src/HOL/Orderings.thy

generalized

+
−(*  Title:      HOL/Orderings.thy
+
−    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
+
−*)
+
−
+
−section \<open>Abstract orderings\<close>
+
−
+
−theory Orderings
+
−imports HOL
+
−keywords "print_orders" :: diag
+
−begin
+
−
+
−ML_file "~~/src/Provers/order.ML"
+
−ML_file "~~/src/Provers/quasi.ML"  (* FIXME unused? *)
+
−
+
−subsection \<open>Abstract ordering\<close>
+
−
+
−locale ordering =
+
−  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50)
+
−   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50)
+
−  assumes strict_iff_order: "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b"
+
−  assumes refl: "a \<^bold>\<le> a" \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
+
−    and antisym: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b"
+
−    and trans: "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c"
+
−begin
+
−
+
−lemma strict_implies_order:
+
−  "a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b"
+
−  by (simp add: strict_iff_order)
+
−
+
−lemma strict_implies_not_eq:
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−  "a \<^bold>< b \<Longrightarrow> a \<noteq> b"
+
−  by (simp add: strict_iff_order)
+
−
+
−lemma not_eq_order_implies_strict:
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−  "a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b"
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−  by (simp add: strict_iff_order)
+
−
+
−lemma order_iff_strict:
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−  "a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b"
+
−  by (auto simp add: strict_iff_order refl)
+
−
+
−lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
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−  "\<not> a \<^bold>< a"
+
−  by (simp add: strict_iff_order)
+
−
+
−lemma asym:
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−  "a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False"
+
−  by (auto simp add: strict_iff_order intro: antisym)
+
−
+
−lemma strict_trans1:
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−  "a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
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−  by (auto simp add: strict_iff_order intro: trans antisym)
+
−
+
−lemma strict_trans2:
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−  "a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c"
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−  by (auto simp add: strict_iff_order intro: trans antisym)
+
−
+
−lemma strict_trans:
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−  "a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
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−  by (auto intro: strict_trans1 strict_implies_order)
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−
+
−end
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−
+
−text \<open>Alternative introduction rule with bias towards strict order\<close>
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−
+
−lemma ordering_strictI:
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−  fixes less_eq (infix "\<^bold>\<le>" 50)
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−    and less (infix "\<^bold><" 50)
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−  assumes less_eq_less: "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b"
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−    assumes asym: "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a"
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−  assumes irrefl: "\<And>a. \<not> a \<^bold>< a"
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−  assumes trans: "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
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−  shows "ordering less_eq less"
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−proof
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−  fix a b
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−  show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b"
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−    by (auto simp add: less_eq_less asym irrefl)
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−next
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−  fix a
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−  show "a \<^bold>\<le> a"
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−    by (auto simp add: less_eq_less)
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−next
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−  fix a b c
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−  assume "a \<^bold>\<le> b" and "b \<^bold>\<le> c" then show "a \<^bold>\<le> c"
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−    by (auto simp add: less_eq_less intro: trans)
+
−next
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−  fix a b
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−  assume "a \<^bold>\<le> b" and "b \<^bold>\<le> a" then show "a = b"
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−    by (auto simp add: less_eq_less asym)
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−qed
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−
+
−lemma ordering_dualI:
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−  fixes less_eq (infix "\<^bold>\<le>" 50)
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−    and less (infix "\<^bold><" 50)
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−  assumes "ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)"
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−  shows "ordering less_eq less"
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−proof -
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−  from assms interpret ordering "\<lambda>a b. b \<^bold>\<le> a" "\<lambda>a b. b \<^bold>< a" .
+
−  show ?thesis
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−    by standard (auto simp: strict_iff_order refl intro: antisym trans)
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−qed
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−
+
−locale ordering_top = ordering +
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−  fixes top :: "'a"  ("\<^bold>\<top>")
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−  assumes extremum [simp]: "a \<^bold>\<le> \<^bold>\<top>"
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−begin
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−
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−lemma extremum_uniqueI:
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−  "\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>"
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−  by (rule antisym) auto
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−
+
−lemma extremum_unique:
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−  "\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>"
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−  by (auto intro: antisym)
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−
+
−lemma extremum_strict [simp]:
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−  "\<not> (\<^bold>\<top> \<^bold>< a)"
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−  using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
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−
+
−lemma not_eq_extremum:
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−  "a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>"
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−  by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
+
−
+
−end
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−
+
−
+
−subsection \<open>Syntactic orders\<close>
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−
+
−class ord =
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−  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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−    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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−begin
+
−
+
−notation
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−  less_eq  ("'(\<le>')") and
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−  less_eq  ("(_/ \<le> _)"  [51, 51] 50) and
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−  less  ("'(<')") and
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−  less  ("(_/ < _)"  [51, 51] 50)
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−
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−abbreviation (input)
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−  greater_eq  (infix "\<ge>" 50)
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−  where "x \<ge> y \<equiv> y \<le> x"
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−
+
−abbreviation (input)
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−  greater  (infix ">" 50)
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−  where "x > y \<equiv> y < x"
+
−
+
−notation (ASCII)
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−  less_eq  ("'(<=')") and
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−  less_eq  ("(_/ <= _)" [51, 51] 50)
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−
