src/HOL/Orderings.thy
author wenzelm
Mon Nov 09 15:48:17 2015 +0100 (2015-11-09)
changeset 61605 1bf7b186542e
parent 61378 3e04c9ca001a
child 61630 608520e0e8e2
permissions -rw-r--r--
qualifier is mandatory by default;
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(*  Title:      HOL/Orderings.thy
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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section \<open>Abstract orderings\<close>
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theory Orderings
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imports HOL
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keywords "print_orders" :: diag
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begin
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ML_file "~~/src/Provers/order.ML"
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ML_file "~~/src/Provers/quasi.ML"  (* FIXME unused? *)
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subsection \<open>Abstract ordering\<close>
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locale ordering =
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50)
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   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50)
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  assumes strict_iff_order: "a \<prec> b \<longleftrightarrow> a \<preceq> b \<and> a \<noteq> b"
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  assumes refl: "a \<preceq> a" -- \<open>not @{text iff}: makes problems due to multiple (dual) interpretations\<close>
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    and antisym: "a \<preceq> b \<Longrightarrow> b \<preceq> a \<Longrightarrow> a = b"
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    and trans: "a \<preceq> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<preceq> c"
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begin
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lemma strict_implies_order:
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  "a \<prec> b \<Longrightarrow> a \<preceq> b"
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  by (simp add: strict_iff_order)
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lemma strict_implies_not_eq:
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  "a \<prec> b \<Longrightarrow> a \<noteq> b"
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  by (simp add: strict_iff_order)
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lemma not_eq_order_implies_strict:
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  "a \<noteq> b \<Longrightarrow> a \<preceq> b \<Longrightarrow> a \<prec> b"
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  by (simp add: strict_iff_order)
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lemma order_iff_strict:
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  "a \<preceq> b \<longleftrightarrow> a \<prec> b \<or> a = b"
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  by (auto simp add: strict_iff_order refl)
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lemma irrefl: -- \<open>not @{text iff}: makes problems due to multiple (dual) interpretations\<close>
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  "\<not> a \<prec> a"
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  by (simp add: strict_iff_order)
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lemma asym:
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  "a \<prec> b \<Longrightarrow> b \<prec> a \<Longrightarrow> False"
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  by (auto simp add: strict_iff_order intro: antisym)
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lemma strict_trans1:
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  "a \<preceq> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c"
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  by (auto simp add: strict_iff_order intro: trans antisym)
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lemma strict_trans2:
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  "a \<prec> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<prec> c"
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  by (auto simp add: strict_iff_order intro: trans antisym)
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lemma strict_trans:
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  "a \<prec> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c"
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  by (auto intro: strict_trans1 strict_implies_order)
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end
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locale ordering_top = ordering +
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  fixes top :: "'a"
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  assumes extremum [simp]: "a \<preceq> top"
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begin
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lemma extremum_uniqueI:
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  "top \<preceq> a \<Longrightarrow> a = top"
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  by (rule antisym) auto
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lemma extremum_unique:
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  "top \<preceq> a \<longleftrightarrow> a = top"
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  by (auto intro: antisym)
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lemma extremum_strict [simp]:
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  "\<not> (top \<prec> 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> top \<longleftrightarrow> a \<prec> top"
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  by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
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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
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notation
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  less_eq  ("op <=") and
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  less_eq  ("(_/ <= _)" [51, 51] 50) and
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  less  ("op <") and
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  less  ("(_/ < _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater_eq  (infix ">=" 50) where
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  "x >= y \<equiv> y <= x"
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notation (input)
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  greater_eq  (infix "\<ge>" 50)
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abbreviation (input)
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  greater  (infix ">" 50) where
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  "x > y \<equiv> y < x"
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end
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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
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text \<open>Reflexivity.\<close>
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
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    -- \<open>This form is useful with the classical reasoner.\<close>
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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_not_le)
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
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unfolding less_le_not_le by blast
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text \<open>Asymmetry.\<close>
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lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
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by (simp add: less_le_not_le)
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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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text \<open>Transitivity.\<close>
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lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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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"
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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 \<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)
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text \<open>Dual order\<close>
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lemma dual_preorder:
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  "class.preorder (op \<ge>) (op >)"
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proof qed (auto simp add: less_le_not_le intro: order_trans)
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end
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subsection \<open>Partial orders\<close>
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class order = preorder +
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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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lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
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  by (auto simp add: less_le_not_le intro: antisym)
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sublocale order: ordering less_eq less +  dual_order: ordering greater_eq greater
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  by standard (auto intro: antisym order_trans simp add: less_le)
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text \<open>Reflexivity.\<close>
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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
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    -- \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close>
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by (fact order.order_iff_strict)
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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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text \<open>Useful for simplification, but too risky to include by default.\<close>
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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 \<open>Transitivity rules for calculational reasoning\<close>
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
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by (fact order.not_eq_order_implies_strict)
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
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by (rule order.not_eq_order_implies_strict)
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text \<open>Asymmetry.\<close>
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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 (fact order.strict_implies_not_eq)
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text \<open>Least value operator\<close>
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definition (in ord)
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  Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
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  "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
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lemma Least_equality:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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  shows "Least P = x"
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unfolding Least_def by (rule the_equality)
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  (blast intro: assms antisym)+
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lemma LeastI2_order:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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    and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
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  shows "Q (Least P)"
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unfolding Least_def by (rule theI2)
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  (blast intro: assms antisym)+
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text \<open>Dual order\<close>
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lemma dual_order:
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  "class.order (op \<ge>) (op >)"
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym)
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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 order_strictI:
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  fixes less (infix "\<sqsubset>" 50)
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    and less_eq (infix "\<sqsubseteq>" 50)
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  assumes less_eq_less: "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b"
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    assumes asym: "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a"
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  assumes irrefl: "\<And>a. \<not> a \<sqsubset> a"
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  assumes trans: "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> c"
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  shows "class.order less_eq less"
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proof
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  fix a b
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  show "a \<sqsubset> b \<longleftrightarrow> a \<sqsubseteq> b \<and> \<not> b \<sqsubseteq> a"
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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 \<sqsubseteq> 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 \<sqsubseteq> b" and "b \<sqsubseteq> c" then show "a \<sqsubseteq> c"
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    by (auto simp add: less_eq_less intro: trans)
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next
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  fix a b
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  assume "a \<sqsubseteq> b" and "b \<sqsubseteq> 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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subsection \<open>Linear (total) orders\<close>
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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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unfolding less_le using less_le linear by blast
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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 linorder_wlog[case_names le sym]:
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  "(\<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"
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  by (cases rule: le_cases[of a b]) 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"
nipkow@23212
   342
by (simp add: neq_iff) blast
nipkow@15524
   343
haftmann@25062
   344
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
nipkow@23212
   345
by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524
   346
haftmann@25062
   347
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
nipkow@23212
   348
by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524
   349
haftmann@25062
   350
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
nipkow@23212
   351
by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524
   352
haftmann@25062
   353
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
nipkow@23212
   354
unfolding not_less .
paulson@16796
   355
haftmann@25062
   356
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
nipkow@23212
   357
unfolding not_less .
paulson@16796
   358
paulson@16796
   359
(*FIXME inappropriate name (or delete altogether)*)
haftmann@25062
   360
lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
nipkow@23212
   361
unfolding not_le .
