src/HOL/Orderings.thy
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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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diff changeset
   328
 "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   329
apply(simp add:not_less le_less)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   330
apply blast
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   331
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   332
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   333
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   334
apply (simp add: less_le)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   335
using linear apply (blast intro: antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   336
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   337
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   338
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   339
by (cut_tac x = x and y = y in less_linear, auto)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   340
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   341
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   342
by (simp add: neq_iff) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   343
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   344
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   345
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   346
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   347
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   348
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   349
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   350
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   351
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   352
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   353
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   354
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   355
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   356
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   357
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   358
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   359
(*FIXME inappropriate name (or delete altogether)*)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   360
lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   361
unfolding not_le .
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   362
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   363
text \<open>Dual order\<close>
22916
haftmann
parents: 22886
diff changeset
   364
26014
00c2c3525bef dual orders and dual lattices
haftmann
parents: 25614
diff changeset
   365
lemma dual_linorder:
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   366
  "class.linorder (op \<ge>) (op >)"
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   367
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
22916
haftmann
parents: 22886
diff changeset
   368
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   369
end
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   370
23948
261bd4678076 using class target
haftmann
parents: 23881
diff changeset
   371
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   372
text \<open>Alternative introduction rule with bias towards strict order\<close>
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   373
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   374
lemma linorder_strictI:
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   375
  fixes less (infix "\<sqsubset>" 50)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   376
    and less_eq (infix "\<sqsubseteq>" 50)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   377
  assumes "class.order less_eq less"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   378
  assumes trichotomy: "\<And>a b. a \<sqsubset> b \<or> a = b \<or> b \<sqsubset> a"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   379
  shows "class.linorder less_eq less"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   380
proof -
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   381
  interpret order less_eq less
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   382
    by (fact \<open>class.order less_eq less\<close>)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   383
  show ?thesis
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   384
  proof
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   385
    fix a b
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   386
    show "a \<sqsubseteq> b \<or> b \<sqsubseteq> a"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   387
      using trichotomy by (auto simp add: le_less)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   388
  qed
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   389
qed
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   390
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   391
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   392
subsection \<open>Reasoning tools setup\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   393
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   394
ML \<open>
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   395
signature ORDERS =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   396
sig
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   397
  val print_structures: Proof.context -> unit
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   398
  val order_tac: Proof.context -> thm list -> int -> tactic
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   399
  val add_struct: string * term list -> string -> attribute
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   400
  val del_struct: string * term list -> attribute
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   401
end;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   402
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   403
structure Orders: ORDERS =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   404
struct
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   405
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   406
(* context data *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   407
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   408
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   409
  s1 = s2 andalso eq_list (op aconv) (ts1, ts2);
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   410
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   411
structure Data = Generic_Data
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   412
(
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   413
  type T = ((string * term list) * Order_Tac.less_arith) list;
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   414
    (* Order structures:
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   415
       identifier of the structure, list of operations and record of theorems
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   416
       needed to set up the transitivity reasoner,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   417
       identifier and operations identify the structure uniquely. *)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   418
  val empty = [];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   419
  val extend = I;
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   420
  fun merge data = AList.join struct_eq (K fst) data;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   421
);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   422
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   423
fun print_structures ctxt =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   424
  let
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   425
    val structs = Data.get (Context.Proof ctxt);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   426
    fun pretty_term t = Pretty.block
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   427
      [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   428
        Pretty.str "::", Pretty.brk 1,
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   429
        Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   430
    fun pretty_struct ((s, ts), _) = Pretty.block
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   431
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   432
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   433
  in
51579
ec3b78ce0758 tuned message;
wenzelm
parents: 51546
diff changeset
   434
    Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs))
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   435
  end;
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   436
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   437
val _ =
59936
b8ffc3dc9e24 @{command_spec} is superseded by @{command_keyword};
wenzelm
parents: 59582
diff changeset
   438
  Outer_Syntax.command @{command_keyword print_orders}
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   439
    "print order structures available to transitivity reasoner"
60097
d20ca79d50e4 discontinued pointless warnings: commands are only defined inside a theory context;
wenzelm
parents: 59936
diff changeset
   440
    (Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of)));
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   441
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   442
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   443
(* tactics *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   444
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   445
fun struct_tac ((s, ops), thms) ctxt facts =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   446
  let
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   447
    val [eq, le, less] = ops;
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   448
    fun decomp thy (@{const Trueprop} $ t) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   449
          let
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   450
            fun excluded t =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   451
              (* exclude numeric types: linear arithmetic subsumes transitivity *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   452
              let val T = type_of t
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   453
              in
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   454
                T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   455
              end;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   456
            fun rel (bin_op $ t1 $ t2) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   457
                  if excluded t1 then NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   458
                  else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   459
                  else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   460
                  else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   461
                  else NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   462
              | rel _ = NONE;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   463
            fun dec (Const (@{const_name Not}, _) $ t) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   464
                  (case rel t of NONE =>
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   465
                    NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   466
                  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   467
              | dec x = rel x;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   468
          in dec t end
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   469
      | decomp _ _ = NONE;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   470
  in
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   471
    (case s of
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   472
      "order" => Order_Tac.partial_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   473
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   474
    | _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner"))
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   475
  end
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   476
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   477
fun order_tac ctxt facts =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   478
  FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt)));
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   479
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   480
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   481
(* attributes *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   482
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   483
fun add_struct s tag =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   484
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   485
