
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  let
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    val structs = Data.get (Context.Proof ctxt);
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    fun pretty_term t = Pretty.block
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      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
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        Pretty.str "::", Pretty.brk 1,
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        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
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    fun pretty_struct ((s, ts), _) = Pretty.block
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      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
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       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
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  in
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    Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
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  end;
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(** Method **)
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fun struct_tac ((s, [eq, le, less]), thms) prems =
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  let
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    fun decomp thy (Trueprop $ t) =
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      let
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        fun excluded t =
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          (* exclude numeric types: linear arithmetic subsumes transitivity *)
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          let val T = type_of t
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          in
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	    T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
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          end;
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	fun rel (bin_op $ t1 $ t2) =
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              if excluded t1 then NONE
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              else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
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              else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
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              else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
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              else NONE
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	  | rel _ = NONE;
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	fun dec (Const (@{const_name Not}, _) $ t) = (case rel t
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	      of NONE => NONE
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	       | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
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          | dec x = rel x;
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      in dec t end;
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  in
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    case s of
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      "order" => Order_Tac.partial_tac decomp thms prems
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    | "linorder" => Order_Tac.linear_tac decomp thms prems
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    | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
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  end
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fun order_tac prems ctxt =
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  FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));
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(** Attribute **)
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parents: 24422
diff changeset
   319
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   320
fun add_struct_thm s tag =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   321
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   322
    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   323
fun del_struct s =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   324
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   325
    (fn _ => Data.map (AList.delete struct_eq s));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   326
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   327
val attribute = Attrib.syntax
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   328
     (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) ||
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   329
          Args.del >> K NONE) --| Args.colon (* FIXME ||
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   330
        Scan.succeed true *) ) -- Scan.lift Args.name --
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   331
      Scan.repeat Args.term
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   332
      >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   333
           | ((NONE, n), ts) => del_struct (n, ts)));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   334
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   335
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   336
(** Diagnostic command **)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   337
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   338
val print = Toplevel.unknown_context o
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   339
  Toplevel.keep (Toplevel.node_case
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   340
    (Context.cases (print_structures o ProofContext.init) print_structures)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   341
    (print_structures o Proof.context_of));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   342
24867
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   343
val _ =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   344
  OuterSyntax.improper_command "print_orders"
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   345
    "print order structures available to transitivity reasoner" OuterKeyword.diag
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   346
    (Scan.succeed (Toplevel.no_timing o print));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   347
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   348
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   349
(** Setup **)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   350
24867
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   351
val setup =
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   352
  Method.add_methods
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   353
    [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #>
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   354
  Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")];
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   355
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   356
end;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   357
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   358
*}
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   359
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   360
setup Orders.setup
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   361
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   362
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   363
text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   364
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   365
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   366
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   367
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   368
(* The type constraint on @{term op =} below is necessary since the operation
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   369
   is not a parameter of the locale. *)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   370
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   371
lemmas
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   372
  [order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   373
  less_irrefl [THEN notE]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   374
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   375
  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   376
  order_refl
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   377
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   378
  [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   379
  less_imp_le
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   380
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   381
  [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   382
  antisym
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   383
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   384
  [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   385
  eq_refl
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   386
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   387
  [order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   388
  sym [THEN eq_refl]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   389
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   390
  [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   391
  less_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   392
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   393
  [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   394
  less_le_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   395
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   396
  [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   397
  le_less_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   398
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   399
  [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   400
  order_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   401
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   402
  [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   403
  le_neq_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   404
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   405
  [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   406
  neq_le_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   407
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   408
  [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   409
  less_imp_neq
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   410
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   411
  [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   412
   eq_neq_eq_imp_neq
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   413
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   414
  [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   415
  not_sym
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   416
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   417
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   418
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   419
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   420
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   421
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   422
lemmas
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   423
  [order del: order "op = :: 'a => 'a => bool" "op <=" "op <"] = _
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   424
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   425
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   426
  [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   427
  less_irrefl [THEN notE]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   428
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   429
  [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   430
  order_refl
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   431
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   432
  [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   433
  less_imp_le
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   434
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   435
  [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   436
  not_less [THEN iffD2]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   437
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   438
  [order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   439
  not_le [THEN iffD2]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   440
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   441
  [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   442
  not_less [THEN iffD1]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   443
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   444
  [order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   445
  not_le [THEN iffD1]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   446
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   447
  [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   448
  antisym
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   449
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   450
  [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   451
  eq_refl
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   452
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   453
  [order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   454
  sym [THEN eq_refl]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   455
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   456
  [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   457
  less_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   458
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   459
  [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   460
  less_le_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   461
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   462
  [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   463
  le_less_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   464
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   465
  [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   466
  order_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   467
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   468
  [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   469
  le_neq_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   470
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   471
  [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   472
  neq_le_trans
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   473
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   474
  [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   475
  less_imp_neq
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   476
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   477
  [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   478
  eq_neq_eq_imp_neq
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   479
lemmas
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   480
  [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   481
  not_sym
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   482
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   483
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   484
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   485
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   486
setup {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   487
let
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   488
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   489
fun prp t thm = (#prop (rep_thm thm) = t);
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   490
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   491
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   492
  let val prems = prems_of_ss ss;
22916
haftmann
parents: 22886
diff changeset
   493
      val less = Const (@{const_name less}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   494
      val t = HOLogic.mk_Trueprop(le $ s $ r);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   495
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   496
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   497