haftmann@21248
   362
wenzelm@60758
   363
text \<open>Dual order\<close>
haftmann@22916
   364
haftmann@26014
   365
lemma dual_linorder:
haftmann@36635
   366
  "class.linorder (op \<ge>) (op >)"
haftmann@36635
   367
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
haftmann@22916
   368
haftmann@21248
   369
end
haftmann@21248
   370
haftmann@23948
   371
wenzelm@60758
   372
text \<open>Alternative introduction rule with bias towards strict order\<close>
haftmann@56545
   373
haftmann@56545
   374
lemma linorder_strictI:
haftmann@56545
   375
  fixes less (infix "\<sqsubset>" 50)
haftmann@56545
   376
    and less_eq (infix "\<sqsubseteq>" 50)
haftmann@56545
   377
  assumes "class.order less_eq less"
haftmann@56545
   378
  assumes trichotomy: "\<And>a b. a \<sqsubset> b \<or> a = b \<or> b \<sqsubset> a"
haftmann@56545
   379
  shows "class.linorder less_eq less"
haftmann@56545
   380
proof -
haftmann@56545
   381
  interpret order less_eq less
wenzelm@60758
   382
    by (fact \<open>class.order less_eq less\<close>)
haftmann@56545
   383
  show ?thesis
haftmann@56545
   384
  proof
haftmann@56545
   385
    fix a b
haftmann@56545
   386
    show "a \<sqsubseteq> b \<or> b \<sqsubseteq> a"
haftmann@56545
   387
      using trichotomy by (auto simp add: le_less)
haftmann@56545
   388
  qed
haftmann@56545
   389
qed
haftmann@56545
   390
haftmann@56545
   391
wenzelm@60758
   392
subsection \<open>Reasoning tools setup\<close>
haftmann@21083
   393
wenzelm@60758
   394
ML \<open>
ballarin@24641
   395
signature ORDERS =
ballarin@24641
   396
sig
ballarin@24641
   397
  val print_structures: Proof.context -> unit
wenzelm@32215
   398
  val order_tac: Proof.context -> thm list -> int -> tactic
wenzelm@58826
   399
  val add_struct: string * term list -> string -> attribute
wenzelm@58826
   400
  val del_struct: string * term list -> attribute
ballarin@24641
   401
end;
haftmann@21091
   402
ballarin@24641
   403
structure Orders: ORDERS =
haftmann@21248
   404
struct
ballarin@24641
   405
wenzelm@56508
   406
(* context data *)
ballarin@24641
   407
ballarin@24641
   408
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
wenzelm@56508
   409
  s1 = s2 andalso eq_list (op aconv) (ts1, ts2);
ballarin@24641
   410
wenzelm@33519
   411
structure Data = Generic_Data
ballarin@24641
   412
(
ballarin@24641
   413
  type T = ((string * term list) * Order_Tac.less_arith) list;
ballarin@24641
   414
    (* Order structures:
ballarin@24641
   415
       identifier of the structure, list of operations and record of theorems
ballarin@24641
   416
       needed to set up the transitivity reasoner,
ballarin@24641
   417
       identifier and operations identify the structure uniquely. *)
ballarin@24641
   418
  val empty = [];
ballarin@24641
   419
  val extend = I;
wenzelm@33519
   420
  fun merge data = AList.join struct_eq (K fst) data;
ballarin@24641
   421
);
ballarin@24641
   422
ballarin@24641
   423
fun print_structures ctxt =
ballarin@24641
   424
  let
ballarin@24641
   425
    val structs = Data.get (Context.Proof ctxt);
ballarin@24641
   426
    fun pretty_term t = Pretty.block
wenzelm@24920
   427
      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
ballarin@24641
   428
        Pretty.str "::", Pretty.brk 1,
wenzelm@24920
   429
        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
ballarin@24641
   430
    fun pretty_struct ((s, ts), _) = Pretty.block
ballarin@24641
   431
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
ballarin@24641
   432
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
ballarin@24641
   433
  in
wenzelm@51579
   434
    Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs))
ballarin@24641
   435
  end;
ballarin@24641
   436
wenzelm@56508
   437
val _ =
wenzelm@59936
   438
  Outer_Syntax.command @{command_keyword print_orders}
wenzelm@56508
   439
    "print order structures available to transitivity reasoner"
wenzelm@60097
   440
    (Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of)));
haftmann@21091
   441
wenzelm@56508
   442
wenzelm@56508
   443
(* tactics *)
wenzelm@56508
   444
wenzelm@56508
   445
fun struct_tac ((s, ops), thms) ctxt facts =
ballarin@24641
   446
  let
wenzelm@56508
   447
    val [eq, le, less] = ops;
berghofe@30107
   448
    fun decomp thy (@{const Trueprop} $ t) =
wenzelm@56508
   449
          let
wenzelm@56508
   450
            fun excluded t =
wenzelm@56508
   451
              (* exclude numeric types: linear arithmetic subsumes transitivity *)
wenzelm@56508
   452
              let val T = type_of t
wenzelm@56508
   453
              in
wenzelm@56508
   454
                T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
wenzelm@56508
   455
              end;
wenzelm@56508
   456
            fun rel (bin_op $ t1 $ t2) =
wenzelm@56508
   457
                  if excluded t1 then NONE
wenzelm@56508
   458
                  else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
wenzelm@56508
   459
                  else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
wenzelm@56508
   460
                  else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
wenzelm@56508
   461
                  else NONE
wenzelm@56508
   462
              | rel _ = NONE;
wenzelm@56508
   463
            fun dec (Const (@{const_name Not}, _) $ t) =
wenzelm@56508
   464
                  (case rel t of NONE =>
wenzelm@56508
   465
                    NONE
wenzelm@56508
   466
                  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
wenzelm@56508
   467
              | dec x = rel x;
wenzelm@56508
   468
          in dec t end
wenzelm@56508
   469
      | decomp _ _ = NONE;
ballarin@24641
   470
  in
wenzelm@56508
   471
    (case s of
wenzelm@56508
   472
      "order" => Order_Tac.partial_tac decomp thms ctxt facts
wenzelm@56508
   473
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
wenzelm@56508
   474
    | _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner"))
ballarin@24641
   475
  end
ballarin@24641
   476
wenzelm@56508
   477
fun order_tac ctxt facts =
wenzelm@56508
   478
  FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt)));
ballarin@24641
   479
ballarin@24641
   480
wenzelm@56508
   481
(* attributes *)
ballarin@24641
   482
wenzelm@58826
   483
fun add_struct s tag =
ballarin@24641
   484
  Thm.declaration_attribute
ballarin@24641
   485
    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
ballarin@24641
   486
fun del_struct s =
ballarin@24641
   487
  Thm.declaration_attribute
ballarin@24641
   488
    (fn _ => Data.map (AList.delete struct_eq s));
ballarin@24641
   489
haftmann@21091
   490
end;
wenzelm@60758
   491
\<close>
haftmann@21091
   492
wenzelm@60758
   493
attribute_setup order = \<open>
wenzelm@58826
   494
  Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
wenzelm@58826
   495
    Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
wenzelm@58826
   496
    Scan.repeat Args.term
wenzelm@58826
   497
    >> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag
wenzelm@58826
   498
         | ((NONE, n), ts) => Orders.del_struct (n, ts))
wenzelm@60758
   499
\<close> "theorems controlling transitivity reasoner"
wenzelm@58826
   500
wenzelm@60758
   501
method_setup order = \<open>
wenzelm@47432
   502
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
wenzelm@60758
   503
\<close> "transitivity reasoner"
ballarin@24641
   504
ballarin@24641
   505
wenzelm@60758
   506
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
ballarin@24641
   507
haftmann@25076
   508
context order
haftmann@25076
   509
begin
haftmann@25076
   510
ballarin@24641
   511
(* The type constraint on @{term op =} below is necessary since the operation
ballarin@24641
   512
   is not a parameter of the locale. *)
haftmann@25076
   513
haftmann@27689
   514
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]
haftmann@27689
   515
  
haftmann@27689
   516
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   517
  
haftmann@27689
   518
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   519
  
haftmann@27689
   520
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   521
haftmann@27689
   522
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   523
haftmann@27689
   524
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   525
haftmann@27689
   526
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   527
  
haftmann@27689
   528
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   529
  
haftmann@27689
   530
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   531
haftmann@27689
   532
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   533
haftmann@27689
   534
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   535
haftmann@27689
   536
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   537
haftmann@27689
   538
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   539
haftmann@27689
   540
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   541
haftmann@27689
   542
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]
ballarin@24641
   543
haftmann@25076
   544
end
haftmann@25076
   545
haftmann@25076
   546
context linorder
haftmann@25076
   547
begin
ballarin@24641
   548
haftmann@27689
   549
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
haftmann@27689
   550
haftmann@27689
   551
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   552
haftmann@27689
   553
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   554
haftmann@27689
   555
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   556
haftmann@27689
   557
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   558
haftmann@27689
   559
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   560
haftmann@27689
   561
