src/HOL/Orderings.thy
author haftmann
Wed Dec 25 15:52:25 2013 +0100 (2013-12-25)
changeset 54860 69b3e46d8fbe
parent 54857 5c05f7c5f8ae
child 54861 00d551179872
permissions -rw-r--r--
tuned structure of min/max lemmas
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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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header {* Abstract orderings *}
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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 {* Abstract ordering *}
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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" -- {* not @{text iff}: makes problems due to multiple (dual) interpretations *}
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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: -- {* not @{text iff}: makes problems due to multiple (dual) interpretations *}
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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 {* Syntactic orders *}
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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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notation (HTML output)
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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 {* Quasi orders *}
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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 {* Reflexivity. *}
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
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    -- {* This form is useful with the classical reasoner. *}
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by (erule ssubst) (rule order_refl)
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lemma less_irrefl [iff]: "\<not> x < x"
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by (simp add: less_le_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 {* Asymmetry. *}
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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 {* Transitivity. *}
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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 {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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text {* Transitivity rules for calculational reasoning *}
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lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
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by (rule less_asym)
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text {* Dual order *}
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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 {* Partial orders *}
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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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end
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sublocale order < order!: ordering less_eq less +  dual_order!: ordering greater_eq greater
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  by default (auto intro: antisym order_trans simp add: less_le)
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context order
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begin
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text {* Reflexivity. *}
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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
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    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
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by (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 {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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by auto
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lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
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by auto
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text {* Transitivity rules for calculational reasoning *}
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
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by (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 {* Asymmetry. *}
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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 {* Least value operator *}
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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 {* Dual order *}
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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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subsection {* Linear (total) orders *}
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class linorder = order +
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  assumes linear: "x \<le> y \<or> y \<le> x"
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begin
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lemma less_linear: "x < y \<or> x = y \<or> y < x"
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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 not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma not_less_iff_gr_or_eq:
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 "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
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apply(simp add:not_less le_less)
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apply blast
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done
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lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
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by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
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unfolding not_less .
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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
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unfolding not_less .
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(*FIXME inappropriate name (or delete altogether)*)
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lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
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unfolding not_le .
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text {* Dual order *}
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lemma dual_linorder:
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  "class.linorder (op \<ge>) (op >)"
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by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
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end
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subsection {* Reasoning tools setup *}
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ML {*
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   351
ballarin@24641
   352
signature ORDERS =
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   353
sig
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   354
  val print_structures: Proof.context -> unit
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   355
  val attrib_setup: theory -> theory
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   356
  val order_tac: Proof.context -> thm list -> int -> tactic
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   357
end;
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   358
ballarin@24641
   359
structure Orders: ORDERS =
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   360
struct
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   361
ballarin@24641
   362
(** Theory and context data **)
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   363
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   364
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
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   365
  (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
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   366
wenzelm@33519
   367
structure Data = Generic_Data
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   368
(
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   369
  type T = ((string * term list) * Order_Tac.less_arith) list;
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   370
    (* Order structures:
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   371
       identifier of the structure, list of operations and record of theorems
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   372
       needed to set up the transitivity reasoner,
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   373
       identifier and operations identify the structure uniquely. *)
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   374
  val empty = [];
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   375
  val extend = I;
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   376
  fun merge data = AList.join struct_eq (K fst) data;
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   377
);
ballarin@24641
   378
ballarin@24641
   379
fun print_structures ctxt =
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   380
  let
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   381
    val structs = Data.get (Context.Proof ctxt);
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   382
    fun pretty_term t = Pretty.block
wenzelm@24920
   383
      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
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   384
        Pretty.str "::", Pretty.brk 1,
wenzelm@24920
   385
        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
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   386
    fun pretty_struct ((s, ts), _) = Pretty.block
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   387
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
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   388
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
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   389
  in
wenzelm@51579
   390
    Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs))
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   391
  end;
ballarin@24641
   392
ballarin@24641
   393
ballarin@24641
   394
(** Method **)
haftmann@21091
   395
wenzelm@32215
   396