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−notation (input)
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−  greater_eq  (infix ">=" 50)
+
−
+
−end
+
−
+
−
+
−subsection \<open>Quasi orders\<close>
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−
+
−class preorder = ord +
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−  assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
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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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−begin
+
−
+
−text \<open>Reflexivity.\<close>
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−
+
−lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
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−    \<comment> \<open>This form is useful with the classical reasoner.\<close>
+
−by (erule ssubst) (rule order_refl)
+
−
+
−lemma less_irrefl [iff]: "\<not> x < x"
+
−by (simp add: less_le_not_le)
+
−
+
−lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
+
−by (simp add: less_le_not_le)
+
−
+
−
+
−text \<open>Asymmetry.\<close>
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−
+
−lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
+
−by (simp add: less_le_not_le)
+
−
+
−lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
+
−by (drule less_not_sym, erule contrapos_np) simp
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−
+
−
+
−text \<open>Transitivity.\<close>
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−
+
−lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+
−by (auto simp add: less_le_not_le intro: order_trans)
+
−
+
−lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+
−by (auto simp add: less_le_not_le intro: order_trans)
+
−
+
−lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
+
−by (auto simp add: less_le_not_le intro: order_trans)
+
−
+
−
+
−text \<open>Useful for simplification, but too risky to include by default.\<close>
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−
+
−lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
+
−by (blast elim: less_asym)
+
−
+
−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 \<open>Transitivity rules for calculational reasoning\<close>
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−
+
−lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
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−by (rule less_asym)
+
−
+
−
+
−text \<open>Dual order\<close>
+
−
+
−lemma dual_preorder:
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−  "class.preorder (\<ge>) (>)"
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−  by standard (auto simp add: less_le_not_le intro: order_trans)
+
−
+
−end
+
−
+
−
+
−subsection \<open>Partial orders\<close>
+
−
+
−class order = preorder +
+
−  assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
+
−begin
+
−
+
−lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
+
−  by (auto simp add: less_le_not_le intro: antisym)
+
−
+
−sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
+
−proof -
+
−  interpret ordering less_eq less
+
−    by standard (auto intro: antisym order_trans simp add: less_le)
+
−  show "ordering less_eq less"
+
−    by (fact ordering_axioms)
+
−  then show "ordering greater_eq greater"
+
−    by (rule ordering_dualI)
+
−qed
+
−
+
−text \<open>Reflexivity.\<close>
+
−
+
−lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
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−    \<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close>
+
−by (fact order.order_iff_strict)
+
−
+
−lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
+
−by (simp add: less_le)
+
−
+
−
+
−text \<open>Useful for simplification, but too risky to include by default.\<close>
+
−
+
−lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
+
−by auto
+
−
+
−lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
+
−by auto
+
−
+
−
+
−text \<open>Transitivity rules for calculational reasoning\<close>
+
−
+
−lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
+
−by (fact order.not_eq_order_implies_strict)
+
−
+
−lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
+
−by (rule order.not_eq_order_implies_strict)
+
−
+
−
+
−text \<open>Asymmetry.\<close>
+
−
+
−lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
+
−by (blast intro: antisym)
+
−
+
−lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
+
−by (blast intro: antisym)
+
−
+
−lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
+
−by (fact order.strict_implies_not_eq)
+
−
+
−
+
−text \<open>Least value operator\<close>
+
−
+
−definition (in ord)
+
−  Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
+
−  "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
+
−
+
−lemma Least_equality:
+
−  assumes "P x"
+
−    and "\<And>y. P y \<Longrightarrow> x \<le> y"
+
−  shows "Least P = x"
+
−unfolding Least_def by (rule the_equality)
+
−  (blast intro: assms antisym)+
+
−
+
−lemma LeastI2_order:
+
−  assumes "P x"
+
−    and "\<And>y. P y \<Longrightarrow> x \<le> y"
+
−    and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
+
−  shows "Q (Least P)"
+
−unfolding Least_def by (rule theI2)
+
−  (blast intro: assms antisym)+
+
−
+
−text \<open>Greatest value operator\<close>
+
−
+
−definition Greatest :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10) where
+
−"Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<ge> y))"
+
−
+
−lemma GreatestI2_order:
+
−  "\<lbrakk> P x;
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−    \<And>y. P y \<Longrightarrow> x \<ge> y;
+
−    \<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk>
+
−  \<Longrightarrow> Q (Greatest P)"
+
−unfolding Greatest_def
+
−by (rule theI2) (blast intro: antisym)+
+
−
+
−lemma Greatest_equality:
+
−  "\<lbrakk> P x;  \<And>y. P y \<Longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Greatest P = x"
+
−unfolding Greatest_def
+
−by (rule the_equality) (blast intro: antisym)+
+
−
+
−end
+
−
+
−lemma ordering_orderI:
+
−  fixes less_eq (infix "\<^bold>\<le>" 50)
+
−    and less (infix "\<^bold><" 50)
+
−  assumes "ordering less_eq less"
+
−  shows "class.order less_eq less"
+
−proof -
+
−  from assms interpret ordering less_eq less .
+
−  show ?thesis
+
−    by standard (auto intro: antisym trans simp add: refl strict_iff_order)
+
−qed
+
−
+
−lemma order_strictI:
+
−  fixes less (infix "\<sqsubset>" 50)
+
−    and less_eq (infix "\<sqsubseteq>" 50)
+
−  assumes "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b"
+
−    assumes "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a"
+
−  assumes "\<And>a. \<not> a \<sqsubset> a"
+
−  assumes "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> c"
+
−  shows "class.order less_eq less"
+
−  by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)
+
−
+
−context order
+
−begin
+
−
+
−text \<open>Dual order\<close>
+
−
+
−lemma dual_order:
+
−  "class.order (\<ge>) (>)"
+
−  using dual_order.ordering_axioms by (rule ordering_orderI)
+
−
+
−end
+
−
+
−
+
−subsection \<open>Linear (total) orders\<close>
+
−
+
−class linorder = order +
+
−  assumes linear: "x \<le> y \<or> y \<le> x"
+
−begin
+
−
+
−lemma less_linear: "x < y \<or> x = y \<or> y < x"
+
−unfolding less_le using less_le linear by blast
+
−
+
−lemma le_less_linear: "x \<le> y \<or> y < x"
+
−by (simp add: le_less less_linear)
+
−
+
−lemma le_cases [case_names le ge]:
+
−  "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
+
−using linear by blast
+
−
+
−lemma (in linorder) le_cases3:
+
−  "\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P;
+
−    \<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+
−by (blast intro: le_cases)
+
−
+
−lemma linorder_cases [case_names less equal greater]:
+
−  "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
+
−using less_linear by blast
+
−
+
−lemma linorder_wlog[case_names le sym]:
+
−  "(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b"
+
−  by (cases rule: le_cases[of a b]) blast+
+
−
+
−lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
+
−apply (simp add: less_le)
+
−using linear apply (blast intro: antisym)
+
−done
+
−
+
−lemma not_less_iff_gr_or_eq:
+
− "\<not>(x < y) \<longleftrightarrow> (x > y \<or> x = y)"
+
−apply(simp add:not_less le_less)
+
−apply blast
+
−done
+
−
+
−lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
+
−apply (simp add: less_le)
+
−using linear apply (blast intro: antisym)
+
−done
+
−
+
−lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
+
−by (cut_tac x = x and y = y in less_linear, auto)
+
−
+
−lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
+
−by (simp add: neq_iff) blast
+
−
+
−lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
+
−by (blast intro: antisym dest: not_less [THEN iffD1])
+
−
+
−lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
+
−by (blast intro: antisym dest: not_less [THEN iffD1])
+
−
+
−lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
+
−by (blast intro: antisym dest: not_less [THEN iffD1])
+
−
+
−lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
+
−unfolding not_less .
+
−
+
−lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
+
−unfolding not_less .
+
−
+
−lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y"
+
−unfolding not_le .