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   498
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   499
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   500
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   501
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   502
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   503
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   504
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   505
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   506
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   507
  let val prems = prems_of_ss ss;
22916
haftmann
parents: 22886
diff changeset
   508
      val le = Const (@{const_name less_eq}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   509
      val t = HOLogic.mk_Trueprop(le $ r $ s);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   510
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   511
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   512
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   513
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   514
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   515
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   516
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   517
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   518
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   519
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   520
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   521
fun add_simprocs procs thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   522
  (Simplifier.change_simpset_of thy (fn ss => ss
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   523
    addsimprocs (map (fn (name, raw_ts, proc) =>
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   524
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   525
fun add_solver name tac thy =
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   526
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
24704
9a95634ab135 Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents: 24641
diff changeset
   527
    (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   528
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   529
in
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   530
  add_simprocs [
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   531
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   532
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   533
     ]
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   534
  #> add_solver "Transitivity" Orders.order_tac
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   535
  (* Adding the transitivity reasoners also as safe solvers showed a slight
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   536
     speed up, but the reasoning strength appears to be not higher (at least
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   537
     no breaking of additional proofs in the entire HOL distribution, as
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   538
     of 5 March 2004, was observed). *)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   539
end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   540
*}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   541
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   542
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   543
subsection {* Dense orders *}
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   544
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   545
class dense_linear_order = linorder + 
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   546
  assumes gt_ex: "\<exists>y. x < y" 
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   547
  and lt_ex: "\<exists>y. y < x"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   548
  and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   549
  (*see further theory Dense_Linear_Order*)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   550
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   551
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   552
lemma interval_empty_iff:
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   553
  "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   554
  by (auto dest: dense)
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   555
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   556
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   557
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   558
subsection {* Name duplicates *}
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   559
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   560
lemmas order_less_le = less_le
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   561
lemmas order_eq_refl = order_class.eq_refl
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   562
lemmas order_less_irrefl = order_class.less_irrefl
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   563
lemmas order_le_less = order_class.le_less
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   564
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   565
lemmas order_less_imp_le = order_class.less_imp_le
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   566
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   567
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   568
lemmas order_neq_le_trans = order_class.neq_le_trans
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   569
lemmas order_le_neq_trans = order_class.le_neq_trans
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   570
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   571
lemmas order_antisym = antisym
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   572
lemmas order_less_not_sym = order_class.less_not_sym
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   573
lemmas order_less_asym = order_class.less_asym
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   574
lemmas order_eq_iff = order_class.eq_iff
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   575
lemmas order_antisym_conv = order_class.antisym_conv
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   576
lemmas order_less_trans = order_class.less_trans
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   577
lemmas order_le_less_trans = order_class.le_less_trans
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   578
lemmas order_less_le_trans = order_class.less_le_trans
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   579
lemmas order_less_imp_not_less = order_class.less_imp_not_less
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   580
lemmas order_less_imp_triv = order_class.less_imp_triv
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   581
lemmas order_less_asym' = order_class.less_asym'
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   582
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   583
lemmas linorder_linear = linear
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   584
lemmas linorder_less_linear = linorder_class.less_linear
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   585
lemmas linorder_le_less_linear = linorder_class.le_less_linear
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   586
lemmas linorder_le_cases = linorder_class.le_cases
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   587
lemmas linorder_not_less = linorder_class.not_less
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   588
lemmas linorder_not_le = linorder_class.not_le
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   589
lemmas linorder_neq_iff = linorder_class.neq_iff
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   590
lemmas linorder_neqE = linorder_class.neqE
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   591
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   592
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   593
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   594
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   595
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   596
subsection {* Bounded quantifiers *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   597
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   598
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   599
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   600
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   601
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   602
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   603
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   604
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   605
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   606
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   607
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   608
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   609
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   610
  "_All_less" :: "[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
   611
  "_Ex_less" :: "[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
   612
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   613
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   614
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   615
  "_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
   616
  "_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
   617
  "_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
   618
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   619
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   620
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   621
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   622
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   623
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   624
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   625
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   626
syntax (HTML output)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   627
  "_All_less" :: "[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
   628
  "_Ex_less" :: "[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
   629
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   630
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   631
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   632
  "_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
   633
  "_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
   634
  "_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
   635
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   636
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   637
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   638
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   639
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   640
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   641
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   642
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   643
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   644
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   645
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   646
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   647
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   648
let
22916
haftmann
parents: 22886
diff changeset
   649
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann
parents: 22886
diff changeset
   650
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
22377
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   651
  val impl = @{const_syntax "op -->"};
61610b1beedf tuned ML setup;
wenzelm
parents: 22348
diff changeset
   652
  val conj = @{const_syntax "op &"};
22916
haftmann
parents: 22886
diff changeset
   653
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   654
  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
   655
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   656
  val trans =
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   657
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   658
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   659
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   660
    ((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
   661
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   662
  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
   663
     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
   664
              | _ => false
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   665
  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
   666
  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
   667
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   668
  fun tr' q = (q,
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   669
    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
   670
      (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
   671
        NONE => raise Match
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   672
      | SOME (l, g) =>
22344
eddeabf16b5d Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents: 22316
diff changeset
   673
          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
   674
          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
   675
          else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   676
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   677
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   678
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   679
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   680
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   681
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   682
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   683
context ord
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   684
begin
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   685
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   686
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   687
  by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   688
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   689
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   690
  by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   691
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   692
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   693
  by (rule subst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   694
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   695
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   696
  by (rule ssubst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   697
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   698
end
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   699
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   700
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
   701
  (!!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
   702
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   703
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   704
  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
   705
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   706
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   707
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   708
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   709
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
   710
  (!!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
   711
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   712
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   713
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   714
  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
   715
  finally (order_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   716
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   717
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   718
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
   719
  (!!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
   720
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   721
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   722
  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
   723
  also assume "f b < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   724
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   725
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   726
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   727
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