fun struct_tac ((s, [eq, le, less]), thms) ctxt prems =
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   397
  let
berghofe@30107
   398
    fun decomp thy (@{const Trueprop} $ t) =
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   399
      let
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   400
        fun excluded t =
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   401
          (* exclude numeric types: linear arithmetic subsumes transitivity *)
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   402
          let val T = type_of t
ballarin@24641
   403
          in
wenzelm@32960
   404
            T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
ballarin@24641
   405
          end;
wenzelm@32960
   406
        fun rel (bin_op $ t1 $ t2) =
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   407
              if excluded t1 then NONE
ballarin@24641
   408
              else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
ballarin@24641
   409
              else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
ballarin@24641
   410
              else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
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   411
              else NONE
wenzelm@32960
   412
          | rel _ = NONE;
wenzelm@32960
   413
        fun dec (Const (@{const_name Not}, _) $ t) = (case rel t
wenzelm@32960
   414
              of NONE => NONE
wenzelm@32960
   415
               | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
ballarin@24741
   416
          | dec x = rel x;
berghofe@30107
   417
      in dec t end
berghofe@30107
   418
      | decomp thy _ = NONE;
ballarin@24641
   419
  in
ballarin@24641
   420
    case s of
wenzelm@32215
   421
      "order" => Order_Tac.partial_tac decomp thms ctxt prems
wenzelm@32215
   422
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt prems
ballarin@24641
   423
    | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
ballarin@24641
   424
  end
ballarin@24641
   425
wenzelm@32215
   426
fun order_tac ctxt prems =
wenzelm@32215
   427
  FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt)));
ballarin@24641
   428
ballarin@24641
   429
ballarin@24641
   430
(** Attribute **)
ballarin@24641
   431
ballarin@24641
   432
fun add_struct_thm s tag =
ballarin@24641
   433
  Thm.declaration_attribute
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   434
    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
ballarin@24641
   435
fun del_struct s =
ballarin@24641
   436
  Thm.declaration_attribute
ballarin@24641
   437
    (fn _ => Data.map (AList.delete struct_eq s));
ballarin@24641
   438
wenzelm@30722
   439
val attrib_setup =
wenzelm@30722
   440
  Attrib.setup @{binding order}
wenzelm@30722
   441
    (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
wenzelm@30722
   442
      Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
wenzelm@30722
   443
      Scan.repeat Args.term
wenzelm@30722
   444
      >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
wenzelm@30722
   445
           | ((NONE, n), ts) => del_struct (n, ts)))
wenzelm@30722
   446
    "theorems controlling transitivity reasoner";
ballarin@24641
   447
ballarin@24641
   448
ballarin@24641
   449
(** Diagnostic command **)
ballarin@24641
   450
wenzelm@24867
   451
val _ =
wenzelm@46961
   452
  Outer_Syntax.improper_command @{command_spec "print_orders"}
wenzelm@46961
   453
    "print order structures available to transitivity reasoner"
wenzelm@51658
   454
    (Scan.succeed (Toplevel.unknown_context o
wenzelm@51658
   455
      Toplevel.keep (print_structures o Toplevel.context_of)));
ballarin@24641
   456
haftmann@21091
   457
end;
ballarin@24641
   458
haftmann@21091
   459
*}
haftmann@21091
   460
wenzelm@47432
   461
setup Orders.attrib_setup
wenzelm@47432
   462
wenzelm@47432
   463
method_setup order = {*
wenzelm@47432
   464
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
wenzelm@47432
   465
*} "transitivity reasoner"
ballarin@24641
   466
ballarin@24641
   467
ballarin@24641
   468
text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
ballarin@24641
   469
haftmann@25076
   470
context order
haftmann@25076
   471
begin
haftmann@25076
   472
ballarin@24641
   473
(* The type constraint on @{term op =} below is necessary since the operation
ballarin@24641
   474
   is not a parameter of the locale. *)
haftmann@25076
   475
haftmann@27689
   476
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]
haftmann@27689
   477
  
haftmann@27689
   478
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   479
  
haftmann@27689
   480
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   481
  
haftmann@27689
   482
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   483
haftmann@27689
   484
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   485
haftmann@27689
   486
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   487
haftmann@27689
   488
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   489
  
haftmann@27689
   490
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   491
  
haftmann@27689
   492
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   493
haftmann@27689
   494
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   495
haftmann@27689
   496
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   497
haftmann@27689
   498
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   499
haftmann@27689
   500
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   501
haftmann@27689
   502
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   503
haftmann@27689
   504
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]
ballarin@24641
   505
haftmann@25076
   506
end
haftmann@25076
   507
haftmann@25076
   508
context linorder
haftmann@25076
   509
begin
ballarin@24641
   510
haftmann@27689
   511
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
haftmann@27689
   512
haftmann@27689
   513
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   514
haftmann@27689
   515
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   516
haftmann@27689
   517
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   518
haftmann@27689
   519
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   520
haftmann@27689
   521
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   522
haftmann@27689
   523
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   524
haftmann@27689
   525
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   526
haftmann@27689
   527
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   528
haftmann@27689
   529
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@25076
   530
haftmann@27689
   531
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   532
haftmann@27689
   533
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   534
haftmann@27689
   535
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   536
haftmann@27689
   537
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   538
haftmann@27689
   539
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   540
haftmann@27689
   541
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   542
haftmann@27689
   543
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   544
haftmann@27689
   545
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   546
haftmann@27689
   547
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
haftmann@27689
   548
haftmann@27689
   549
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
ballarin@24641
   550
haftmann@25076
   551
end
haftmann@25076
   552
ballarin@24641
   553
haftmann@21083
   554
setup {*
haftmann@21083
   555
let
haftmann@21083
   556
wenzelm@44058
   557
fun prp t thm = Thm.prop_of thm = t;  (* FIXME aconv!? *)
nipkow@15524
   558
wenzelm@51717
   559
fun prove_antisym_le ctxt ((le as Const(_,T)) $ r $ s) =
wenzelm@51717
   560
  let val prems = Simplifier.prems_of ctxt;
haftmann@22916
   561
      val less = Const (@{const_name less}, T);
haftmann@21083
   562
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   563
  in case find_first (prp t) prems of
haftmann@21083
   564
       NONE =>
haftmann@21083
   565
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   566
         in case find_first (prp t) prems of
haftmann@21083
   567
              NONE => NONE
haftmann@24422
   568
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
haftmann@21083
   569
         end
haftmann@24422
   570
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
haftmann@21083
   571
  end
haftmann@21083
   572
  handle THM _ => NONE;
nipkow@15524
   573
wenzelm@51717
   574
fun prove_antisym_less ctxt (NotC $ ((less as Const(_,T)) $ r $ s)) =
wenzelm@51717
   575
  let val prems = Simplifier.prems_of ctxt;
haftmann@22916
   576
      val le = Const (@{const_name less_eq}, T);
haftmann@21083
   577
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   578
  in case find_first (prp t) prems of
haftmann@21083
   579
       NONE =>
haftmann@21083
   580
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   581
         in case find_first (prp t) prems of
haftmann@21083
   582
              NONE => NONE
haftmann@24422
   583
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
haftmann@21083
   584
         end
haftmann@24422
   585
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
haftmann@21083
   586
  end
haftmann@21083
   587
  handle THM _ => NONE;
nipkow@15524
   588
haftmann@21248
   589
fun add_simprocs procs thy =
wenzelm@51717
   590
  map_theory_simpset (fn ctxt => ctxt
haftmann@21248
   591
    addsimprocs (map (fn (name, raw_ts, proc) =>
wenzelm@38715
   592
      Simplifier.simproc_global thy name raw_ts proc) procs)) thy;
wenzelm@42795
   593
wenzelm@26496
   594
fun add_solver name tac =
wenzelm@51717
   595
  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
wenzelm@51717
   596
    mk_solver name (fn ctxt => tac ctxt (Simplifier.prems_of ctxt)));
haftmann@21083
   597
haftmann@21083
   598
in
haftmann@21248
   599
  add_simprocs [
haftmann@21248
   600
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   601
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   602
     ]
ballarin@24641
   603
  #> add_solver "Transitivity" Orders.order_tac
haftmann@21248
   604
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   605
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   606
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   607