+
−
+
−lemma linorder_less_wlog[case_names less refl sym]:
+
−     "\<lbrakk>\<And>a b. a < b \<Longrightarrow> P a b;  \<And>a. P a a;  \<And>a b. P b a \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
+
−  using antisym_conv3 by blast
+
−
+
−text \<open>Dual order\<close>
+
−
+
−lemma dual_linorder:
+
−  "class.linorder (\<ge>) (>)"
+
−by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
+
−
+
−end
+
−
+
−
+
−text \<open>Alternative introduction rule with bias towards strict order\<close>
+
−
+
−lemma linorder_strictI:
+
−  fixes less_eq (infix "\<^bold>\<le>" 50)
+
−    and less (infix "\<^bold><" 50)
+
−  assumes "class.order less_eq less"
+
−  assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> b \<^bold>< a"
+
−  shows "class.linorder less_eq less"
+
−proof -
+
−  interpret order less_eq less
+
−    by (fact \<open>class.order less_eq less\<close>)
+
−  show ?thesis
+
−  proof
+
−    fix a b
+
−    show "a \<^bold>\<le> b \<or> b \<^bold>\<le> a"
+
−      using trichotomy by (auto simp add: le_less)
+
−  qed
+
−qed
+
−
+
−
+
−subsection \<open>Reasoning tools setup\<close>
+
−
+
−ML \<open>
+
−signature ORDERS =
+
−sig
+
−  val print_structures: Proof.context -> unit
+
−  val order_tac: Proof.context -> thm list -> int -> tactic
+
−  val add_struct: string * term list -> string -> attribute
+
−  val del_struct: string * term list -> attribute
+
−end;
+
−
+
−structure Orders: ORDERS =
+
−struct
+
−
+
−(* context data *)
+
−
+
−fun struct_eq ((s1: string, ts1), (s2, ts2)) =
+
−  s1 = s2 andalso eq_list (op aconv) (ts1, ts2);
+
−
+
−structure Data = Generic_Data
+
−(
+
−  type T = ((string * term list) * Order_Tac.less_arith) list;
+
−    (* Order structures:
+
−       identifier of the structure, list of operations and record of theorems
+
−       needed to set up the transitivity reasoner,
+
−       identifier and operations identify the structure uniquely. *)
+
−  val empty = [];
+
−  val extend = I;
+
−  fun merge data = AList.join struct_eq (K fst) data;
+
−);
+
−
+
−fun print_structures ctxt =
+
−  let
+
−    val structs = Data.get (Context.Proof ctxt);
+
−    fun pretty_term t = Pretty.block
+
−      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
+
−        Pretty.str "::", Pretty.brk 1,
+
−        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
+
−    fun pretty_struct ((s, ts), _) = Pretty.block
+
−      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
+
−       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
+
−  in
+
−    Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs))
+
−  end;
+
−
+
−val _ =
+
−  Outer_Syntax.command @{command_keyword print_orders}
+
−    "print order structures available to transitivity reasoner"
+
−    (Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of)));
+
−
+
−
+
−(* tactics *)
+
−
+
−fun struct_tac ((s, ops), thms) ctxt facts =
+
−  let
+
−    val [eq, le, less] = ops;
+
−    fun decomp thy (@{const Trueprop} $ t) =
+
−          let
+
−            fun excluded t =
+
−              (* exclude numeric types: linear arithmetic subsumes transitivity *)
+
−              let val T = type_of t
+
−              in
+
−                T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
+
−              end;
+
−            fun rel (bin_op $ t1 $ t2) =
+
−                  if excluded t1 then NONE
+
−                  else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
+
−                  else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
+
−                  else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
+
−                  else NONE
+
−              | rel _ = NONE;
+
−            fun dec (Const (@{const_name Not}, _) $ t) =
+
−                  (case rel t of NONE =>
+
−                    NONE
+
−                  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
+
−              | dec x = rel x;
+
−          in dec t end
+
−      | decomp _ _ = NONE;
+
−  in
+
−    (case s of
+
−      "order" => Order_Tac.partial_tac decomp thms ctxt facts
+
−    | "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
+
−    | _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner"))
+
−  end
+
−
+
−fun order_tac ctxt facts =
+
−  FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt)));
+
−
+
−
+
−(* attributes *)
+
−
+
−fun add_struct s tag =
+
−  Thm.declaration_attribute
+
−    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
+
−fun del_struct s =
+
−  Thm.declaration_attribute
+
−    (fn _ => Data.map (AList.delete struct_eq s));
+
−
+
−end;
+
−\<close>
+
−
+
−attribute_setup order = \<open>
+
−  Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
+
−    Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
+
−    Scan.repeat Args.term
+
−    >> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag
+
−         | ((NONE, n), ts) => Orders.del_struct (n, ts))
+
−\<close> "theorems controlling transitivity reasoner"
+
−
+
−method_setup order = \<open>
+
−  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
+
−\<close> "transitivity reasoner"
+
−
+
−
+
−text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
+
−
+
−context order
+
−begin
+
−
+
−(* The type constraint on @{term (=}) below is necessary since the operation
+
−   is not a parameter of the locale. *)
+
−
+
−declare less_irrefl [THEN notE, order add less_reflE: order "(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "(<=)" "(<)"]
+
−
+
−declare order_refl  [order add le_refl: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_imp_le [order add less_imp_le: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare antisym [order add eqI: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare eq_refl [order add eqD1: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare sym [THEN eq_refl, order add eqD2: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_trans [order add less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_le_trans [order add less_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare le_less_trans [order add le_less_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare order_trans [order add le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare le_neq_trans [order add le_neq_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare neq_le_trans [order add neq_le_trans: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_imp_neq [order add less_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_sym [order add not_sym: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−end
+
−
+
−context linorder
+
−begin
+
−
+
−declare [[order del: order "(=) :: 'a => 'a => bool" "(<=)" "(<)"]]
+
−
+
−declare less_irrefl [THEN notE, order add less_reflE: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare order_refl [order add le_refl: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_imp_le [order add less_imp_le: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_less [THEN iffD2, order add not_lessI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_le [THEN iffD2, order add not_leI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_less [THEN iffD1, order add not_lessD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_le [THEN iffD1, order add not_leD: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare antisym [order add eqI: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare eq_refl [order add eqD1: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare sym [THEN eq_refl, order add eqD2: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_trans [order add less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_le_trans [order add less_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare le_less_trans [order add le_less_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare order_trans [order add le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare le_neq_trans [order add le_neq_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare neq_le_trans [order add neq_le_trans: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare less_imp_neq [order add less_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−declare not_sym [order add not_sym: linorder "(=) :: 'a => 'a => bool" "(<=)" "(<)"]