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   562
haftmann@27689
   563
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   564
haftmann@27689
   565
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   566
haftmann@27689
   567
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@25076
   568
haftmann@27689
   569
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   570
haftmann@27689
   571
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   572
haftmann@27689
   573
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   574
haftmann@27689
   575
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   576
haftmann@27689
   577
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   578
haftmann@27689
   579
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   580
haftmann@27689
   581
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   582
haftmann@27689
   583
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   584
haftmann@27689
   585
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   586
haftmann@27689
   587
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
ballarin@24641
   588
haftmann@25076
   589
end
haftmann@25076
   590
wenzelm@60758
   591
setup \<open>
wenzelm@56509
   592
  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
wenzelm@56509
   593
    mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt)))
wenzelm@56509
   594
  (*Adding the transitivity reasoners also as safe solvers showed a slight
wenzelm@56509
   595
    speed up, but the reasoning strength appears to be not higher (at least
wenzelm@56509
   596
    no breaking of additional proofs in the entire HOL distribution, as
wenzelm@56509
   597
    of 5 March 2004, was observed).*)
wenzelm@60758
   598
\<close>
nipkow@15524
   599
wenzelm@60758
   600
ML \<open>
wenzelm@56509
   601
local
wenzelm@56509
   602
  fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
wenzelm@56509
   603
in
nipkow@15524
   604
wenzelm@56509
   605
fun antisym_le_simproc ctxt ct =
wenzelm@59582
   606
  (case Thm.term_of ct of
wenzelm@56509
   607
    (le as Const (_, T)) $ r $ s =>
wenzelm@56509
   608
     (let
wenzelm@56509
   609
        val prems = Simplifier.prems_of ctxt;
wenzelm@56509
   610
        val less = Const (@{const_name less}, T);
wenzelm@56509
   611
        val t = HOLogic.mk_Trueprop(le $ s $ r);
wenzelm@56509
   612
      in
wenzelm@56509
   613
        (case find_first (prp t) prems of
wenzelm@56509
   614
          NONE =>
wenzelm@56509
   615
            let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
wenzelm@56509
   616
              (case find_first (prp t) prems of
wenzelm@56509
   617
                NONE => NONE
wenzelm@56509
   618
              | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})))
wenzelm@56509
   619
             end
wenzelm@56509
   620
         | SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
wenzelm@56509
   621
      end handle THM _ => NONE)
wenzelm@56509
   622
  | _ => NONE);
nipkow@15524
   623
wenzelm@56509
   624
fun antisym_less_simproc ctxt ct =
wenzelm@59582
   625
  (case Thm.term_of ct of
wenzelm@56509
   626
    NotC $ ((less as Const(_,T)) $ r $ s) =>
wenzelm@56509
   627
     (let
wenzelm@56509
   628
       val prems = Simplifier.prems_of ctxt;
wenzelm@56509
   629
       val le = Const (@{const_name less_eq}, T);
wenzelm@56509
   630
       val t = HOLogic.mk_Trueprop(le $ r $ s);
wenzelm@56509
   631
      in
wenzelm@56509
   632
        (case find_first (prp t) prems of
wenzelm@56509
   633
          NONE =>
wenzelm@56509
   634
            let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
wenzelm@56509
   635
              (case find_first (prp t) prems of
wenzelm@56509
   636
                NONE => NONE
wenzelm@56509
   637
              | SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
wenzelm@56509
   638
            end
wenzelm@56509
   639
        | SOME thm => SOME (mk_meta_eq (thm RS @{thm linorder_class.antisym_conv2})))
wenzelm@56509
   640
      end handle THM _ => NONE)
wenzelm@56509
   641
  | _ => NONE);
haftmann@21083
   642
wenzelm@56509
   643
end;
wenzelm@60758
   644
\<close>
nipkow@15524
   645
wenzelm@56509
   646
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
wenzelm@56509
   647
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
wenzelm@56509
   648
nipkow@15524
   649
wenzelm@60758
   650
subsection \<open>Bounded quantifiers\<close>
haftmann@21083
   651
haftmann@21083
   652
syntax
wenzelm@21180
   653
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   654
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   655
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   656
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   657
wenzelm@21180
   658
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   659
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   660
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   661
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   662
haftmann@21083
   663
syntax (xsymbols)
wenzelm@21180
   664
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   665
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   666
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   667
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   668
wenzelm@21180
   669
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   670
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   671
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   672
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   673
haftmann@21083
   674
syntax (HOL)
wenzelm@21180
   675
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   676
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   677
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   678
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   679
haftmann@21083
   680
translations
haftmann@21083
   681
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   682
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   683
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   684
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   685
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   686
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   687
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   688
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   689
wenzelm@60758
   690
print_translation \<open>
haftmann@21083
   691
let
wenzelm@42287
   692
  val All_binder = Mixfix.binder_name @{const_syntax All};
wenzelm@42287
   693
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
haftmann@38786
   694
  val impl = @{const_syntax HOL.implies};
haftmann@38795
   695
  val conj = @{const_syntax HOL.conj};
haftmann@22916
   696
  val less = @{const_syntax less};
haftmann@22916
   697
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   698
wenzelm@21180
   699
  val trans =
wenzelm@35115
   700
   [((All_binder, impl, less),
wenzelm@35115
   701
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
wenzelm@35115
   702
    ((All_binder, impl, less_eq),
wenzelm@35115
   703
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
wenzelm@35115
   704
    ((Ex_binder, conj, less),
wenzelm@35115
   705
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
wenzelm@35115
   706
    ((Ex_binder, conj, less_eq),
wenzelm@35115
   707
    (@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];
wenzelm@21180
   708
wenzelm@35115
   709
  fun matches_bound v t =
wenzelm@35115
   710
    (case t of
wenzelm@35364
   711
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
wenzelm@35115
   712
    | _ => false);
wenzelm@35115
   713
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
wenzelm@49660
   714
  fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
wenzelm@21180
   715
wenzelm@52143
   716
  fun tr' q = (q, fn _ =>
wenzelm@52143
   717
    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
wenzelm@35364
   718
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@35115
   719
        (case AList.lookup (op =) trans (q, c, d) of
wenzelm@35115
   720
          NONE => raise Match
wenzelm@35115
   721
        | SOME (l, g) =>
wenzelm@49660
   722
            if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
wenzelm@49660
   723
            else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
wenzelm@35115
   724
            else raise Match)
wenzelm@52143
   725
      | _ => raise Match));
wenzelm@21524
   726
in [tr' All_binder, tr' Ex_binder] end
wenzelm@60758
   727
\<close>
haftmann@21083
   728
haftmann@21083
   729
wenzelm@60758
   730
subsection \<open>Transitivity reasoning\<close>
haftmann@21383
   731
haftmann@25193
   732
context ord
haftmann@25193
   733
begin
haftmann@21383
   734
haftmann@25193
   735
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
haftmann@25193
   736
  by (rule subst)
haftmann@21383
   737
haftmann@25193
   738
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
haftmann@25193
   739
  by (rule ssubst)
haftmann@21383
   740
haftmann@25193
   741
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
haftmann@25193
   742
  by (rule subst)
haftmann@25193
   743
haftmann@25193
   744
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
haftmann@25193
   745
  by (rule ssubst)
haftmann@25193
   746
haftmann@25193
   747
end
haftmann@21383
   748
haftmann@21383
   749
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   750
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   751
proof -
haftmann@21383
   752
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   753
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   754
  also assume "f b < c"
haftmann@34250
   755
  finally (less_trans) show ?thesis .