     of 5 March 2004, was observed). *)
haftmann@21083
   608
end
haftmann@21083
   609
*}
nipkow@15524
   610
nipkow@15524
   611
haftmann@21083
   612
subsection {* Bounded quantifiers *}
haftmann@21083
   613
haftmann@21083
   614
syntax
wenzelm@21180
   615
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   616
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   617
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   618
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   619
wenzelm@21180
   620
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   621
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   622
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   623
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   624
haftmann@21083
   625
syntax (xsymbols)
wenzelm@21180
   626
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   627
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   628
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   629
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   630
wenzelm@21180
   631
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   632
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   633
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   634
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   635
haftmann@21083
   636
syntax (HOL)
wenzelm@21180
   637
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   638
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   639
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   640
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   641
haftmann@21083
   642
syntax (HTML output)
wenzelm@21180
   643
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   644
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   645
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   646
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   647
wenzelm@21180
   648
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   649
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   650
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   651
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   652
haftmann@21083
   653
translations
haftmann@21083
   654
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   655
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   656
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   657
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   658
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   659
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   660
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   661
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   662
haftmann@21083
   663
print_translation {*
haftmann@21083
   664
let
wenzelm@42287
   665
  val All_binder = Mixfix.binder_name @{const_syntax All};
wenzelm@42287
   666
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
haftmann@38786
   667
  val impl = @{const_syntax HOL.implies};
haftmann@38795
   668
  val conj = @{const_syntax HOL.conj};
haftmann@22916
   669
  val less = @{const_syntax less};
haftmann@22916
   670
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   671
wenzelm@21180
   672
  val trans =
wenzelm@35115
   673
   [((All_binder, impl, less),
wenzelm@35115
   674
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
wenzelm@35115
   675
    ((All_binder, impl, less_eq),
wenzelm@35115
   676
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
wenzelm@35115
   677
    ((Ex_binder, conj, less),
wenzelm@35115
   678
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
wenzelm@35115
   679
    ((Ex_binder, conj, less_eq),
wenzelm@35115
   680
    (@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];
wenzelm@21180
   681
wenzelm@35115
   682
  fun matches_bound v t =
wenzelm@35115
   683
    (case t of
wenzelm@35364
   684
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
wenzelm@35115
   685
    | _ => false);
wenzelm@35115
   686
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
wenzelm@49660
   687
  fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
wenzelm@21180
   688
wenzelm@52143
   689
  fun tr' q = (q, fn _ =>
wenzelm@52143
   690
    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
wenzelm@35364
   691
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@35115
   692
        (case AList.lookup (op =) trans (q, c, d) of
wenzelm@35115
   693
          NONE => raise Match
wenzelm@35115
   694
        | SOME (l, g) =>
wenzelm@49660
   695
            if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
wenzelm@49660
   696
            else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
wenzelm@35115
   697
            else raise Match)
wenzelm@52143
   698
      | _ => raise Match));
wenzelm@21524
   699
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   700
*}
haftmann@21083
   701
haftmann@21083
   702
haftmann@21383
   703
subsection {* Transitivity reasoning *}
haftmann@21383
   704
haftmann@25193
   705
context ord
haftmann@25193
   706
begin
haftmann@21383
   707
haftmann@25193
   708
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
haftmann@25193
   709
  by (rule subst)
haftmann@21383
   710
haftmann@25193
   711
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
haftmann@25193
   712
  by (rule ssubst)
haftmann@21383
   713
haftmann@25193
   714
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
haftmann@25193
   715
  by (rule subst)
haftmann@25193
   716
haftmann@25193
   717
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
haftmann@25193
   718
  by (rule ssubst)
haftmann@25193
   719
haftmann@25193
   720
end
haftmann@21383
   721
haftmann@21383
   722
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   723
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   724
proof -
haftmann@21383
   725
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   726
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   727
  also assume "f b < c"
haftmann@34250
   728
  finally (less_trans) show ?thesis .
haftmann@21383
   729
qed
haftmann@21383
   730
haftmann@21383
   731
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   732
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   733
proof -
haftmann@21383
   734
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   735
  assume "a < f b"
haftmann@21383
   736
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@34250
   737
  finally (less_trans) show ?thesis .
haftmann@21383
   738
qed
haftmann@21383
   739
haftmann@21383
   740
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   741
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   742
proof -
haftmann@21383
   743
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   744
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   745
  also assume "f b < c"
haftmann@34250
   746
  finally (le_less_trans) show ?thesis .
haftmann@21383
   747
qed
haftmann@21383
   748
haftmann@21383
   749
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   750
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   751
proof -
haftmann@21383
   752
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   753
  assume "a <= f b"
haftmann@21383
   754
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@34250
   755
  finally (le_less_trans) show ?thesis .
haftmann@21383
   756
qed
haftmann@21383
   757
haftmann@21383
   758
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   759
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   760
proof -
haftmann@21383
   761
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   762
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   763
  also assume "f b <= c"
haftmann@34250
   764
  finally (less_le_trans) show ?thesis .
haftmann@21383
   765
qed
haftmann@21383
   766
haftmann@21383
   767
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   768
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   769
proof -
haftmann@21383
   770
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   771
  assume "a < f b"
haftmann@21383
   772
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@34250
   773
  finally (less_le_trans) show ?thesis .
haftmann@21383
   774
qed
haftmann@21383
   775
haftmann@21383
   776
lemma order_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@21383
   782
  finally (order_trans) show ?thesis .
haftmann@21383
   783
qed
haftmann@21383
   784
haftmann@21383
   785
lemma order_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@21383
   791
  finally (order_trans) show ?thesis .
haftmann@21383
   792
qed
haftmann@21383
   793
haftmann@21383
   794
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   795
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   796
proof -
haftmann@21383
   797
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   798
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   799
  also assume "f b = c"
haftmann@21383
   800
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   801
qed
haftmann@21383
   802
haftmann@21383
   803
lemma ord_eq_le_subst: "a = f b ==> b <= 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 (ord_eq_le_trans) show ?thesis .
haftmann@21383
   810
qed
haftmann@21383
   811
haftmann@21383
   812
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
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 (ord_less_eq_trans) show ?thesis .
haftmann@21383
   819
qed
haftmann@21383
   820
haftmann@21383
   821
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   822
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   823
proof -
haftmann@21383
   824
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   825
  assume "a = f b"
haftmann@21383
   826
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   827
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   828
qed
haftmann@21383
   829
haftmann@21383
   830
text {*
haftmann@21383
   831
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   832
*}
haftmann@21383
   833
haftmann@27682
   834
lemmas [trans] =
haftmann@21383
   835
  order_less_subst2
haftmann@21383
   836
  order_less_subst1
haftmann@21383
   837
  order_le_less_subst2
haftmann@21383
   838
  order_le_less_subst1
haftmann@21383
   839
  order_less_le_subst2
haftmann@21383
   840
  order_less_le_subst1
haftmann@21383
   841
  order_subst2
haftmann@21383
   842
  order_subst1