+
−
+
−end
+
−
+
−setup \<open>
+
−  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
+
−    mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt)))
+
−  (*Adding the transitivity reasoners also as safe solvers showed a slight
+
−    speed up, but the reasoning strength appears to be not higher (at least
+
−    no breaking of additional proofs in the entire HOL distribution, as
+
−    of 5 March 2004, was observed).*)
+
−\<close>
+
−
+
−ML \<open>
+
−local
+
−  fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
+
−in
+
−
+
−fun antisym_le_simproc ctxt ct =
+
−  (case Thm.term_of ct of
+
−    (le as Const (_, T)) $ r $ s =>
+
−     (let
+
−        val prems = Simplifier.prems_of ctxt;
+
−        val less = Const (@{const_name less}, T);
+
−        val t = HOLogic.mk_Trueprop(le $ s $ r);
+
−      in
+
−        (case find_first (prp t) prems of
+
−          NONE =>
+
−            let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
+
−              (case find_first (prp t) prems of
+
−                NONE => NONE
+
−              | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})))
+
−             end
+
−         | SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
+
−      end handle THM _ => NONE)
+
−  | _ => NONE);
+
−
+
−fun antisym_less_simproc ctxt ct =
+
−  (case Thm.term_of ct of
+
−    NotC $ ((less as Const(_,T)) $ r $ s) =>
+
−     (let
+
−       val prems = Simplifier.prems_of ctxt;
+
−       val le = Const (@{const_name less_eq}, T);
+
−       val t = HOLogic.mk_Trueprop(le $ r $ s);
+
−      in
+
−        (case find_first (prp t) prems of
+
−          NONE =>
+
−            let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
+
−              (case find_first (prp t) prems of
+
−                NONE => NONE
+
−              | SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
+
−            end
+
−        | SOME thm => SOME (mk_meta_eq (thm RS @{thm linorder_class.antisym_conv2})))
+
−      end handle THM _ => NONE)
+
−  | _ => NONE);
+
−
+
−end;
+
−\<close>
+
−
+
−simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
+
−simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
+
−
+
−
+
−subsection \<open>Bounded quantifiers\<close>
+
−
+
−syntax (ASCII)
+
−  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
+
−  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
+
−  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
+
−
+
−  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
+
−  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
+
−  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
+
−
+
−  "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3ALL _~=_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3EX _~=_./ _)"  [0, 0, 10] 10)
+
−
+
−syntax
+
−  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
+
−  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
+
−  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
+
−
+
−  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
+
−  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
+
−  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
+
−
+
−  "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<noteq>_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<noteq>_./ _)"  [0, 0, 10] 10)
+
−
+
−syntax (input)
+
−  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
+
−  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
+
−  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
+
−  "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3! _~=_./ _)"  [0, 0, 10] 10)
+
−  "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3? _~=_./ _)"  [0, 0, 10] 10)
+
−
+
−translations
+
−  "\<forall>x<y. P" \<rightharpoonup> "\<forall>x. x < y \<longrightarrow> P"
+
−  "\<exists>x<y. P" \<rightharpoonup> "\<exists>x. x < y \<and> P"
+
−  "\<forall>x\<le>y. P" \<rightharpoonup> "\<forall>x. x \<le> y \<longrightarrow> P"
+
−  "\<exists>x\<le>y. P" \<rightharpoonup> "\<exists>x. x \<le> y \<and> P"
+
−  "\<forall>x>y. P" \<rightharpoonup> "\<forall>x. x > y \<longrightarrow> P"
+
−  "\<exists>x>y. P" \<rightharpoonup> "\<exists>x. x > y \<and> P"
+
−  "\<forall>x\<ge>y. P" \<rightharpoonup> "\<forall>x. x \<ge> y \<longrightarrow> P"
+
−  "\<exists>x\<ge>y. P" \<rightharpoonup> "\<exists>x. x \<ge> y \<and> P"
+
−  "\<forall>x\<noteq>y. P" \<rightharpoonup> "\<forall>x. x \<noteq> y \<longrightarrow> P"
+
−  "\<exists>x\<noteq>y. P" \<rightharpoonup> "\<exists>x. x \<noteq> y \<and> P"
+
−
+
−print_translation \<open>
+
−let
+
−  val All_binder = Mixfix.binder_name @{const_syntax All};
+
−  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
+
−  val impl = @{const_syntax HOL.implies};
+
−  val conj = @{const_syntax HOL.conj};
+
−  val less = @{const_syntax less};
+
−  val less_eq = @{const_syntax less_eq};
+
−
+
−  val trans =
+
−   [((All_binder, impl, less),
+
−    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
+
−    ((All_binder, impl, less_eq),
+
−    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
+
−    ((Ex_binder, conj, less),
+
−    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
+
−    ((Ex_binder, conj, less_eq),
+
−    (@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];
+
−
+
−  fun matches_bound v t =
+
−    (case t of
+
−      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
+
−    | _ => false);
+
−  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
+
−  fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
+
−
+
−  fun tr' q = (q, fn _ =>
+
−    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
+
−        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
+
−        (case AList.lookup (=) trans (q, c, d) of
+
−          NONE => raise Match
+
−        | SOME (l, g) =>
+
−            if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
+
−            else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
+
−            else raise Match)
+
−      | _ => raise Match));
+
−in [tr' All_binder, tr' Ex_binder] end
+
−\<close>
+
−
+
−
+
−subsection \<open>Transitivity reasoning\<close>
+
−
+
−context ord
+
−begin
+
−
+
−lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
+
−  by (rule subst)
+
−
+
−lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
+
−  by (rule ssubst)
+
−
+
−lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
+
−  by (rule subst)
+
−
+
−lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
+
−  by (rule ssubst)
+
−
+
−end
+
−
+
−lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
+
−  (!!x y. x < y ==> f x < f y) ==> f a < c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a < b" hence "f a < f b" by (rule r)
+
−  also assume "f b < c"
+
−  finally (less_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
+
−  (!!x y. x < y ==> f x < f y) ==> a < f c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a < f b"
+
−  also assume "b < c" hence "f b < f c" by (rule r)
+
−  finally (less_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a <= b" hence "f a <= f b" by (rule r)
+
−  also assume "f b < c"
+
−  finally (le_less_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
+
−  (!!x y. x < y ==> f x < f y) ==> a < f c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a <= f b"
+
−  also assume "b < c" hence "f b < f c" by (rule r)
+
−  finally (le_less_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
+
−  (!!x y. x < y ==> f x < f y) ==> f a < c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a < b" hence "f a < f b" by (rule r)
+
−  also assume "f b <= c"
+
−  finally (less_le_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a < f b"
+
−  also assume "b <= c" hence "f b <= f c" by (rule r)
+
−  finally (less_le_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a <= f b"
+
−  also assume "b <= c" hence "f b <= f c" by (rule r)
+
−  finally (order_trans) show ?thesis .