   728
  (!!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
   729
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   730
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   731
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   732
  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
   733
  finally (order_le_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   734
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   735
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   736
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
   737
  (!!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
   738
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   739
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   740
  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
   741
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   742
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   743
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   744
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   745
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
   746
  (!!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
   747
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   748
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   749
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   750
  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
   751
  finally (order_less_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   752
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   753
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   754
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
   755
  (!!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
   756
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   757
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   758
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   759
  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
   760
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   761
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   762
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   763
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
   764
  (!!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
   765
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   766
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   767
  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
   768
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   769
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   770
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   771
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   772
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   773
  (!!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
   774
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   775
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   776
  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
   777
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   778
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   779
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   780
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   781
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   782
  (!!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
   783
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   784
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   785
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   786
  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
   787
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   788
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   789
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   790
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   791
  (!!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
   792
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   793
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   794
  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
   795
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   796
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   797
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   798
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   799
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   800
  (!!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
   801
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   802
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   803
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   804
  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
   805
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   806
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   807
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   808
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   809
  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
   810
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   811
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   812
lemmas order_trans_rules [trans] =
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   813
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   814
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   815
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   816
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   817
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   818
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   819
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   820
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   821
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   822
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   823
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   824
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   825
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   826
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   827
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   828
  mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   829
  order_neq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   830
  order_le_neq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   831
  order_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   832
  order_less_asym'
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   833
  order_le_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   834
  order_less_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   835
  order_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   836
  order_antisym
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   837
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   838
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   839
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   841
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   842
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   843
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   844
(* FIXME cleanup *)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   845
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   846
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   847
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   848
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   849
lemma xt1:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   850
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   851
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   852
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   853
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   854
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   855
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   856
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   857
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   858
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   859
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   860
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   861
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   862
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   863
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   864
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   865
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   866
  by auto
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   867
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   868
lemma xt2:
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   869
  "(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
   870
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   871
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   872
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   873
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   874
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   875
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   876
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   877
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   878
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   879
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   880
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   881
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   882
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   883
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   884
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   885
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   886
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   887
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   888
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   889
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   890
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   891
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   892
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   893
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   894
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   895
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   896
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   897
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   898
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   899
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   900
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   901
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   902
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   903
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   904
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   905
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   906
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   907
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   908
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   909
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   910
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   911
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   912
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   913
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   914
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   915
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   916
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   917
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   918
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   919
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   920
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   921
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   922
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   923
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   924
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   925
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   926
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   927
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   928
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   929
subsection {* Order on bool *}
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   930
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   931
instantiation bool :: order 
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   932
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   933
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   934
definition
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   935
  le_bool_def [code func del]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   936
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   937
definition
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   938
  less_bool_def [code func del]: "(P\<Colon>bool) < Q \<longleftrightarrow> P \<le> Q \<and> P \<noteq> Q"
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   939
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   940
instance
22916
haftmann
parents: 22886
diff changeset
   941
  by intro_classes (auto simp add: le_bool_def less_bool_def)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   942
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   943
end
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   944
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   945
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   946
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   947
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   948
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   949
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   950
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   951
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
   952
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   953
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   954
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   955
by (simp add: le_bool_def)
21546
268b6bed0cc8 removed HOL structure
haftmann
parents: 21524
diff changeset
   956
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   957
lemma [code func]:
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   958
  "False \<le> b \<longleftrightarrow> True"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   959
  "True \<le> b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   960
  "False < b \<longleftrightarrow> b"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   961
  "True < b \<longleftrightarrow> False"
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   962
  unfolding le_bool_def less_bool_def by simp_all
ab505d281015 adjusted code lemmas
haftmann
parents: 22344
diff changeset
   963
22424
8a5412121687 *** empty log message ***
haftmann
parents: 22384
diff changeset
   964
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   965
subsection {* Order on sets *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   966
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   967
instance set :: (type) order
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   968
  by (intro_classes,
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   969
      (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
   970
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   971
lemmas basic_trans_rules [trans] =
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   972
  order_trans_rules set_rev_mp set_mp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   973
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   974
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   975
subsection {* Order on functions *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   976
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   977
instantiation "fun" :: (type, ord) ord
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   978
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   979
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   980
definition
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   981
  le_fun_def [code func del]: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   982
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   983
definition
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   984
  less_fun_def [code func del]: "(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> f \<noteq> g"
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   985
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   986
instance ..