haftmann@21383
   843
  ord_le_eq_subst
haftmann@21383
   844
  ord_eq_le_subst
haftmann@21383
   845
  ord_less_eq_subst
haftmann@21383
   846
  ord_eq_less_subst
haftmann@21383
   847
  forw_subst
haftmann@21383
   848
  back_subst
haftmann@21383
   849
  rev_mp
haftmann@21383
   850
  mp
haftmann@27682
   851
haftmann@27682
   852
lemmas (in order) [trans] =
haftmann@27682
   853
  neq_le_trans
haftmann@27682
   854
  le_neq_trans
haftmann@27682
   855
haftmann@27682
   856
lemmas (in preorder) [trans] =
haftmann@27682
   857
  less_trans
haftmann@27682
   858
  less_asym'
haftmann@27682
   859
  le_less_trans
haftmann@27682
   860
  less_le_trans
haftmann@21383
   861
  order_trans
haftmann@27682
   862
haftmann@27682
   863
lemmas (in order) [trans] =
haftmann@27682
   864
  antisym
haftmann@27682
   865
haftmann@27682
   866
lemmas (in ord) [trans] =
haftmann@27682
   867
  ord_le_eq_trans
haftmann@27682
   868
  ord_eq_le_trans
haftmann@27682
   869
  ord_less_eq_trans
haftmann@27682
   870
  ord_eq_less_trans
haftmann@27682
   871
haftmann@27682
   872
lemmas [trans] =
haftmann@27682
   873
  trans
haftmann@27682
   874
haftmann@27682
   875
lemmas order_trans_rules =
haftmann@27682
   876
  order_less_subst2
haftmann@27682
   877
  order_less_subst1
haftmann@27682
   878
  order_le_less_subst2
haftmann@27682
   879
  order_le_less_subst1
haftmann@27682
   880
  order_less_le_subst2
haftmann@27682
   881
  order_less_le_subst1
haftmann@27682
   882
  order_subst2
haftmann@27682
   883
  order_subst1
haftmann@27682
   884
  ord_le_eq_subst
haftmann@27682
   885
  ord_eq_le_subst
haftmann@27682
   886
  ord_less_eq_subst
haftmann@27682
   887
  ord_eq_less_subst
haftmann@27682
   888
  forw_subst
haftmann@27682
   889
  back_subst
haftmann@27682
   890
  rev_mp
haftmann@27682
   891
  mp
haftmann@27682
   892
  neq_le_trans
haftmann@27682
   893
  le_neq_trans
haftmann@27682
   894
  less_trans
haftmann@27682
   895
  less_asym'
haftmann@27682
   896
  le_less_trans
haftmann@27682
   897
  less_le_trans
haftmann@27682
   898
  order_trans
haftmann@27682
   899
  antisym
haftmann@21383
   900
  ord_le_eq_trans
haftmann@21383
   901
  ord_eq_le_trans
haftmann@21383
   902
  ord_less_eq_trans
haftmann@21383
   903
  ord_eq_less_trans
haftmann@21383
   904
  trans
haftmann@21383
   905
haftmann@21083
   906
text {* These support proving chains of decreasing inequalities
haftmann@21083
   907
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   908
blanchet@45221
   909
lemma xt1 [no_atp]:
haftmann@21083
   910
  "a = b ==> b > c ==> a > c"
haftmann@21083
   911
  "a > b ==> b = c ==> a > c"
haftmann@21083
   912
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   913
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   914
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   915
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   916
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   917
  "(x::'a::order) >= y ==> y > z ==> x > z"
wenzelm@23417
   918
  "(a::'a::order) > b ==> b > a ==> P"
haftmann@21083
   919
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   920
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   921
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   922
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   923
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   924
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   925
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@25076
   926
  by auto
haftmann@21083
   927
blanchet@45221
   928
lemma xt2 [no_atp]:
haftmann@21083
   929
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   930
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   931
blanchet@45221
   932
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
haftmann@21083
   933
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   934
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   935
blanchet@45221
   936
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   937
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   938
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   939
blanchet@45221
   940
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   941
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   942
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   943
blanchet@45221
   944
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   945
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   946
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   947
blanchet@45221
   948
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   949
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   950
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   951
blanchet@45221
   952
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   953
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   954
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   955
blanchet@45221
   956
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   957
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   958
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   959
blanchet@54147
   960
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   961
haftmann@21083
   962
(* 
haftmann@21083
   963
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   964
  for the wrong thing in an Isar proof.
haftmann@21083
   965
haftmann@21083
   966
  The extra transitivity rules can be used as follows: 
haftmann@21083
   967
haftmann@21083
   968
lemma "(a::'a::order) > z"
haftmann@21083
   969
proof -
haftmann@21083
   970
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   971
    sorry
haftmann@21083
   972
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   973
    sorry
haftmann@21083
   974
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   975
    sorry
haftmann@21083
   976
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   977
    sorry
haftmann@21083
   978
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   979
    sorry
haftmann@21083
   980
  also (xtrans) have "?rhs > z"
haftmann@21083
   981
    sorry
haftmann@21083
   982
  finally (xtrans) show ?thesis .
haftmann@21083
   983
qed
haftmann@21083
   984
haftmann@21083
   985
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   986
  leave out the "(xtrans)" above.
haftmann@21083
   987
*)
haftmann@21083
   988
haftmann@23881
   989
haftmann@54860
   990
subsection {* Monotonicity *}
haftmann@21083
   991
haftmann@25076
   992
context order
haftmann@25076
   993
begin
haftmann@25076
   994
haftmann@30298
   995
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
haftmann@25076
   996
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
haftmann@25076
   997
haftmann@25076
   998
lemma monoI [intro?]:
haftmann@25076
   999
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
haftmann@25076
  1000
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
haftmann@25076
  1001
  unfolding mono_def by iprover
haftmann@21216
  1002
haftmann@25076
  1003
lemma monoD [dest?]:
haftmann@25076
  1004
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
haftmann@25076
  1005
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
haftmann@25076
  1006
  unfolding mono_def by iprover
haftmann@25076
  1007
haftmann@51263
  1008
lemma monoE:
haftmann@51263
  1009
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
haftmann@51263
  1010
  assumes "mono f"
haftmann@51263
  1011
  assumes "x \<le> y"
haftmann@51263
  1012
  obtains "f x \<le> f y"
haftmann@51263
  1013
proof
haftmann@51263
  1014
  from assms show "f x \<le> f y" by (simp add: mono_def)
haftmann@51263
  1015
qed
haftmann@51263
  1016
haftmann@30298
  1017
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
haftmann@30298
  1018
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
haftmann@30298
  1019
haftmann@30298
  1020
lemma strict_monoI [intro?]:
haftmann@30298
  1021
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
haftmann@30298
  1022
  shows "strict_mono f"
haftmann@30298
  1023
  using assms unfolding strict_mono_def by auto
haftmann@30298
  1024
haftmann@30298
  1025
lemma strict_monoD [dest?]:
haftmann@30298
  1026
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
haftmann@30298
  1027
  unfolding strict_mono_def by auto
haftmann@30298
  1028
haftmann@30298
  1029
lemma strict_mono_mono [dest?]:
haftmann@30298
  1030
  assumes "strict_mono f"
haftmann@30298
  1031
  shows "mono f"
haftmann@30298
  1032
proof (rule monoI)
haftmann@30298
  1033
  fix x y
haftmann@30298
  1034
  assume "x \<le> y"
haftmann@30298
  1035
  show "f x \<le> f y"
haftmann@30298
  1036
  proof (cases "x = y")
haftmann@30298
  1037
    case True then show ?thesis by simp
haftmann@30298
  1038
  next
haftmann@30298
  1039
    case False with `x \<le> y` have "x < y" by simp
haftmann@30298
  1040
    with assms strict_monoD have "f x < f y" by auto
haftmann@30298
  1041
    then show ?thesis by simp
haftmann@30298
  1042
  qed
haftmann@30298
  1043
qed
haftmann@30298
  1044
haftmann@25076
  1045
end
haftmann@25076
  1046
haftmann@25076
  1047
context linorder
haftmann@25076
  1048
begin
haftmann@25076
  1049
haftmann@51263
  1050
lemma mono_invE:
haftmann@51263
  1051
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
haftmann@51263
  1052
  assumes "mono f"
haftmann@51263
  1053
  assumes "f x < f y"
haftmann@51263
  1054
  obtains "x \<le> y"
haftmann@51263
  1055
proof
haftmann@51263
  1056
  show "x \<le> y"
haftmann@51263
  1057
  proof (rule ccontr)
haftmann@51263
  1058
    assume "\<not> x \<le> y"
haftmann@51263
  1059
    then have "y \<le> x" by simp
haftmann@51263
  1060
    with `mono f` obtain "f y \<le> f x" by (rule monoE)
haftmann@51263
  1061
    with `f x < f y` show False by simp
haftmann@51263
  1062
  qed
haftmann@51263
  1063
qed
haftmann@51263
  1064
haftmann@30298
  1065
lemma strict_mono_eq:
haftmann@30298
  1066
  assumes "strict_mono f"
haftmann@30298
  1067
  shows "f x = f y \<longleftrightarrow> x = y"
haftmann@30298
  1068
proof
haftmann@30298
  1069
  assume "f x = f y"
haftmann@30298
  1070
  show "x = y" proof (cases x y rule: linorder_cases)
haftmann@30298
  1071
    case less with assms strict_monoD have "f x < f y" by auto
haftmann@30298
  1072
    with `f x = f y` show ?thesis by simp
haftmann@30298
  1073
  next
haftmann@30298
  1074
    case equal then show ?thesis .