+
−qed
+
−
+
−lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a <= b" hence "f a <= f b" by (rule r)
+
−  also assume "f b <= c"
+
−  finally (order_trans) show ?thesis .
+
−qed
+
−
+
−lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a <= b" hence "f a <= f b" by (rule r)
+
−  also assume "f b = c"
+
−  finally (ord_le_eq_trans) show ?thesis .
+
−qed
+
−
+
−lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
+
−  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
+
−proof -
+
−  assume r: "!!x y. x <= y ==> f x <= f y"
+
−  assume "a = f b"
+
−  also assume "b <= c" hence "f b <= f c" by (rule r)
+
−  finally (ord_eq_le_trans) show ?thesis .
+
−qed
+
−
+
−lemma ord_less_eq_subst: "a < b ==> f b = c ==>
+
−  (!!x y. x < y ==> f x < f y) ==> f a < c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a < b" hence "f a < f b" by (rule r)
+
−  also assume "f b = c"
+
−  finally (ord_less_eq_trans) show ?thesis .
+
−qed
+
−
+
−lemma ord_eq_less_subst: "a = f b ==> b < c ==>
+
−  (!!x y. x < y ==> f x < f y) ==> a < f c"
+
−proof -
+
−  assume r: "!!x y. x < y ==> f x < f y"
+
−  assume "a = f b"
+
−  also assume "b < c" hence "f b < f c" by (rule r)
+
−  finally (ord_eq_less_trans) show ?thesis .
+
−qed
+
−
+
−text \<open>
+
−  Note that this list of rules is in reverse order of priorities.
+
−\<close>
+
−
+
−lemmas [trans] =
+
−  order_less_subst2
+
−  order_less_subst1
+
−  order_le_less_subst2
+
−  order_le_less_subst1
+
−  order_less_le_subst2
+
−  order_less_le_subst1
+
−  order_subst2
+
−  order_subst1
+
−  ord_le_eq_subst
+
−  ord_eq_le_subst
+
−  ord_less_eq_subst
+
−  ord_eq_less_subst
+
−  forw_subst
+
−  back_subst
+
−  rev_mp
+
−  mp
+
−
+
−lemmas (in order) [trans] =
+
−  neq_le_trans
+
−  le_neq_trans
+
−
+
−lemmas (in preorder) [trans] =
+
−  less_trans
+
−  less_asym'
+
−  le_less_trans
+
−  less_le_trans
+
−  order_trans
+
−
+
−lemmas (in order) [trans] =
+
−  antisym
+
−
+
−lemmas (in ord) [trans] =
+
−  ord_le_eq_trans
+
−  ord_eq_le_trans
+
−  ord_less_eq_trans
+
−  ord_eq_less_trans
+
−
+
−lemmas [trans] =
+
−  trans
+
−
+
−lemmas order_trans_rules =
+
−  order_less_subst2
+
−  order_less_subst1
+
−  order_le_less_subst2
+
−  order_le_less_subst1
+
−  order_less_le_subst2
+
−  order_less_le_subst1
+
−  order_subst2
+
−  order_subst1
+
−  ord_le_eq_subst
+
−  ord_eq_le_subst
+
−  ord_less_eq_subst
+
−  ord_eq_less_subst
+
−  forw_subst
+
−  back_subst
+
−  rev_mp
+
−  mp
+
−  neq_le_trans
+
−  le_neq_trans
+
−  less_trans
+
−  less_asym'
+
−  le_less_trans
+
−  less_le_trans
+
−  order_trans
+
−  antisym
+
−  ord_le_eq_trans
+
−  ord_eq_le_trans
+
−  ord_less_eq_trans
+
−  ord_eq_less_trans
+
−  trans
+
−
+
−text \<open>These support proving chains of decreasing inequalities
+
−    a >= b >= c ... in Isar proofs.\<close>
+
−
+
−lemma xt1 [no_atp]:
+
−  "a = b \<Longrightarrow> b > c \<Longrightarrow> a > c"
+
−  "a > b \<Longrightarrow> b = c \<Longrightarrow> a > c"
+
−  "a = b \<Longrightarrow> b \<ge> c \<Longrightarrow> a \<ge> c"
+
−  "a \<ge> b \<Longrightarrow> b = c \<Longrightarrow> a \<ge> c"
+
−  "(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> x \<Longrightarrow> x = y"
+
−  "(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> z \<Longrightarrow> x \<ge> z"
+
−  "(x::'a::order) > y \<Longrightarrow> y \<ge> z \<Longrightarrow> x > z"
+
−  "(x::'a::order) \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z"
+
−  "(a::'a::order) > b \<Longrightarrow> b > a \<Longrightarrow> P"
+
−  "(x::'a::order) > y \<Longrightarrow> y > z \<Longrightarrow> x > z"
+
−  "(a::'a::order) \<ge> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a > b"
+
−  "(a::'a::order) \<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b"
+
−  "a = f b \<Longrightarrow> b > c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> a > f c"
+
−  "a > b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> f a > c"
+
−  "a = f b \<Longrightarrow> b \<ge> c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> a \<ge> f c"
+
−  "a \<ge> b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> f a \<ge> c"
+
−  by auto
+
−
+
−lemma xt2 [no_atp]:
+
−  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
+
−by (subgoal_tac "f b >= f c", force, force)
+
−
+
−lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
+
−    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
+
−by (subgoal_tac "f a >= f b", force, force)
+
−
+
−lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
+
−  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
+
−by (subgoal_tac "f b >= f c", force, force)
+
−
+
−lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
+
−    (!!x y. x > y ==> f x > f y) ==> f a > c"
+
−by (subgoal_tac "f a > f b", force, force)
+
−
+
−lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
+
−    (!!x y. x > y ==> f x > f y) ==> a > f c"
+
−by (subgoal_tac "f b > f c", force, force)
+
−
+
−lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
+
−    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
+
−by (subgoal_tac "f a >= f b", force, force)
+
−
+
−lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
+
−    (!!x y. x > y ==> f x > f y) ==> a > f c"
+
−by (subgoal_tac "f b > f c", force, force)
+
−
+
−lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
+
−    (!!x y. x > y ==> f x > f y) ==> f a > c"
+
−by (subgoal_tac "f a > f b", force, force)
+
−
+
−lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
+
−
+
−(*
+
−  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
+
−  for the wrong thing in an Isar proof.