haftmann@21383
   756
qed
haftmann@21383
   757
haftmann@21383
   758
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   759
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   760
proof -
haftmann@21383
   761
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   762
  assume "a < f b"
haftmann@21383
   763
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@34250
   764
  finally (less_trans) show ?thesis .
haftmann@21383
   765
qed
haftmann@21383
   766
haftmann@21383
   767
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   768
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   769
proof -
haftmann@21383
   770
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   771
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   772
  also assume "f b < c"
haftmann@34250
   773
  finally (le_less_trans) show ?thesis .
haftmann@21383
   774
qed
haftmann@21383
   775
haftmann@21383
   776
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   777
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   778
proof -
haftmann@21383
   779
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   780
  assume "a <= f b"
haftmann@21383
   781
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@34250
   782
  finally (le_less_trans) show ?thesis .
haftmann@21383
   783
qed
haftmann@21383
   784
haftmann@21383
   785
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   786
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   787
proof -
haftmann@21383
   788
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   789
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   790
  also assume "f b <= c"
haftmann@34250
   791
  finally (less_le_trans) show ?thesis .
haftmann@21383
   792
qed
haftmann@21383
   793
haftmann@21383
   794
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   795
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   796
proof -
haftmann@21383
   797
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   798
  assume "a < f b"
haftmann@21383
   799
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@34250
   800
  finally (less_le_trans) show ?thesis .
haftmann@21383
   801
qed
haftmann@21383
   802
haftmann@21383
   803
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   804
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   805
proof -
haftmann@21383
   806
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   807
  assume "a <= f b"
haftmann@21383
   808
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   809
  finally (order_trans) show ?thesis .
haftmann@21383
   810
qed
haftmann@21383
   811
haftmann@21383
   812
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   813
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   814
proof -
haftmann@21383
   815
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   816
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   817
  also assume "f b <= c"
haftmann@21383
   818
  finally (order_trans) show ?thesis .
haftmann@21383
   819
qed
haftmann@21383
   820
haftmann@21383
   821
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   822
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   823
proof -
haftmann@21383
   824
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   825
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   826
  also assume "f b = c"
haftmann@21383
   827
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   828
qed
haftmann@21383
   829
haftmann@21383
   830
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   831
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   832
proof -
haftmann@21383
   833
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   834
  assume "a = f b"
haftmann@21383
   835
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   836
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   837
qed
haftmann@21383
   838
haftmann@21383
   839
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   840
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   841
proof -
haftmann@21383
   842
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   843
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   844
  also assume "f b = c"
haftmann@21383
   845
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   846
qed
haftmann@21383
   847
haftmann@21383
   848
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   849
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   850
proof -
haftmann@21383
   851
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   852
  assume "a = f b"
haftmann@21383
   853
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   854
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   855
qed
haftmann@21383
   856
wenzelm@60758
   857
text \<open>
haftmann@21383
   858
  Note that this list of rules is in reverse order of priorities.
wenzelm@60758
   859
\<close>
haftmann@21383
   860
haftmann@27682
   861
lemmas [trans] =
haftmann@21383
   862
  order_less_subst2
haftmann@21383
   863
  order_less_subst1
haftmann@21383
   864
  order_le_less_subst2
haftmann@21383
   865
  order_le_less_subst1
haftmann@21383
   866
  order_less_le_subst2
haftmann@21383
   867
  order_less_le_subst1
haftmann@21383
   868
  order_subst2
haftmann@21383
   869
  order_subst1
haftmann@21383
   870
  ord_le_eq_subst
haftmann@21383
   871
  ord_eq_le_subst
haftmann@21383
   872
  ord_less_eq_subst
haftmann@21383
   873
  ord_eq_less_subst
haftmann@21383
   874
  forw_subst
haftmann@21383
   875
  back_subst
haftmann@21383
   876
  rev_mp
haftmann@21383
   877
  mp
haftmann@27682
   878
haftmann@27682
   879
lemmas (in order) [trans] =
haftmann@27682
   880
  neq_le_trans
haftmann@27682
   881
  le_neq_trans
haftmann@27682
   882
haftmann@27682
   883
lemmas (in preorder) [trans] =
haftmann@27682
   884
  less_trans
haftmann@27682
   885
  less_asym'
haftmann@27682
   886
  le_less_trans
haftmann@27682
   887
  less_le_trans
haftmann@21383
   888
  order_trans
haftmann@27682
   889
haftmann@27682
   890
lemmas (in order) [trans] =
haftmann@27682
   891
  antisym
haftmann@27682
   892
haftmann@27682
   893
lemmas (in ord) [trans] =
haftmann@27682
   894
  ord_le_eq_trans
haftmann@27682
   895
  ord_eq_le_trans
haftmann@27682
   896
  ord_less_eq_trans
haftmann@27682
   897
  ord_eq_less_trans
haftmann@27682
   898
haftmann@27682
   899
lemmas [trans] =
haftmann@27682
   900
  trans
haftmann@27682
   901
haftmann@27682
   902
lemmas order_trans_rules =
haftmann@27682
   903
  order_less_subst2
haftmann@27682
   904
  order_less_subst1
haftmann@27682
   905
  order_le_less_subst2
haftmann@27682
   906
  order_le_less_subst1
haftmann@27682
   907
  order_less_le_subst2
haftmann@27682
   908
  order_less_le_subst1
haftmann@27682
   909
  order_subst2
haftmann@27682
   910
  order_subst1
haftmann@27682
   911
  ord_le_eq_subst
haftmann@27682
   912
  ord_eq_le_subst
haftmann@27682
   913
  ord_less_eq_subst
haftmann@27682
   914
  ord_eq_less_subst
haftmann@27682
   915
  forw_subst
haftmann@27682
   916
  back_subst
haftmann@27682
   917
  rev_mp
haftmann@27682
   918
  mp
haftmann@27682
   919
  neq_le_trans
haftmann@27682
   920
  le_neq_trans
haftmann@27682
   921
  less_trans
haftmann@27682
   922
  less_asym'
haftmann@27682
   923
  le_less_trans
haftmann@27682
   924
  less_le_trans
haftmann@27682
   925
  order_trans
haftmann@27682
   926
  antisym
haftmann@21383
   927
  ord_le_eq_trans
haftmann@21383
   928
  ord_eq_le_trans
haftmann@21383
   929
  ord_less_eq_trans
haftmann@21383
   930
  ord_eq_less_trans
haftmann@21383
   931
  trans
haftmann@21383
   932
wenzelm@60758
   933
text \<open>These support proving chains of decreasing inequalities
wenzelm@60758
   934
    a >= b >= c ... in Isar proofs.\<close>
haftmann@21083
   935
blanchet@45221
   936
lemma xt1 [no_atp]:
haftmann@21083
   937
  "a = b ==> b > c ==> a > c"
haftmann@21083
   938
  "a > b ==> b = c ==> a > c"
haftmann@21083
   939
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   940
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   941
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   942
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   943
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   944
  "(x::'a::order) >= y ==> y > z ==> x > z"
wenzelm@23417
   945
  "(a::'a::order) > b ==> b > a ==> P"
haftmann@21083
   946
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   947
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   948
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   949
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   950
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   951
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   952
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@25076
   953
  by auto
haftmann@21083
   954
blanchet@45221
   955
lemma xt2 [no_atp]:
haftmann@21083
   956
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   957
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   958
blanchet@45221
   959
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
haftmann@21083
   960
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   961
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   962
blanchet@45221
   963
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   964
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   965
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   966
blanchet@45221
   967
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   968
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   969
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   970
blanchet@45221
   971
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   972
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   973
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   974
blanchet@45221
   975
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   976
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   977
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   978
blanchet@45221
   979
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   980
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   981
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   982
blanchet@45221
   983
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   984
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   985
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   986
blanchet@54147
   987
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   988
haftmann@21083
   989
(* 
haftmann@21083
   990
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   991
  for the wrong thing in an Isar proof.