haftmann@30298
  1075
  next
haftmann@30298
  1076
    case greater with assms strict_monoD have "f y < f x" by auto
haftmann@30298
  1077
    with `f x = f y` show ?thesis by simp
haftmann@30298
  1078
  qed
haftmann@30298
  1079
qed simp
haftmann@30298
  1080
haftmann@30298
  1081
lemma strict_mono_less_eq:
haftmann@30298
  1082
  assumes "strict_mono f"
haftmann@30298
  1083
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
haftmann@30298
  1084
proof
haftmann@30298
  1085
  assume "x \<le> y"
haftmann@30298
  1086
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
haftmann@30298
  1087
next
haftmann@30298
  1088
  assume "f x \<le> f y"
haftmann@30298
  1089
  show "x \<le> y" proof (rule ccontr)
haftmann@30298
  1090
    assume "\<not> x \<le> y" then have "y < x" by simp
haftmann@30298
  1091
    with assms strict_monoD have "f y < f x" by auto
haftmann@30298
  1092
    with `f x \<le> f y` show False by simp
haftmann@30298
  1093
  qed
haftmann@30298
  1094
qed
haftmann@30298
  1095
  
haftmann@30298
  1096
lemma strict_mono_less:
haftmann@30298
  1097
  assumes "strict_mono f"
haftmann@30298
  1098
  shows "f x < f y \<longleftrightarrow> x < y"
haftmann@30298
  1099
  using assms
haftmann@30298
  1100
    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
haftmann@30298
  1101
haftmann@54860
  1102
end
haftmann@54860
  1103
haftmann@54860
  1104
haftmann@54860
  1105
subsection {* min and max *}
haftmann@54860
  1106
haftmann@54860
  1107
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@54860
  1108
  "min a b = (if a \<le> b then a else b)"
haftmann@54860
  1109
haftmann@54860
  1110
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@54860
  1111
  "max a b = (if a \<le> b then b else a)"
haftmann@54860
  1112
haftmann@54860
  1113
context linorder
haftmann@54860
  1114
begin
haftmann@54860
  1115
haftmann@54860
  1116
lemma min_le_iff_disj:
haftmann@54860
  1117
  "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
haftmann@54860
  1118
unfolding min_def using linear by (auto intro: order_trans)
haftmann@54860
  1119
haftmann@54860
  1120
lemma le_max_iff_disj:
haftmann@54860
  1121
  "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
haftmann@54860
  1122
unfolding max_def using linear by (auto intro: order_trans)
haftmann@54860
  1123
haftmann@54860
  1124
lemma min_less_iff_disj:
haftmann@54860
  1125
  "min x y < z \<longleftrightarrow> x < z \<or> y < z"
haftmann@54860
  1126
unfolding min_def le_less using less_linear by (auto intro: less_trans)
haftmann@54860
  1127
haftmann@54860
  1128
lemma less_max_iff_disj:
haftmann@54860
  1129
  "z < max x y \<longleftrightarrow> z < x \<or> z < y"
haftmann@54860
  1130
unfolding max_def le_less using less_linear by (auto intro: less_trans)
haftmann@54860
  1131
haftmann@54860
  1132
lemma min_less_iff_conj [simp]:
haftmann@54860
  1133
  "z < min x y \<longleftrightarrow> z < x \<and> z < y"
haftmann@54860
  1134
unfolding min_def le_less using less_linear by (auto intro: less_trans)
haftmann@54860
  1135
haftmann@54860
  1136
lemma max_less_iff_conj [simp]:
haftmann@54860
  1137
  "max x y < z \<longleftrightarrow> x < z \<and> y < z"
haftmann@54860
  1138
unfolding max_def le_less using less_linear by (auto intro: less_trans)
haftmann@54860
  1139
haftmann@54860
  1140
lemma split_min [no_atp]:
haftmann@54860
  1141
  "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
haftmann@54860
  1142
by (simp add: min_def)
haftmann@54860
  1143
haftmann@54860
  1144
lemma split_max [no_atp]:
haftmann@54860
  1145
  "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
haftmann@54860
  1146
by (simp add: max_def)
haftmann@54860
  1147
haftmann@25076
  1148
lemma min_of_mono:
haftmann@25076
  1149
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
wenzelm@25377
  1150
  shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
haftmann@25076
  1151
  by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
haftmann@25076
  1152
haftmann@25076
  1153
lemma max_of_mono:
haftmann@25076
  1154
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
wenzelm@25377
  1155
  shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
haftmann@25076
  1156
  by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
haftmann@25076
  1157
haftmann@25076
  1158
end
haftmann@21083
  1159
noschinl@45931
  1160
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
nipkow@23212
  1161
by (simp add: min_def)
haftmann@21383
  1162
haftmann@54857
  1163
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
nipkow@23212
  1164
by (simp add: max_def)
haftmann@21383
  1165
noschinl@45931
  1166
lemma min_absorb2: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> min x y = y"
noschinl@45931
  1167
by (simp add:min_def)
noschinl@45893
  1168
noschinl@45931
  1169
lemma max_absorb1: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> max x y = x"
noschinl@45893
  1170
by (simp add: max_def)
noschinl@45893
  1171
noschinl@45893
  1172
haftmann@43813
  1173
subsection {* (Unique) top and bottom elements *}
haftmann@28685
  1174
haftmann@52729
  1175
class bot =
haftmann@43853
  1176
  fixes bot :: 'a ("\<bottom>")
haftmann@52729
  1177
haftmann@52729
  1178
class order_bot = order + bot +
haftmann@51487
  1179
  assumes bot_least: "\<bottom> \<le> a"
haftmann@51487
  1180
haftmann@52729
  1181
sublocale order_bot < bot!: ordering_top greater_eq greater bot
haftmann@51546
  1182
  by default (fact bot_least)
haftmann@51487
  1183
haftmann@52729
  1184
context order_bot
haftmann@43814
  1185
begin
haftmann@43814
  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@51487
  1210
haftmann@52729
  1211
sublocale order_top < top!: ordering_top less_eq less top