+
−
+
−  The extra transitivity rules can be used as follows:
+
−
+
−lemma "(a::'a::order) > z"
+
−proof -
+
−  have "a >= b" (is "_ >= ?rhs")
+
−    sorry
+
−  also have "?rhs >= c" (is "_ >= ?rhs")
+
−    sorry
+
−  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
+
−    sorry
+
−  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
+
−    sorry
+
−  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
+
−    sorry
+
−  also (xtrans) have "?rhs > z"
+
−    sorry
+
−  finally (xtrans) show ?thesis .
+
−qed
+
−
+
−  Alternatively, one can use "declare xtrans [trans]" and then
+
−  leave out the "(xtrans)" above.
+
−*)
+
−
+
−
+
−subsection \<open>Monotonicity\<close>
+
−
+
−context order
+
−begin
+
−
+
−definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
+
−  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
+
−
+
−lemma monoI [intro?]:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
+
−  unfolding mono_def by iprover
+
−
+
−lemma monoD [dest?]:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
+
−  unfolding mono_def by iprover
+
−
+
−lemma monoE:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  assumes "mono f"
+
−  assumes "x \<le> y"
+
−  obtains "f x \<le> f y"
+
−proof
+
−  from assms show "f x \<le> f y" by (simp add: mono_def)
+
−qed
+
−
+
−definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
+
−  "antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
+
−
+
−lemma antimonoI [intro?]:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
+
−  unfolding antimono_def by iprover
+
−
+
−lemma antimonoD [dest?]:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
+
−  unfolding antimono_def by iprover
+
−
+
−lemma antimonoE:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  assumes "antimono f"
+
−  assumes "x \<le> y"
+
−  obtains "f x \<ge> f y"
+
−proof
+
−  from assms show "f x \<ge> f y" by (simp add: antimono_def)
+
−qed
+
−
+
−definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
+
−  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
+
−
+
−lemma strict_monoI [intro?]:
+
−  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
+
−  shows "strict_mono f"
+
−  using assms unfolding strict_mono_def by auto
+
−
+
−lemma strict_monoD [dest?]:
+
−  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
+
−  unfolding strict_mono_def by auto
+
−
+
−lemma strict_mono_mono [dest?]:
+
−  assumes "strict_mono f"
+
−  shows "mono f"
+
−proof (rule monoI)
+
−  fix x y
+
−  assume "x \<le> y"
+
−  show "f x \<le> f y"
+
−  proof (cases "x = y")
+
−    case True then show ?thesis by simp
+
−  next
+
−    case False with \<open>x \<le> y\<close> have "x < y" by simp
+
−    with assms strict_monoD have "f x < f y" by auto
+
−    then show ?thesis by simp
+
−  qed
+
−qed
+
−
+
−end
+
−
+
−context linorder
+
−begin
+
−
+
−lemma mono_invE:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  assumes "mono f"
+
−  assumes "f x < f y"
+
−  obtains "x \<le> y"
+
−proof
+
−  show "x \<le> y"
+
−  proof (rule ccontr)
+
−    assume "\<not> x \<le> y"
+
−    then have "y \<le> x" by simp
+
−    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
+
−    with \<open>f x < f y\<close> show False by simp
+
−  qed
+
−qed
+
−
+
−lemma mono_strict_invE:
+
−  fixes f :: "'a \<Rightarrow> 'b::order"
+
−  assumes "mono f"
+
−  assumes "f x < f y"
+
−  obtains "x < y"
+
−proof
+
−  show "x < y"
+
−  proof (rule ccontr)
+
−    assume "\<not> x < y"
+
−    then have "y \<le> x" by simp
+
−    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
+
−    with \<open>f x < f y\<close> show False by simp
+
−  qed
+
−qed
+
−
+
−lemma strict_mono_eq:
+
−  assumes "strict_mono f"
+
−  shows "f x = f y \<longleftrightarrow> x = y"
+
−proof
+
−  assume "f x = f y"
+
−  show "x = y" proof (cases x y rule: linorder_cases)
+
−    case less with assms strict_monoD have "f x < f y" by auto
+
−    with \<open>f x = f y\<close> show ?thesis by simp
+
−  next
+
−    case equal then show ?thesis .
+
−  next
+
−    case greater with assms strict_monoD have "f y < f x" by auto
+
−    with \<open>f x = f y\<close> show ?thesis by simp
+
−  qed
+
−qed simp
+
−
+
−lemma strict_mono_less_eq:
+
−  assumes "strict_mono f"
+
−  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
+
−proof
+
−  assume "x \<le> y"
+
−  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
+
−next
+
−  assume "f x \<le> f y"
+
−  show "x \<le> y" proof (rule ccontr)
+
−    assume "\<not> x \<le> y" then have "y < x" by simp
+
−    with assms strict_monoD have "f y < f x" by auto
+
−    with \<open>f x \<le> f y\<close> show False by simp
+
−  qed
+
−qed
+
−
+
−lemma strict_mono_less:
+
−  assumes "strict_mono f"
+
−  shows "f x < f y \<longleftrightarrow> x < y"
+
−  using assms
+
−    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
+
−
+
−end
+
−
+
−
+
−subsection \<open>min and max -- fundamental\<close>
+
−
+
−definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
+
−  "min a b = (if a \<le> b then a else b)"
+
−
+
−definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
+
−  "max a b = (if a \<le> b then b else a)"
+
−
+
−lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
+
−  by (simp add: min_def)
+
−
+
−lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
+
−  by (simp add: max_def)
+
−
+
−lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
+
−  by (simp add:min_def)
+
−
+
−lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x"
+
−  by (simp add: max_def)
+
−
+
−lemma max_min_same [simp]:
+
−  fixes x y :: "'a :: linorder"
+
−  shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"
+
−by(auto simp add: max_def min_def)
+
−
+
−
+
−subsection \<open>(Unique) top and bottom elements\<close>
+
−
+
−class bot =
+
−  fixes bot :: 'a ("\<bottom>")
+
−
+
−class order_bot = order + bot +
+
−  assumes bot_least: "\<bottom> \<le> a"
+
−begin
+
−
+
−sublocale bot: ordering_top greater_eq greater bot
+
−  by standard (fact bot_least)
+
−
+
−lemma le_bot:
+
−  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
+
−  by (fact bot.extremum_uniqueI)
+
−
+
−lemma bot_unique:
+
−  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
+
−  by (fact bot.extremum_unique)
+
−
+
−lemma not_less_bot:
+
−  "\<not> a < \<bottom>"
+
−  by (fact bot.extremum_strict)
+
−
+
−lemma bot_less:
+