haftmann@21083
   992
haftmann@21083
   993
  The extra transitivity rules can be used as follows: 
haftmann@21083
   994
haftmann@21083
   995
lemma "(a::'a::order) > z"
haftmann@21083
   996
proof -
haftmann@21083
   997
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   998
    sorry
haftmann@21083
   999
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
  1000
    sorry
haftmann@21083
  1001
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
  1002
    sorry
haftmann@21083
  1003
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
  1004
    sorry
haftmann@21083
  1005
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
  1006
    sorry
haftmann@21083
  1007
  also (xtrans) have "?rhs > z"
haftmann@21083
  1008
    sorry
haftmann@21083
  1009
  finally (xtrans) show ?thesis .
haftmann@21083
  1010
qed
haftmann@21083
  1011
haftmann@21083
  1012
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
  1013
  leave out the "(xtrans)" above.
haftmann@21083
  1014
*)
haftmann@21083
  1015
haftmann@23881
  1016
wenzelm@60758
  1017
subsection \<open>Monotonicity\<close>
haftmann@21083
  1018
haftmann@25076
  1019
context order
haftmann@25076
  1020
begin
haftmann@25076
  1021
wenzelm@61076
  1022
definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
haftmann@25076
  1023
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
haftmann@25076
  1024
haftmann@25076
  1025
lemma monoI [intro?]:
wenzelm@61076
  1026
  fixes f :: "'a \<Rightarrow> 'b::order"
haftmann@25076
  1027
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
haftmann@25076
  1028
  unfolding mono_def by iprover
haftmann@21216
  1029
haftmann@25076
  1030
lemma monoD [dest?]:
wenzelm@61076
  1031
  fixes f :: "'a \<Rightarrow> 'b::order"
haftmann@25076
  1032
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
haftmann@25076
  1033
  unfolding mono_def by iprover
haftmann@25076
  1034
haftmann@51263
  1035
lemma monoE:
wenzelm@61076
  1036
  fixes f :: "'a \<Rightarrow> 'b::order"
haftmann@51263
  1037
  assumes "mono f"
haftmann@51263
  1038
  assumes "x \<le> y"
haftmann@51263
  1039
  obtains "f x \<le> f y"
haftmann@51263
  1040
proof
haftmann@51263
  1041
  from assms show "f x \<le> f y" by (simp add: mono_def)
haftmann@51263
  1042
qed
haftmann@51263
  1043
wenzelm@61076
  1044
definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
hoelzl@56020
  1045
  "antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
hoelzl@56020
  1046
hoelzl@56020
  1047
lemma antimonoI [intro?]:
wenzelm@61076
  1048
  fixes f :: "'a \<Rightarrow> 'b::order"
hoelzl@56020
  1049
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
hoelzl@56020
  1050
  unfolding antimono_def by iprover
hoelzl@56020
  1051
hoelzl@56020
  1052
lemma antimonoD [dest?]:
wenzelm@61076
  1053
  fixes f :: "'a \<Rightarrow> 'b::order"
hoelzl@56020
  1054
  shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
hoelzl@56020
  1055
  unfolding antimono_def by iprover
hoelzl@56020
  1056
hoelzl@56020
  1057
lemma antimonoE:
wenzelm@61076
  1058
  fixes f :: "'a \<Rightarrow> 'b::order"
hoelzl@56020
  1059
  assumes "antimono f"
hoelzl@56020
  1060
  assumes "x \<le> y"
hoelzl@56020
  1061
  obtains "f x \<ge> f y"
hoelzl@56020
  1062
proof
hoelzl@56020
  1063
  from assms show "f x \<ge> f y" by (simp add: antimono_def)
hoelzl@56020
  1064
qed
hoelzl@56020
  1065
wenzelm@61076
  1066
definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
haftmann@30298
  1067
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
haftmann@30298
  1068
haftmann@30298
  1069
lemma strict_monoI [intro?]:
haftmann@30298
  1070
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
haftmann@30298
  1071
  shows "strict_mono f"
haftmann@30298
  1072
  using assms unfolding strict_mono_def by auto
haftmann@30298
  1073
haftmann@30298
  1074
lemma strict_monoD [dest?]:
haftmann@30298
  1075
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
haftmann@30298
  1076
  unfolding strict_mono_def by auto
haftmann@30298
  1077
haftmann@30298
  1078
lemma strict_mono_mono [dest?]:
haftmann@30298
  1079
  assumes "strict_mono f"
haftmann@30298
  1080
  shows "mono f"
haftmann@30298
  1081
proof (rule monoI)
haftmann@30298
  1082
  fix x y
haftmann@30298
  1083
  assume "x \<le> y"
haftmann@30298
  1084
  show "f x \<le> f y"
haftmann@30298
  1085
  proof (cases "x = y")
haftmann@30298
  1086
    case True then show ?thesis by simp
haftmann@30298
  1087
  next
wenzelm@60758
  1088
    case False with \<open>x \<le> y\<close> have "x < y" by simp
haftmann@30298
  1089
    with assms strict_monoD have "f x < f y" by auto
haftmann@30298
  1090
    then show ?thesis by simp
haftmann@30298
  1091
  qed
haftmann@30298
  1092
qed
haftmann@30298
  1093
haftmann@25076
  1094
end
haftmann@25076
  1095
haftmann@25076
  1096
context linorder
haftmann@25076
  1097
begin
haftmann@25076
  1098
haftmann@51263
  1099
lemma mono_invE:
wenzelm@61076
  1100
  fixes f :: "'a \<Rightarrow> 'b::order"
haftmann@51263
  1101
  assumes "mono f"
haftmann@51263
  1102
  assumes "f x < f y"
haftmann@51263
  1103
  obtains "x \<le> y"
haftmann@51263
  1104
proof
haftmann@51263
  1105
  show "x \<le> y"
haftmann@51263
  1106
  proof (rule ccontr)
haftmann@51263
  1107
    assume "\<not> x \<le> y"
haftmann@51263
  1108
    then have "y \<le> x" by simp
wenzelm@60758
  1109
    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
wenzelm@60758
  1110
    with \<open>f x < f y\<close> show False by simp
haftmann@51263
  1111
  qed
haftmann@51263
  1112
qed
haftmann@51263
  1113
haftmann@30298
  1114
lemma strict_mono_eq:
haftmann@30298
  1115
  assumes "strict_mono f"
haftmann@30298
  1116
  shows "f x = f y \<longleftrightarrow> x = y"
haftmann@30298
  1117
proof
haftmann@30298
  1118
  assume "f x = f y"
haftmann@30298
  1119
  show "x = y" proof (cases x y rule: linorder_cases)
haftmann@30298
  1120
    case less with assms strict_monoD have "f x < f y" by auto
wenzelm@60758
  1121
    with \<open>f x = f y\<close> show ?thesis by simp
haftmann@30298
  1122
  next
haftmann@30298
  1123
    case equal then show ?thesis .