haftmann@51546
  1212
  by default (fact top_greatest)
haftmann@51487
  1213
haftmann@52729
  1214
context order_top
haftmann@43814
  1215
begin
haftmann@43814
  1216
haftmann@43853
  1217
lemma top_le:
haftmann@43853
  1218
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
haftmann@51487
  1219
  by (fact top.extremum_uniqueI)
haftmann@43853
  1220
haftmann@43816
  1221
lemma top_unique:
haftmann@43853
  1222
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
haftmann@51487
  1223
  by (fact top.extremum_unique)
haftmann@43853
  1224
haftmann@51487
  1225
lemma not_top_less:
haftmann@51487
  1226
  "\<not> \<top> < a"
haftmann@51487
  1227
  by (fact top.extremum_strict)
haftmann@43816
  1228
haftmann@43814
  1229
lemma less_top:
haftmann@43853
  1230
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
haftmann@51487
  1231
  by (fact top.not_eq_extremum)
haftmann@43814
  1232
haftmann@43814
  1233
end
haftmann@28685
  1234
haftmann@28685
  1235
haftmann@27823
  1236
subsection {* Dense orders *}
haftmann@27823
  1237
hoelzl@53216
  1238
class dense_order = order +
hoelzl@51329
  1239
  assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
hoelzl@51329
  1240
hoelzl@53216
  1241
class dense_linorder = linorder + dense_order
hoelzl@35579
  1242
begin
haftmann@27823
  1243
hoelzl@35579
  1244
lemma dense_le:
hoelzl@35579
  1245
  fixes y z :: 'a
hoelzl@35579
  1246
  assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
hoelzl@35579
  1247
  shows "y \<le> z"
hoelzl@35579
  1248
proof (rule ccontr)
hoelzl@35579
  1249
  assume "\<not> ?thesis"
hoelzl@35579
  1250
  hence "z < y" by simp
hoelzl@35579
  1251
  from dense[OF this]
hoelzl@35579
  1252
  obtain x where "x < y" and "z < x" by safe
hoelzl@35579
  1253
  moreover have "x \<le> z" using assms[OF `x < y`] .
hoelzl@35579
  1254
  ultimately show False by auto
hoelzl@35579
  1255
qed
hoelzl@35579
  1256
hoelzl@35579
  1257
lemma dense_le_bounded:
hoelzl@35579
  1258
  fixes x y z :: 'a
hoelzl@35579
  1259
  assumes "x < y"
hoelzl@35579
  1260
  assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
hoelzl@35579
  1261
  shows "y \<le> z"
hoelzl@35579
  1262
proof (rule dense_le)
hoelzl@35579
  1263
  fix w assume "w < y"
hoelzl@35579
  1264
  from dense[OF `x < y`] obtain u where "x < u" "u < y" by safe
hoelzl@35579
  1265
  from linear[of u w]
hoelzl@35579
  1266
  show "w \<le> z"
hoelzl@35579
  1267
  proof (rule disjE)
hoelzl@35579
  1268
    assume "u \<le> w"
hoelzl@35579
  1269
    from less_le_trans[OF `x < u` `u \<le> w`] `w < y`
hoelzl@35579
  1270
    show "w \<le> z" by (rule *)
hoelzl@35579
  1271
  next
hoelzl@35579
  1272
    assume "w \<le> u"
hoelzl@35579
  1273
    from `w \<le> u` *[OF `x < u` `u < y`]
hoelzl@35579
  1274
    show "w \<le> z" by (rule order_trans)
hoelzl@35579
  1275
  qed
hoelzl@35579
  1276
qed
hoelzl@35579
  1277
hoelzl@51329
  1278
lemma dense_ge:
hoelzl@51329
  1279
  fixes y z :: 'a
hoelzl@51329
  1280
  assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
hoelzl@51329
  1281
  shows "y \<le> z"
hoelzl@51329
  1282
proof (rule ccontr)
hoelzl@51329
  1283
  assume "\<not> ?thesis"
hoelzl@51329
  1284
  hence "z < y" by simp
hoelzl@51329
  1285
  from dense[OF this]
hoelzl@51329
  1286
  obtain x where "x < y" and "z < x" by safe
hoelzl@51329
  1287
  moreover have "y \<le> x" using assms[OF `z < x`] .
hoelzl@51329
  1288
  ultimately show False by auto
hoelzl@51329
  1289
qed
hoelzl@51329
  1290
hoelzl@51329
  1291
lemma dense_ge_bounded:
hoelzl@51329
  1292
  fixes x y z :: 'a
hoelzl@51329
  1293
  assumes "z < x"
hoelzl@51329
  1294
  assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
hoelzl@51329
  1295
  shows "y \<le> z"
hoelzl@51329
  1296
proof (rule dense_ge)
hoelzl@51329
  1297
  fix w assume "z < w"
hoelzl@51329
  1298
  from dense[OF `z < x`] obtain u where "z < u" "u < x" by safe
hoelzl@51329
  1299
  from linear[of u w]
hoelzl@51329
  1300
  show "y \<le> w"
hoelzl@51329
  1301
  proof (rule disjE)
hoelzl@51329
  1302
    assume "w \<le> u"
hoelzl@51329
  1303
    from `z < w` le_less_trans[OF `w \<le> u` `u < x`]
hoelzl@51329
  1304
    show "y \<le> w" by (rule *)
hoelzl@51329
  1305
  next
hoelzl@51329
  1306
    assume "u \<le> w"
hoelzl@51329
  1307
    from *[OF `z < u` `u < x`] `u \<le> w`
hoelzl@51329
  1308
    show "y \<le> w" by (rule order_trans)
hoelzl@51329
  1309
  qed
hoelzl@51329
  1310
qed
hoelzl@51329
  1311
hoelzl@35579
  1312
end
haftmann@27823
  1313
hoelzl@51329
  1314
class no_top = order + 
hoelzl@51329
  1315
  assumes gt_ex: "\<exists>y. x < y"
hoelzl@51329
  1316
hoelzl@51329
  1317
class no_bot = order + 
hoelzl@51329
  1318
  assumes lt_ex: "\<exists>y. y < x"
hoelzl@51329
  1319
hoelzl@53216
  1320
class unbounded_dense_linorder = dense_linorder + no_top + no_bot
hoelzl@51329
  1321
haftmann@51546
  1322
haftmann@27823
  1323
subsection {* Wellorders *}
haftmann@27823
  1324
haftmann@27823
  1325
class wellorder = linorder +
haftmann@27823
  1326
  assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
haftmann@27823
  1327
begin
haftmann@27823
  1328
haftmann@27823
  1329
lemma wellorder_Least_lemma:
haftmann@27823
  1330
  fixes k :: 'a
haftmann@27823
  1331
  assumes "P k"
haftmann@34250
  1332
  shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
haftmann@27823
  1333
proof -
haftmann@27823
  1334
  have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
haftmann@27823
  1335
  using assms proof (induct k rule: less_induct)
haftmann@27823
  1336
    case (less x) then have "P x" by simp
haftmann@27823
  1337
    show ?case proof (rule classical)
haftmann@27823
  1338
      assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