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   987
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   988
end
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   989
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   990
instance "fun" :: (type, order) order
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   991
  by default
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   992
    (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
   993
       intro: order_trans order_antisym)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   994
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   995
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
   996
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   997
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   998
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
   999
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1000
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1001
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
  1002
  unfolding le_fun_def by simp
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1003
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1004
text {*
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1005
  Handy introduction and elimination rules for @{text "\<le>"}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1006
  on unary and binary predicates
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1007
*}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1008
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1009
lemma predicate1I [Pure.intro!, intro!]:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1010
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1011
  shows "P \<le> Q"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1012
  apply (rule le_funI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1013
  apply (rule le_boolI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1014
  apply (rule PQ)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1015
  apply assumption
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1016
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1017
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1018
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
  1019
  apply (erule le_funE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1020
  apply (erule le_boolE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1021
  apply assumption+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1022
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1023
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1024
lemma predicate2I [Pure.intro!, intro!]:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1025
  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
  1026
  shows "P \<le> Q"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1027
  apply (rule le_funI)+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1028
  apply (rule le_boolI)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1029
  apply (rule PQ)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1030
  apply assumption
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1031
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1032
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1033
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
  1034
  apply (erule le_funE)+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1035
  apply (erule le_boolE)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1036
  apply assumption+
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1037
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1038
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1039
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1040
  by (rule predicate1D)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1041
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1042
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
  1043
  by (rule predicate2D)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1044
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1045
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1046
subsection {* Monotonicity, least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1047
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1048
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1049
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1050
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1051
definition
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1052
  mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1053
where
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1054
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1055
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1056
lemma monoI [intro?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1057
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1058
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1059
  unfolding mono_def by iprover
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
  1060
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1061
lemma monoD [dest?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1062
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1063
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1064
  unfolding mono_def by iprover
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1065
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1066
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1067
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1068
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1069
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1070
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1071
lemma min_of_mono:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1072
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
25377
dcde128c84a2 Orderings.min/max: no need to qualify consts;
wenzelm
parents: 25207
diff changeset
  1073
  shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1074
  by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1075
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1076
lemma max_of_mono:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1077
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
25377
dcde128c84a2 Orderings.min/max: no need to qualify consts;
wenzelm
parents: 25207
diff changeset
  1078
  shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1079
  by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1080
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1081
end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1082
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1083
lemma LeastI2_order:
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1084
  "[| P (x::'a::order);
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1085
      !!y. P y ==> x <= y;
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1086
      !!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
  1087
   ==> Q (Least P)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1088
apply (unfold Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1089
apply (rule theI2)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1090
  apply (blast intro: order_antisym)+
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1091
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1092
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1093
lemma Least_mono:
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1094
  "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
  1095
    ==> (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
  1096
    -- {* Courtesy of Stephan Merz *}
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1097
  apply clarify
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1098
  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
  1099
  apply (rule LeastI2_order)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1100
  apply (auto elim: monoD intro!: order_antisym)
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1101
  done
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1102
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1103
lemma Least_equality:
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1104
  "[| 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
  1105
apply (simp add: Least_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1106
apply (rule the_equality)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1107
apply (auto intro!: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1108
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1109
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1110
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1111
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1112
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1113
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1114
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1115
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1116
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
  1117
apply (simp add: min_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1118
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1119
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1120
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1121
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
  1122
apply (simp add: max_def)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1123
apply (blast intro: order_antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1124
done
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1125
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
  1126
end