haftmann@30298
  1124
  next
haftmann@30298
  1125
    case greater with assms strict_monoD have "f y < f x" by auto
wenzelm@60758
  1126
    with \<open>f x = f y\<close> show ?thesis by simp
haftmann@30298
  1127
  qed
haftmann@30298
  1128
qed simp
haftmann@30298
  1129
haftmann@30298
  1130
lemma strict_mono_less_eq:
haftmann@30298
  1131
  assumes "strict_mono f"
haftmann@30298
  1132
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
haftmann@30298
  1133
proof
haftmann@30298
  1134
  assume "x \<le> y"
haftmann@30298
  1135
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
haftmann@30298
  1136
next
haftmann@30298
  1137
  assume "f x \<le> f y"
haftmann@30298
  1138
  show "x \<le> y" proof (rule ccontr)
haftmann@30298
  1139
    assume "\<not> x \<le> y" then have "y < x" by simp
haftmann@30298
  1140
    with assms strict_monoD have "f y < f x" by auto
wenzelm@60758
  1141
    with \<open>f x \<le> f y\<close> show False by simp
haftmann@30298
  1142
  qed
haftmann@30298
  1143
qed
haftmann@30298
  1144
  
haftmann@30298
  1145
lemma strict_mono_less:
haftmann@30298
  1146
  assumes "strict_mono f"
haftmann@30298
  1147
  shows "f x < f y \<longleftrightarrow> x < y"
haftmann@30298
  1148
  using assms
haftmann@30298
  1149
    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
haftmann@30298
  1150
haftmann@54860
  1151
end
haftmann@54860
  1152
haftmann@54860
  1153
wenzelm@60758
  1154
subsection \<open>min and max -- fundamental\<close>
haftmann@54860
  1155
haftmann@54860
  1156
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@54860
  1157
  "min a b = (if a \<le> b then a else b)"
haftmann@54860
  1158
haftmann@54860
  1159
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@54860
  1160
  "max a b = (if a \<le> b then b else a)"
haftmann@54860
  1161
noschinl@45931
  1162
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
haftmann@54861
  1163
  by (simp add: min_def)
haftmann@21383
  1164
haftmann@54857
  1165
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
haftmann@54861
  1166
  by (simp add: max_def)
haftmann@21383
  1167
wenzelm@61076
  1168
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
haftmann@54861
  1169
  by (simp add:min_def)
noschinl@45893
  1170
wenzelm@61076
  1171
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x"
haftmann@54861
  1172
  by (simp add: max_def)
noschinl@45893
  1173
noschinl@45893
  1174
wenzelm@60758
  1175
subsection \<open>(Unique) top and bottom elements\<close>
haftmann@28685
  1176
haftmann@52729
  1177
class bot =
haftmann@43853
  1178
  fixes bot :: 'a ("\<bottom>")
haftmann@52729
  1179
haftmann@52729
  1180
class order_bot = order + bot +
haftmann@51487
  1181
  assumes bot_least: "\<bottom> \<le> a"
haftmann@54868
  1182
begin
haftmann@51487
  1183
wenzelm@61605
  1184
sublocale bot: ordering_top greater_eq greater bot
wenzelm@61169
  1185
  by standard (fact bot_least)
haftmann@51487
  1186
haftmann@43853
  1187
lemma le_bot:
haftmann@43853
  1188
  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
haftmann@51487
  1189
  by (fact bot.extremum_uniqueI)
haftmann@43853
  1190
haftmann@43816
  1191
lemma bot_unique:
haftmann@43853
  1192
  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
haftmann@51487
  1193
  by (fact bot.extremum_unique)
haftmann@43853
  1194
haftmann@51487
  1195
lemma not_less_bot:
haftmann@51487
  1196
  "\<not> a < \<bottom>"
haftmann@51487
  1197
  by (fact bot.extremum_strict)
haftmann@43816
  1198
haftmann@43814
  1199
lemma bot_less:
haftmann@43853
  1200
  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
haftmann@51487
  1201
  by (fact bot.not_eq_extremum)
haftmann@43814
  1202
haftmann@43814
  1203
end
haftmann@41082
  1204
haftmann@52729
  1205
class top =
haftmann@43853
  1206
  fixes top :: 'a ("\<top>")
haftmann@52729
  1207
haftmann@52729
  1208
class order_top = order + top +
haftmann@51487
  1209
  assumes top_greatest: "a \<le> \<top>"
haftmann@54868
  1210
begin
haftmann@51487
  1211
wenzelm@61605
  1212
sublocale top: ordering_top less_eq less top
wenzelm@61169
  1213
  by standard (fact top_greatest)
haftmann@51487
  1214
haftmann@43853
  1215
lemma top_le:
haftmann@43853
  1216
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
haftmann@51487
  1217
  by (fact top.extremum_uniqueI)
haftmann@43853
  1218
haftmann@43816
  1219
lemma top_unique:
haftmann@43853
  1220
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
haftmann@51487
  1221
  by (fact top.extremum_unique)
haftmann@43853
  1222
haftmann@51487
  1223
lemma not_top_less:
haftmann@51487
  1224
  "\<not> \<top> < a"
haftmann@51487
  1225
  by (fact top.extremum_strict)
haftmann@43816
  1226
haftmann@43814
  1227
lemma less_top:
haftmann@43853
  1228
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
haftmann@51487
  1229
  by (fact top.not_eq_extremum)
haftmann@43814
  1230
haftmann@43814
  1231
end
haftmann@28685
  1232
haftmann@28685
  1233
wenzelm@60758
  1234
subsection \<open>Dense orders\<close>
haftmann@27823
  1235
hoelzl@53216
  1236
class dense_order = order +
hoelzl@51329
  1237
  assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
hoelzl@51329
  1238
hoelzl@53216
  1239
class dense_linorder = linorder + dense_order
hoelzl@35579
  1240
begin
haftmann@27823
  1241
hoelzl@35579
  1242
lemma dense_le:
hoelzl@35579
  1243
  fixes y z :: 'a
hoelzl@35579
  1244
  assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
hoelzl@35579
  1245
  shows "y \<le> z"
hoelzl@35579
  1246
proof (rule ccontr)
hoelzl@35579
  1247
  assume "\<not> ?thesis"
hoelzl@35579
  1248
  hence "z < y" by simp
hoelzl@35579
  1249
  from dense[OF this]
hoelzl@35579
  1250
  obtain x where "x < y" and "z < x" by safe
wenzelm@60758
  1251
  moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
hoelzl@35579
  1252
  ultimately show False by auto
hoelzl@35579
  1253
qed
hoelzl@35579
  1254
hoelzl@35579
  1255
lemma dense_le_bounded:
hoelzl@35579
  1256
  fixes x y z :: 'a
hoelzl@35579
  1257
  assumes "x < y"
hoelzl@35579
  1258
  assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
hoelzl@35579
  1259
  shows "y \<le> z"
hoelzl@35579
  1260
proof (rule dense_le)
hoelzl@35579
  1261
  fix w assume "w < y"
wenzelm@60758
  1262
  from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
hoelzl@35579
  1263
  from linear[of u w]
hoelzl@35579
  1264
  show "w \<le> z"
hoelzl@35579
  1265
  proof (rule disjE)
hoelzl@35579
  1266
    assume "u \<le> w"
wenzelm@60758
  1267
    from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
hoelzl@35579
  1268
    show "w \<le> z" by (rule *)
hoelzl@35579
  1269
  next
hoelzl@35579
  1270
    assume "w \<le> u"
wenzelm@60758
  1271
    from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
hoelzl@35579
  1272
    show "w \<le> z" by (rule order_trans)
hoelzl@35579
  1273
  qed
hoelzl@35579
  1274
qed
hoelzl@35579
  1275
hoelzl@51329
  1276
lemma dense_ge:
hoelzl@51329
  1277
  fixes y z :: 'a
hoelzl@51329
  1278
  assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
hoelzl@51329
  1279
  shows "y \<le> z"
hoelzl@51329
  1280
proof (rule ccontr)
hoelzl@51329
  1281
  assume "\<not> ?thesis"
hoelzl@51329
  1282
  hence "z < y" by simp
hoelzl@51329
  1283
  from dense[OF this]
hoelzl@51329
  1284
  obtain x where "x < y" and "z < x" by safe
wenzelm@60758
  1285
  moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
hoelzl@51329
  1286
  ultimately show False by auto
hoelzl@51329
  1287
qed
hoelzl@51329
  1288
hoelzl@51329
  1289
lemma dense_ge_bounded:
hoelzl@51329
  1290
  fixes x y z :: 'a
hoelzl@51329
  1291
  assumes "z < x"
hoelzl@51329
  1292
  assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
hoelzl@51329
  1293
  shows "y \<le> z"