haftmann@27823
  1339
      have "\<And>y. P y \<Longrightarrow> x \<le> y"
haftmann@27823
  1340
      proof (rule classical)
haftmann@27823
  1341
        fix y
hoelzl@38705
  1342
        assume "P y" and "\<not> x \<le> y"
haftmann@27823
  1343
        with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
haftmann@27823
  1344
          by (auto simp add: not_le)
haftmann@27823
  1345
        with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
haftmann@27823
  1346
          by auto
haftmann@27823
  1347
        then show "x \<le> y" by auto
haftmann@27823
  1348
      qed
haftmann@27823
  1349
      with `P x` have Least: "(LEAST a. P a) = x"
haftmann@27823
  1350
        by (rule Least_equality)
haftmann@27823
  1351
      with `P x` show ?thesis by simp
haftmann@27823
  1352
    qed
haftmann@27823
  1353
  qed
haftmann@27823
  1354
  then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
haftmann@27823
  1355
qed
haftmann@27823
  1356
haftmann@27823
  1357
-- "The following 3 lemmas are due to Brian Huffman"
haftmann@27823
  1358
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
haftmann@27823
  1359
  by (erule exE) (erule LeastI)
haftmann@27823
  1360
haftmann@27823
  1361
lemma LeastI2:
haftmann@27823
  1362
  "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
haftmann@27823
  1363
  by (blast intro: LeastI)
haftmann@27823
  1364
haftmann@27823
  1365
lemma LeastI2_ex:
haftmann@27823
  1366
  "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
haftmann@27823
  1367
  by (blast intro: LeastI_ex)
haftmann@27823
  1368
hoelzl@38705
  1369
lemma LeastI2_wellorder:
hoelzl@38705
  1370
  assumes "P a"
hoelzl@38705
  1371
  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
hoelzl@38705
  1372
  shows "Q (Least P)"
hoelzl@38705
  1373
proof (rule LeastI2_order)
hoelzl@38705
  1374
  show "P (Least P)" using `P a` by (rule LeastI)
hoelzl@38705
  1375
next
hoelzl@38705
  1376
  fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
hoelzl@38705
  1377
next
hoelzl@38705
  1378
  fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
hoelzl@38705
  1379
qed
hoelzl@38705
  1380
haftmann@27823
  1381
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
haftmann@27823
  1382
apply (simp (no_asm_use) add: not_le [symmetric])
haftmann@27823
  1383
apply (erule contrapos_nn)
haftmann@27823
  1384
apply (erule Least_le)
haftmann@27823
  1385
done
haftmann@27823
  1386
hoelzl@38705
  1387
end
haftmann@27823
  1388
haftmann@28685
  1389
haftmann@46631
  1390
subsection {* Order on @{typ bool} *}
haftmann@28685
  1391
haftmann@52729
  1392
instantiation bool :: "{order_bot, order_top, linorder}"
haftmann@28685
  1393
begin
haftmann@28685
  1394
haftmann@28685
  1395
definition
haftmann@41080
  1396
  le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
haftmann@28685
  1397
haftmann@28685
  1398
definition
haftmann@41080
  1399
  [simp]: "(P\<Colon>bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
haftmann@28685
  1400
haftmann@28685
  1401
definition
haftmann@46631
  1402
  [simp]: "\<bottom> \<longleftrightarrow> False"
haftmann@28685
  1403
haftmann@28685
  1404
definition
haftmann@46631
  1405
  [simp]: "\<top> \<longleftrightarrow> True"
haftmann@28685
  1406
haftmann@28685
  1407
instance proof
haftmann@41080
  1408
qed auto
haftmann@28685
  1409
nipkow@15524
  1410
end
haftmann@28685
  1411
haftmann@28685
  1412
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@41080
  1413
  by simp
haftmann@28685
  1414
haftmann@28685
  1415
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@41080
  1416
  by simp
haftmann@28685
  1417
haftmann@28685
  1418
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41080
  1419
  by simp
haftmann@28685
  1420
haftmann@28685
  1421
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@41080
  1422
  by simp
haftmann@32899
  1423
haftmann@46631
  1424
lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
haftmann@41080
  1425
  by simp
haftmann@32899
  1426
haftmann@46631
  1427
lemma top_boolI: \<top>
haftmann@41080
  1428
  by simp
haftmann@28685
  1429
haftmann@28685
  1430
lemma [code]:
haftmann@28685
  1431
  "False \<le> b \<longleftrightarrow> True"
haftmann@28685
  1432
  "True \<le> b \<longleftrightarrow> b"
haftmann@28685
  1433
  "False < b \<longleftrightarrow> b"
haftmann@28685
  1434
  "True < b \<longleftrightarrow> False"
haftmann@41080
  1435
  by simp_all
haftmann@28685
  1436
haftmann@28685
  1437
haftmann@46631
  1438
subsection {* Order on @{typ "_ \<Rightarrow> _"} *}
haftmann@28685
  1439
haftmann@28685
  1440
instantiation "fun" :: (type, ord) ord
haftmann@28685
  1441
begin
haftmann@28685
  1442
haftmann@28685
  1443
definition
haftmann@37767
  1444
  le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
haftmann@28685
  1445
haftmann@28685
  1446
definition
haftmann@41080
  1447
  "(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
haftmann@28685
  1448
haftmann@28685
  1449
instance ..
haftmann@28685
  1450
haftmann@28685
  1451
end
haftmann@28685
  1452
haftmann@28685
  1453
instance "fun" :: (type, preorder) preorder proof
haftmann@28685
  1454
qed (auto simp add: le_fun_def less_fun_def
huffman@44921
  1455
  intro: order_trans antisym)
haftmann@28685
  1456
haftmann@28685
  1457
instance "fun" :: (type, order) order proof
huffman@44921
  1458
qed (auto simp add: le_fun_def intro: antisym)
haftmann@28685
  1459
haftmann@41082
  1460
instantiation "fun" :: (type, bot) bot
haftmann@41082
  1461
begin
haftmann@41082
  1462
haftmann@41082
  1463
definition
haftmann@46631
  1464
  "\<bottom> = (\<lambda>x. \<bottom>)"
haftmann@41082
  1465
haftmann@52729
  1466
instance ..