−  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
+
−  by (fact bot.not_eq_extremum)
+
−
+
−lemma max_bot[simp]: "max bot x = x"
+
−by(simp add: max_def bot_unique)
+
−
+
−lemma max_bot2[simp]: "max x bot = x"
+
−by(simp add: max_def bot_unique)
+
−
+
−lemma min_bot[simp]: "min bot x = bot"
+
−by(simp add: min_def bot_unique)
+
−
+
−lemma min_bot2[simp]: "min x bot = bot"
+
−by(simp add: min_def bot_unique)
+
−
+
−end
+
−
+
−class top =
+
−  fixes top :: 'a ("\<top>")
+
−
+
−class order_top = order + top +
+
−  assumes top_greatest: "a \<le> \<top>"
+
−begin
+
−
+
−sublocale top: ordering_top less_eq less top
+
−  by standard (fact top_greatest)
+
−
+
−lemma top_le:
+
−  "\<top> \<le> a \<Longrightarrow> a = \<top>"
+
−  by (fact top.extremum_uniqueI)
+
−
+
−lemma top_unique:
+
−  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
+
−  by (fact top.extremum_unique)
+
−
+
−lemma not_top_less:
+
−  "\<not> \<top> < a"
+
−  by (fact top.extremum_strict)
+
−
+
−lemma less_top:
+
−  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
+
−  by (fact top.not_eq_extremum)
+
−
+
−lemma max_top[simp]: "max top x = top"
+
−by(simp add: max_def top_unique)
+
−
+
−lemma max_top2[simp]: "max x top = top"
+
−by(simp add: max_def top_unique)
+
−
+
−lemma min_top[simp]: "min top x = x"
+
−by(simp add: min_def top_unique)
+
−
+
−lemma min_top2[simp]: "min x top = x"
+
−by(simp add: min_def top_unique)
+
−
+
−end
+
−
+
−
+
−subsection \<open>Dense orders\<close>
+
−
+
−class dense_order = order +
+
−  assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
+
−
+
−class dense_linorder = linorder + dense_order
+
−begin
+
−
+
−lemma dense_le:
+
−  fixes y z :: 'a
+
−  assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
+
−  shows "y \<le> z"
+
−proof (rule ccontr)
+
−  assume "\<not> ?thesis"
+
−  hence "z < y" by simp
+
−  from dense[OF this]
+
−  obtain x where "x < y" and "z < x" by safe
+
−  moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
+
−  ultimately show False by auto
+
−qed
+
−
+
−lemma dense_le_bounded:
+
−  fixes x y z :: 'a
+
−  assumes "x < y"
+
−  assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
+
−  shows "y \<le> z"
+
−proof (rule dense_le)
+
−  fix w assume "w < y"
+
−  from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
+
−  from linear[of u w]
+
−  show "w \<le> z"
+
−  proof (rule disjE)
+
−    assume "u \<le> w"
+
−    from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
+
−    show "w \<le> z" by (rule *)
+
−  next
+
−    assume "w \<le> u"
+
−    from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
+
−    show "w \<le> z" by (rule order_trans)
+
−  qed
+
−qed
+
−
+
−lemma dense_ge:
+
−  fixes y z :: 'a
+
−  assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
+
−  shows "y \<le> z"
+
−proof (rule ccontr)
+
−  assume "\<not> ?thesis"
+
−  hence "z < y" by simp
+
−  from dense[OF this]
+
−  obtain x where "x < y" and "z < x" by safe
+
−  moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
+
−  ultimately show False by auto
+
−qed
+
−
+
−lemma dense_ge_bounded:
+
−  fixes x y z :: 'a
+
−  assumes "z < x"
+
−  assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
+
−  shows "y \<le> z"
+
−proof (rule dense_ge)
+
−  fix w assume "z < w"
+
−  from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
+
−  from linear[of u w]
+
−  show "y \<le> w"
+
−  proof (rule disjE)
+
−    assume "w \<le> u"
+
−    from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
+
−    show "y \<le> w" by (rule *)
+
−  next
+
−    assume "u \<le> w"
+
−    from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
+
−    show "y \<le> w" by (rule order_trans)
+
−  qed
+
−qed
+
−
+
−end
+
−
+
−class no_top = order +
+
−  assumes gt_ex: "\<exists>y. x < y"
+
−
+
−class no_bot = order +
+
−  assumes lt_ex: "\<exists>y. y < x"
+
−
+
−class unbounded_dense_linorder = dense_linorder + no_top + no_bot
+
−
+
−
+
−subsection \<open>Wellorders\<close>
+
−
+
−class wellorder = linorder +
+
−  assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
+
−begin
+
−
+
−lemma wellorder_Least_lemma:
+
−  fixes k :: 'a
+
−  assumes "P k"
+
−  shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
+
−proof -
+
−  have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
+
−  using assms proof (induct k rule: less_induct)
+
−    case (less x) then have "P x" by simp
+
−    show ?case proof (rule classical)
+
−      assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
+
−      have "\<And>y. P y \<Longrightarrow> x \<le> y"
+
−      proof (rule classical)
+
−        fix y
+
−        assume "P y" and "\<not> x \<le> y"
+
−        with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
+
−          by (auto simp add: not_le)
+
−        with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
+
−          by auto
+
−        then show "x \<le> y" by auto
+
−      qed
+
−      with \<open>P x\<close> have Least: "(LEAST a. P a) = x"
+
−        by (rule Least_equality)
+
−      with \<open>P x\<close> show ?thesis by simp
+
−    qed
+
−  qed
+
−  then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
+
−qed
+
−
+
−\<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close>
+
−lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
+
−  by (erule exE) (erule LeastI)
+
−
+
−lemma LeastI2:
+
−  "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
+
−  by (blast intro: LeastI)
+
−
+
−lemma LeastI2_ex:
+
−  "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
+
−  by (blast intro: LeastI_ex)
+
−
+
−lemma LeastI2_wellorder:
+
−  assumes "P a"
+
−  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
+
−  shows "Q (Least P)"
+
−proof (rule LeastI2_order)
+
−  show "P (Least P)" using \<open>P a\<close> by (rule LeastI)
+
−next
+
−  fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
+
−next
+
−  fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
+
−qed
+
−
+
−lemma LeastI2_wellorder_ex:
+
−  assumes "\<exists>x. P x"
+
−  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
+
−  shows "Q (Least P)"
+
−using assms by clarify (blast intro!: LeastI2_wellorder)
+
−
+
−lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
+
−apply (simp add: not_le [symmetric])
+
−apply (erule contrapos_nn)
+
−apply (erule Least_le)
+
−done
+
−
+
−lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs")