hoelzl@51329
  1294
proof (rule dense_ge)
hoelzl@51329
  1295
  fix w assume "z < w"
wenzelm@60758
  1296
  from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
hoelzl@51329
  1297
  from linear[of u w]
hoelzl@51329
  1298
  show "y \<le> w"
hoelzl@51329
  1299
  proof (rule disjE)
hoelzl@51329
  1300
    assume "w \<le> u"
wenzelm@60758
  1301
    from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
hoelzl@51329
  1302
    show "y \<le> w" by (rule *)
hoelzl@51329
  1303
  next
hoelzl@51329
  1304
    assume "u \<le> w"
wenzelm@60758
  1305
    from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
hoelzl@51329
  1306
    show "y \<le> w" by (rule order_trans)
hoelzl@51329
  1307
  qed
hoelzl@51329
  1308
qed
hoelzl@51329
  1309
hoelzl@35579
  1310
end
haftmann@27823
  1311
hoelzl@51329
  1312
class no_top = order + 
hoelzl@51329
  1313
  assumes gt_ex: "\<exists>y. x < y"
hoelzl@51329
  1314
hoelzl@51329
  1315
class no_bot = order + 
hoelzl@51329
  1316
  assumes lt_ex: "\<exists>y. y < x"
hoelzl@51329
  1317
hoelzl@53216
  1318
class unbounded_dense_linorder = dense_linorder + no_top + no_bot
hoelzl@51329
  1319
haftmann@51546
  1320
wenzelm@60758
  1321
subsection \<open>Wellorders\<close>
haftmann@27823
  1322
haftmann@27823
  1323
class wellorder = linorder +
haftmann@27823
  1324
  assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
haftmann@27823
  1325
begin
haftmann@27823
  1326
haftmann@27823
  1327
lemma wellorder_Least_lemma:
haftmann@27823
  1328
  fixes k :: 'a
haftmann@27823
  1329
  assumes "P k"
haftmann@34250
  1330
  shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
haftmann@27823
  1331
proof -
haftmann@27823
  1332
  have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
haftmann@27823
  1333
  using assms proof (induct k rule: less_induct)
haftmann@27823
  1334
    case (less x) then have "P x" by simp
haftmann@27823
  1335
    show ?case proof (rule classical)
haftmann@27823
  1336
      assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
haftmann@27823
  1337
      have "\<And>y. P y \<Longrightarrow> x \<le> y"
haftmann@27823
  1338
      proof (rule classical)
haftmann@27823
  1339
        fix y
hoelzl@38705
  1340
        assume "P y" and "\<not> x \<le> y"
haftmann@27823
  1341
        with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
haftmann@27823
  1342
          by (auto simp add: not_le)
haftmann@27823
  1343
        with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
haftmann@27823
  1344
          by auto
haftmann@27823
  1345
        then show "x \<le> y" by auto
haftmann@27823
  1346
      qed
wenzelm@60758
  1347
      with \<open>P x\<close> have Least: "(LEAST a. P a) = x"
haftmann@27823
  1348
        by (rule Least_equality)
wenzelm@60758
  1349
      with \<open>P x\<close> show ?thesis by simp
haftmann@27823
  1350
    qed
haftmann@27823
  1351
  qed
haftmann@27823
  1352
  then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
haftmann@27823
  1353
qed
haftmann@27823
  1354
haftmann@27823
  1355
-- "The following 3 lemmas are due to Brian Huffman"
haftmann@27823
  1356
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
haftmann@27823
  1357
  by (erule exE) (erule LeastI)
haftmann@27823
  1358
haftmann@27823
  1359
lemma LeastI2:
haftmann@27823
  1360
  "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
haftmann@27823
  1361
  by (blast intro: LeastI)
haftmann@27823
  1362
haftmann@27823
  1363
lemma LeastI2_ex:
haftmann@27823
  1364
  "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
haftmann@27823
  1365
  by (blast intro: LeastI_ex)
haftmann@27823
  1366
hoelzl@38705
  1367
lemma LeastI2_wellorder:
hoelzl@38705
  1368
  assumes "P a"
hoelzl@38705
  1369
  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
hoelzl@38705
  1370
  shows "Q (Least P)"
hoelzl@38705
  1371
proof (rule LeastI2_order)
wenzelm@60758
  1372
  show "P (Least P)" using \<open>P a\<close> by (rule LeastI)
hoelzl@38705
  1373
next
hoelzl@38705
  1374
  fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
hoelzl@38705
  1375
next
hoelzl@38705
  1376
  fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
hoelzl@38705
  1377
qed
hoelzl@38705
  1378
haftmann@27823
  1379
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
haftmann@27823
  1380
apply (simp (no_asm_use) add: not_le [symmetric])
haftmann@27823
  1381
apply (erule contrapos_nn)
haftmann@27823
  1382
apply (erule Least_le)
haftmann@27823
  1383
done
haftmann@27823
  1384
hoelzl@38705
  1385
end
haftmann@27823
  1386
haftmann@28685
  1387
wenzelm@60758
  1388
subsection \<open>Order on @{typ bool}\<close>
haftmann@28685
  1389
haftmann@52729
  1390
instantiation bool :: "{order_bot, order_top, linorder}"
haftmann@28685
  1391
begin
haftmann@28685
  1392
haftmann@28685
  1393
definition
haftmann@41080
  1394
  le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
haftmann@28685
  1395
haftmann@28685
  1396
definition
wenzelm@61076
  1397
  [simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
haftmann@28685
  1398
haftmann@28685
  1399
definition
haftmann@46631
  1400
  [simp]: "\<bottom> \<longleftrightarrow> False"
haftmann@28685
  1401
haftmann@28685
  1402
definition
haftmann@46631
  1403
  [simp]: "\<top> \<longleftrightarrow> True"
haftmann@28685
  1404
haftmann@28685
  1405
instance proof
haftmann@41080
  1406
qed auto
haftmann@28685
  1407
nipkow@15524
  1408
end
haftmann@28685
  1409
haftmann@28685
  1410
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@41080
  1411
  by simp
haftmann@28685
  1412
haftmann@28685
  1413
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@41080
  1414
  by simp
haftmann@28685
  1415
haftmann@28685
  1416
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41080
  1417
  by simp
haftmann@28685
  1418
haftmann@28685
  1419
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@41080
  1420
  by simp
haftmann@32899
  1421
haftmann@46631
  1422
lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
haftmann@41080
  1423
  by simp
haftmann@32899
  1424
haftmann@46631
  1425
lemma top_boolI: \<top>
haftmann@41080
  1426
  by simp
haftmann@28685
  1427
haftmann@28685
  1428
lemma [code]:
haftmann@28685
  1429
  "False \<le> b \<longleftrightarrow> True"
haftmann@28685
  1430
  "True \<le> b \<longleftrightarrow> b"
haftmann@28685
  1431
  "False < b \<longleftrightarrow> b"
haftmann@28685
  1432
  "True < b \<longleftrightarrow> False"
haftmann@41080
  1433
  by simp_all
haftmann@28685
  1434
haftmann@28685
  1435
wenzelm@60758
  1436
subsection \<open>Order on @{typ "_ \<Rightarrow> _"}\<close>
haftmann@28685
  1437
haftmann@28685
  1438
instantiation "fun" :: (type, ord) ord
haftmann@28685
  1439
begin
haftmann@28685
  1440
haftmann@28685
  1441
definition
haftmann@37767
  1442
  le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
haftmann@28685
  1443
haftmann@28685
  1444
definition
wenzelm@61076
  1445
  "(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
haftmann@28685
  1446
haftmann@28685
  1447
instance ..
haftmann@28685
  1448
haftmann@28685
  1449
end
haftmann@28685
  1450
haftmann@28685
  1451
instance "fun" :: (type, preorder) preorder proof
haftmann@28685
  1452
qed (auto simp add: le_fun_def less_fun_def
huffman@44921
  1453
  intro: order_trans antisym)
haftmann@28685
  1454
haftmann@28685
  1455
instance "fun" :: (type, order) order proof
huffman@44921
  1456
qed (auto simp add: le_fun_def intro: antisym)
haftmann@28685
  1457
haftmann@41082
  1458
instantiation "fun" :: (type, bot) bot
haftmann@41082
  1459
begin
haftmann@41082
  1460
haftmann@41082
  1461
definition
haftmann@46631
  1462
  "\<bottom> = (\<lambda>x. \<bottom>)"
haftmann@41082
  1463
haftmann@52729
  1464
instance ..