haftmann@52729
  1467
haftmann@52729
  1468
end
haftmann@52729
  1469
haftmann@52729
  1470
instantiation "fun" :: (type, order_bot) order_bot
haftmann@52729
  1471
begin
haftmann@52729
  1472
haftmann@49769
  1473
lemma bot_apply [simp, code]:
haftmann@46631
  1474
  "\<bottom> x = \<bottom>"
haftmann@41082
  1475
  by (simp add: bot_fun_def)
haftmann@41082
  1476
haftmann@41082
  1477
instance proof
noschinl@46884
  1478
qed (simp add: le_fun_def)
haftmann@41082
  1479
haftmann@41082
  1480
end
haftmann@41082
  1481
haftmann@28685
  1482
instantiation "fun" :: (type, top) top
haftmann@28685
  1483
begin
haftmann@28685
  1484
haftmann@28685
  1485
definition
haftmann@46631
  1486
  [no_atp]: "\<top> = (\<lambda>x. \<top>)"
haftmann@28685
  1487
haftmann@52729
  1488
instance ..
haftmann@52729
  1489
haftmann@52729
  1490
end
haftmann@52729
  1491
haftmann@52729
  1492
instantiation "fun" :: (type, order_top) order_top
haftmann@52729
  1493
begin
haftmann@52729
  1494
haftmann@49769
  1495
lemma top_apply [simp, code]:
haftmann@46631
  1496
  "\<top> x = \<top>"
haftmann@41080
  1497
  by (simp add: top_fun_def)
haftmann@41080
  1498
haftmann@28685
  1499
instance proof
noschinl@46884
  1500
qed (simp add: le_fun_def)
haftmann@28685
  1501
haftmann@28685
  1502
end
haftmann@28685
  1503
haftmann@28685
  1504
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
haftmann@28685
  1505
  unfolding le_fun_def by simp
haftmann@28685
  1506
haftmann@28685
  1507
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@28685
  1508
  unfolding le_fun_def by simp
haftmann@28685
  1509
haftmann@28685
  1510
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
haftmann@54860
  1511
  by (rule le_funE)
haftmann@28685
  1512
haftmann@34250
  1513
haftmann@46631
  1514
subsection {* Order on unary and binary predicates *}
haftmann@46631
  1515
haftmann@46631
  1516
lemma predicate1I:
haftmann@46631
  1517
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
haftmann@46631
  1518
  shows "P \<le> Q"
haftmann@46631
  1519
  apply (rule le_funI)
haftmann@46631
  1520
  apply (rule le_boolI)
haftmann@46631
  1521
  apply (rule PQ)
haftmann@46631
  1522
  apply assumption
haftmann@46631
  1523
  done
haftmann@46631
  1524
haftmann@46631
  1525
lemma predicate1D:
haftmann@46631
  1526
  "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
haftmann@46631
  1527
  apply (erule le_funE)
haftmann@46631
  1528
  apply (erule le_boolE)
haftmann@46631
  1529
  apply assumption+
haftmann@46631
  1530
  done
haftmann@46631
  1531
haftmann@46631
  1532
lemma rev_predicate1D:
haftmann@46631
  1533
  "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
haftmann@46631
  1534
  by (rule predicate1D)
haftmann@46631
  1535
haftmann@46631
  1536
lemma predicate2I:
haftmann@46631
  1537
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
haftmann@46631
  1538
  shows "P \<le> Q"
haftmann@46631
  1539
  apply (rule le_funI)+
haftmann@46631
  1540
  apply (rule le_boolI)
haftmann@46631
  1541
  apply (rule PQ)
haftmann@46631
  1542
  apply assumption
haftmann@46631
  1543
  done
haftmann@46631
  1544
haftmann@46631
  1545
lemma predicate2D:
haftmann@46631
  1546
  "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
haftmann@46631
  1547
  apply (erule le_funE)+
haftmann@46631
  1548
  apply (erule le_boolE)
haftmann@46631
  1549
  apply assumption+
haftmann@46631
  1550
  done
haftmann@46631
  1551
haftmann@46631
  1552
lemma rev_predicate2D:
haftmann@46631
  1553
  "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
haftmann@46631
  1554
  by (rule predicate2D)
haftmann@46631
  1555
haftmann@46631
  1556
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
haftmann@46631
  1557
  by (simp add: bot_fun_def)
haftmann@46631
  1558
haftmann@46631
  1559
lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
haftmann@46631
  1560
  by (simp add: bot_fun_def)
haftmann@46631
  1561
haftmann@46631
  1562
lemma top1I: "\<top> x"
haftmann@46631
  1563
  by (simp add: top_fun_def)
haftmann@46631
  1564
haftmann@46631
  1565
lemma top2I: "\<top> x y"
haftmann@46631
  1566
  by (simp add: top_fun_def)
haftmann@46631
  1567
haftmann@46631
  1568
haftmann@34250
  1569
subsection {* Name duplicates *}
haftmann@34250
  1570
haftmann@34250
  1571
lemmas order_eq_refl = preorder_class.eq_refl
haftmann@34250
  1572
lemmas order_less_irrefl = preorder_class.less_irrefl
haftmann@34250
  1573
lemmas order_less_imp_le = preorder_class.less_imp_le
haftmann@34250
  1574
lemmas order_less_not_sym = preorder_class.less_not_sym
haftmann@34250
  1575
lemmas order_less_asym = preorder_class.less_asym
haftmann@34250
  1576
lemmas order_less_trans = preorder_class.less_trans
haftmann@34250
  1577
lemmas order_le_less_trans = preorder_class.le_less_trans
haftmann@34250
  1578
lemmas order_less_le_trans = preorder_class.less_le_trans
haftmann@34250
  1579
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
haftmann@34250
  1580
lemmas order_less_imp_triv = preorder_class.less_imp_triv
haftmann@34250
  1581
lemmas order_less_asym' = preorder_class.less_asym'
haftmann@34250
  1582
haftmann@34250
  1583
lemmas order_less_le = order_class.less_le
haftmann@34250
  1584
lemmas order_le_less = order_class.le_less
haftmann@34250
  1585
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
haftmann@34250
  1586
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
haftmann@34250
  1587
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
haftmann@34250
  1588
lemmas order_neq_le_trans = order_class.neq_le_trans
haftmann@34250
  1589
lemmas order_le_neq_trans = order_class.le_neq_trans
haftmann@34250
  1590
lemmas order_antisym = order_class.antisym
haftmann@34250
  1591
lemmas order_eq_iff = order_class.eq_iff
haftmann@34250
  1592
lemmas order_antisym_conv = order_class.antisym_conv
haftmann@34250
  1593
haftmann@34250
  1594
lemmas linorder_linear = linorder_class.linear
haftmann@34250
  1595
lemmas linorder_less_linear = linorder_class.less_linear
haftmann@34250
  1596
lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@34250
  1597
lemmas linorder_le_cases = linorder_class.le_cases
haftmann@34250
  1598
lemmas linorder_not_less = linorder_class.not_less
haftmann@34250
  1599
lemmas linorder_not_le = linorder_class.not_le
haftmann@34250
  1600
lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@34250
  1601
lemmas linorder_neqE = linorder_class.neqE
haftmann@34250
  1602
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@34250
  1603
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@34250
  1604
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
haftmann@34250
  1605
haftmann@28685
  1606
end
haftmann@51487
  1607