+
−proof
+
−  assume ?rhs thus ?lhs by blast
+
−next
+
−  assume H: ?lhs then obtain n where n: "P n" by blast
+
−  let ?x = "Least P"
+
−  { fix m assume m: "m < ?x"
+
−    from not_less_Least[OF m] have "\<not> P m" . }
+
−  with LeastI_ex[OF H] show ?rhs by blast
+
−qed
+
−
+
−end
+
−
+
−
+
−subsection \<open>Order on @{typ bool}\<close>
+
−
+
−instantiation bool :: "{order_bot, order_top, linorder}"
+
−begin
+
−
+
−definition
+
−  le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
+
−
+
−definition
+
−  [simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
+
−
+
−definition
+
−  [simp]: "\<bottom> \<longleftrightarrow> False"
+
−
+
−definition
+
−  [simp]: "\<top> \<longleftrightarrow> True"
+
−
+
−instance proof
+
−qed auto
+
−
+
−end
+
−
+
−lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
+
−  by simp
+
−
+
−lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
+
−  by simp
+
−
+
−lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
+
−  by simp
+
−
+
−lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
+
−  by simp
+
−
+
−lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
+
−  by simp
+
−
+
−lemma top_boolI: \<top>
+
−  by simp
+
−
+
−lemma [code]:
+
−  "False \<le> b \<longleftrightarrow> True"
+
−  "True \<le> b \<longleftrightarrow> b"
+
−  "False < b \<longleftrightarrow> b"
+
−  "True < b \<longleftrightarrow> False"
+
−  by simp_all
+
−
+
−
+
−subsection \<open>Order on @{typ "_ \<Rightarrow> _"}\<close>
+
−
+
−instantiation "fun" :: (type, ord) ord
+
−begin
+
−
+
−definition
+
−  le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
+
−
+
−definition
+
−  "(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instance "fun" :: (type, preorder) preorder proof
+
−qed (auto simp add: le_fun_def less_fun_def
+
−  intro: order_trans antisym)
+
−
+
−instance "fun" :: (type, order) order proof
+
−qed (auto simp add: le_fun_def intro: antisym)
+
−
+
−instantiation "fun" :: (type, bot) bot
+
−begin
+
−
+
−definition
+
−  "\<bottom> = (\<lambda>x. \<bottom>)"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation "fun" :: (type, order_bot) order_bot
+
−begin
+
−
+
−lemma bot_apply [simp, code]:
+
−  "\<bottom> x = \<bottom>"
+
−  by (simp add: bot_fun_def)
+
−
+
−instance proof
+
−qed (simp add: le_fun_def)
+
−
+
−end
+
−
+
−instantiation "fun" :: (type, top) top
+
−begin
+
−
+
−definition
+
−  [no_atp]: "\<top> = (\<lambda>x. \<top>)"
+
−
+
−instance ..
+
−
+
−end
+
−
+
−instantiation "fun" :: (type, order_top) order_top
+
−begin
+
−
+
−lemma top_apply [simp, code]:
+
−  "\<top> x = \<top>"
+
−  by (simp add: top_fun_def)
+
−
+
−instance proof
+
−qed (simp add: le_fun_def)
+
−
+
−end
+
−
+
−lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
+
−  unfolding le_fun_def by simp
+
−
+
−lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
+
−  unfolding le_fun_def by simp
+
−
+
−lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
+
−  by (rule le_funE)
+
−
+
−lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))"
+
−  unfolding mono_def le_fun_def by auto
+
−
+
−
+
−subsection \<open>Order on unary and binary predicates\<close>
+
−
+
−lemma predicate1I:
+
−  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
+
−  shows "P \<le> Q"
+
−  apply (rule le_funI)
+
−  apply (rule le_boolI)
+
−  apply (rule PQ)
+
−  apply assumption
+
−  done
+
−
+
−lemma predicate1D:
+
−  "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
+
−  apply (erule le_funE)
+
−  apply (erule le_boolE)
+
−  apply assumption+
+
−  done
+
−
+
−lemma rev_predicate1D:
+
−  "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
+
−  by (rule predicate1D)
+
−
+
−lemma predicate2I:
+
−  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
+
−  shows "P \<le> Q"
+
−  apply (rule le_funI)+
+
−  apply (rule le_boolI)
+
−  apply (rule PQ)
+
−  apply assumption
+
−  done
+
−
+
−lemma predicate2D:
+
−  "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
+
−  apply (erule le_funE)+
+
−  apply (erule le_boolE)
+
−  apply assumption+
+
−  done
+
−
+
−lemma rev_predicate2D:
+
−  "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
+
−  by (rule predicate2D)
+
−
+
−lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
+
−  by (simp add: bot_fun_def)
+
−
+
−lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
+
−  by (simp add: bot_fun_def)
+
−
+
−lemma top1I: "\<top> x"
+
−  by (simp add: top_fun_def)
+
−
+
−lemma top2I: "\<top> x y"
+
−  by (simp add: top_fun_def)
+
−
+
−
+
−subsection \<open>Name duplicates\<close>
+
−
+
−lemmas order_eq_refl = preorder_class.eq_refl
+
−lemmas order_less_irrefl = preorder_class.less_irrefl
+
−lemmas order_less_imp_le = preorder_class.less_imp_le
+
−lemmas order_less_not_sym = preorder_class.less_not_sym
+
−lemmas order_less_asym = preorder_class.less_asym
+
−lemmas order_less_trans = preorder_class.less_trans
+
−lemmas order_le_less_trans = preorder_class.le_less_trans
+
−lemmas order_less_le_trans = preorder_class.less_le_trans
+
−lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
+
−lemmas order_less_imp_triv = preorder_class.less_imp_triv
+
−lemmas order_less_asym' = preorder_class.less_asym'
+
−
+
−lemmas order_less_le = order_class.less_le
+
−lemmas order_le_less = order_class.le_less
+
−lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
+
−lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
+
−lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
+
−lemmas order_neq_le_trans = order_class.neq_le_trans
+
−lemmas order_le_neq_trans = order_class.le_neq_trans
+
−lemmas order_antisym = order_class.antisym
+
−lemmas order_eq_iff = order_class.eq_iff
+
−lemmas order_antisym_conv = order_class.antisym_conv
+
−
+
−lemmas linorder_linear = linorder_class.linear
+
−lemmas linorder_less_linear = linorder_class.less_linear
+
−lemmas linorder_le_less_linear = linorder_class.le_less_linear
+
−lemmas linorder_le_cases = linorder_class.le_cases
+
−lemmas linorder_not_less = linorder_class.not_less
+
−lemmas linorder_not_le = linorder_class.not_le
+
−lemmas linorder_neq_iff = linorder_class.neq_iff
+
−lemmas linorder_neqE = linorder_class.neqE
+
−lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
+
−lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
+
−lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
+
−
+
−end