haftmann@52729
  1465
haftmann@52729
  1466
end
haftmann@52729
  1467
haftmann@52729
  1468
instantiation "fun" :: (type, order_bot) order_bot
haftmann@52729
  1469
begin
haftmann@52729
  1470
haftmann@49769
  1471
lemma bot_apply [simp, code]:
haftmann@46631
  1472
  "\<bottom> x = \<bottom>"
haftmann@41082
  1473
  by (simp add: bot_fun_def)
haftmann@41082
  1474
haftmann@41082
  1475
instance proof
noschinl@46884
  1476
qed (simp add: le_fun_def)
haftmann@41082
  1477
haftmann@41082
  1478
end
haftmann@41082
  1479
haftmann@28685
  1480
instantiation "fun" :: (type, top) top
haftmann@28685
  1481
begin
haftmann@28685
  1482
haftmann@28685
  1483
definition
haftmann@46631
  1484
  [no_atp]: "\<top> = (\<lambda>x. \<top>)"
haftmann@28685
  1485
haftmann@52729
  1486
instance ..
haftmann@52729
  1487
haftmann@52729
  1488
end
haftmann@52729
  1489
haftmann@52729
  1490
instantiation "fun" :: (type, order_top) order_top
haftmann@52729
  1491
begin
haftmann@52729
  1492
haftmann@49769
  1493
lemma top_apply [simp, code]:
haftmann@46631
  1494
  "\<top> x = \<top>"
haftmann@41080
  1495
  by (simp add: top_fun_def)
haftmann@41080
  1496
haftmann@28685
  1497
instance proof
noschinl@46884
  1498
qed (simp add: le_fun_def)
haftmann@28685
  1499
haftmann@28685
  1500
end
haftmann@28685
  1501
haftmann@28685
  1502
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
haftmann@28685
  1503
  unfolding le_fun_def by simp
haftmann@28685
  1504
haftmann@28685
  1505
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@28685
  1506
  unfolding le_fun_def by simp
haftmann@28685
  1507
haftmann@28685
  1508
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
haftmann@54860
  1509
  by (rule le_funE)
haftmann@28685
  1510
hoelzl@59000
  1511
lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))"
hoelzl@59000
  1512
  unfolding mono_def le_fun_def by auto
hoelzl@59000
  1513
haftmann@34250
  1514
wenzelm@60758
  1515
subsection \<open>Order on unary and binary predicates\<close>
haftmann@46631
  1516
haftmann@46631
  1517
lemma predicate1I:
haftmann@46631
  1518
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
haftmann@46631
  1519
  shows "P \<le> Q"
haftmann@46631
  1520
  apply (rule le_funI)
haftmann@46631
  1521
  apply (rule le_boolI)
haftmann@46631
  1522
  apply (rule PQ)
haftmann@46631
  1523
  apply assumption
haftmann@46631
  1524
  done
haftmann@46631
  1525
haftmann@46631
  1526
lemma predicate1D:
haftmann@46631
  1527
  "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
haftmann@46631
  1528
  apply (erule le_funE)
haftmann@46631
  1529
  apply (erule le_boolE)
haftmann@46631
  1530
  apply assumption+
haftmann@46631
  1531
  done
haftmann@46631
  1532
haftmann@46631
  1533
lemma rev_predicate1D:
haftmann@46631
  1534
  "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
haftmann@46631
  1535
  by (rule predicate1D)
haftmann@46631
  1536
haftmann@46631
  1537
lemma predicate2I:
haftmann@46631
  1538
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
haftmann@46631
  1539
  shows "P \<le> Q"
haftmann@46631
  1540
  apply (rule le_funI)+
haftmann@46631
  1541
  apply (rule le_boolI)
haftmann@46631
  1542
  apply (rule PQ)
haftmann@46631
  1543
  apply assumption
haftmann@46631
  1544
  done
haftmann@46631
  1545
haftmann@46631
  1546
lemma predicate2D:
haftmann@46631
  1547
  "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
haftmann@46631
  1548
  apply (erule le_funE)+
haftmann@46631
  1549
  apply (erule le_boolE)
haftmann@46631
  1550
  apply assumption+
haftmann@46631
  1551
  done
haftmann@46631
  1552
haftmann@46631
  1553
lemma rev_predicate2D:
haftmann@46631
  1554
  "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
haftmann@46631
  1555
  by (rule predicate2D)
haftmann@46631
  1556
haftmann@46631
  1557
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
haftmann@46631
  1558
  by (simp add: bot_fun_def)
haftmann@46631
  1559
haftmann@46631
  1560
lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
haftmann@46631
  1561
  by (simp add: bot_fun_def)
haftmann@46631
  1562
haftmann@46631
  1563
lemma top1I: "\<top> x"
haftmann@46631
  1564
  by (simp add: top_fun_def)
haftmann@46631
  1565
haftmann@46631
  1566
lemma top2I: "\<top> x y"
haftmann@46631
  1567
  by (simp add: top_fun_def)
haftmann@46631
  1568
haftmann@46631
  1569
wenzelm@60758
  1570
subsection \<open>Name duplicates\<close>
haftmann@34250
  1571
haftmann@34250
  1572
lemmas order_eq_refl = preorder_class.eq_refl
haftmann@34250
  1573
lemmas order_less_irrefl = preorder_class.less_irrefl
haftmann@34250
  1574
lemmas order_less_imp_le = preorder_class.less_imp_le
haftmann@34250
  1575
lemmas order_less_not_sym = preorder_class.less_not_sym
haftmann@34250
  1576
lemmas order_less_asym = preorder_class.less_asym
haftmann@34250
  1577
lemmas order_less_trans = preorder_class.less_trans
haftmann@34250
  1578
lemmas order_le_less_trans = preorder_class.le_less_trans
haftmann@34250
  1579
lemmas order_less_le_trans = preorder_class.less_le_trans
haftmann@34250
  1580
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
haftmann@34250
  1581
lemmas order_less_imp_triv = preorder_class.less_imp_triv
haftmann@34250
  1582
lemmas order_less_asym' = preorder_class.less_asym'
haftmann@34250
  1583
haftmann@34250
  1584
lemmas order_less_le = order_class.less_le
haftmann@34250
  1585
lemmas order_le_less = order_class.le_less
haftmann@34250
  1586
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
haftmann@34250
  1587
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
haftmann@34250
  1588
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
haftmann@34250
  1589
lemmas order_neq_le_trans = order_class.neq_le_trans
haftmann@34250
  1590
lemmas order_le_neq_trans = order_class.le_neq_trans
haftmann@34250
  1591
lemmas order_antisym = order_class.antisym
haftmann@34250
  1592
lemmas order_eq_iff = order_class.eq_iff
haftmann@34250
  1593
lemmas order_antisym_conv = order_class.antisym_conv
haftmann@34250
  1594
haftmann@34250
  1595
lemmas linorder_linear = linorder_class.linear
haftmann@34250
  1596
lemmas linorder_less_linear = linorder_class.less_linear
haftmann@34250
  1597
lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@34250
  1598
lemmas linorder_le_cases = linorder_class.le_cases
haftmann@34250
  1599
lemmas linorder_not_less = linorder_class.not_less
haftmann@34250
  1600
lemmas linorder_not_le = linorder_class.not_le
haftmann@34250
  1601
lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@34250
  1602
lemmas linorder_neqE = linorder_class.neqE
haftmann@34250
  1603
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@34250
  1604
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@34250
  1605
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
haftmann@34250
  1606
haftmann@28685
  1607
end