    (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   486
fun del_struct s =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   487
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   488
    (fn _ => Data.map (AList.delete struct_eq s));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   489
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   490
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   491
\<close>
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   492
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   493
attribute_setup order = \<open>
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   494
  Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   495
    Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   496
    Scan.repeat Args.term
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   497
    >> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   498
         | ((NONE, n), ts) => Orders.del_struct (n, ts))
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   499
\<close> "theorems controlling transitivity reasoner"
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   500
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   501
method_setup order = \<open>
47432
e1576d13e933 more standard method setup;
wenzelm
parents: 46976
diff changeset
   502
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   503
\<close> "transitivity reasoner"
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   504
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   505
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   506
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   507
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   508
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   509
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   510
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   511
(* The type constraint on @{term op =} below is necessary since the operation
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   512
   is not a parameter of the locale. *)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   513
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   514
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   515
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   516
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   517
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   518
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   519
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   520
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   521
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   522
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   523
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   524
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   525
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   526
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   527
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   528
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   529
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   530
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   531
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   532
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   533
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   534
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   535
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   536
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   537
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   538
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   539
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   540
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   541
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   542
declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   543
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   544
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   545
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   546
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   547
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   548
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   549
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   550
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   551
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   552
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   553
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   554
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   555
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   556
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   557
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   558
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   559
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   560
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   561
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   562
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   563
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   564
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   565
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   566
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   567
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   568
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   569
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   570
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   571
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   572
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   573
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   574
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   575
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   576
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   577
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   578
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   579
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   580
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   581
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   582
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   583
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   584
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   585
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   586
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   587
declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   588
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   589
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   590
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   591
setup \<open>
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   592
  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   593
    mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt)))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   594
  (*Adding the transitivity reasoners also as safe solvers showed a slight
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   595
    speed up, but the reasoning strength appears to be not higher (at least
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   596
    no breaking of additional proofs in the entire HOL distribution, as
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   597
    of 5 March 2004, was observed).*)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   598
\<close>
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   599
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   600
ML \<open>
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   601
local
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   602
  fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   603
in
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   604
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   605
fun antisym_le_simproc ctxt ct =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59000
diff changeset
   606
  (case Thm.term_of ct of
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   607
    (le as Const (_, T)) $ r $ s =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   608
     (let
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   609
        val prems = Simplifier.prems_of ctxt;
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   610
        val less = Const (@{const_name less}, T);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   611
        val t = HOLogic.mk_Trueprop(le $ s $ r);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   612
      in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   613
        (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   614
          NONE =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   615
            let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   616
              (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   617
                NONE => NONE
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   618
              | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   619
             end
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   620
         | SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   621
      end handle THM _ => NONE)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   622
  | _ => NONE);
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   623
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   624
fun antisym_less_simproc ctxt ct =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59000
diff changeset
   625
  (case Thm.term_of ct of
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   626
    NotC $ ((less as Const(_,T)) $ r $ s) =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   627
     (let
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   628
       val prems = Simplifier.prems_of ctxt;
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   629
       val le = Const (@{const_name less_eq}, T);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   630
       val t = HOLogic.mk_Trueprop(le $ r $ s);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   631
      in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   632
        (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   633
          NONE =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   634
            let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   635
              (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   636
                NONE => NONE
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   637
              | SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   638
            end
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   639
        | SOME thm => SOME (mk_meta_eq (thm RS @{thm linorder_class.antisym_conv2})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   640
      end handle THM _ => NONE)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   641
  | _ => NONE);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   642
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   643
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   644
\<close>
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   645
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   646
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   647
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   648
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   649
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   650
subsection \<open>Bounded quantifiers\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   651
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   652
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   653
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   654
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   655
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   656
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   657
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   658
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   659
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   660
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   661
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   662
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   663
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   664
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   665
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   666
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   667
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   668
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   669
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   670
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   671
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   672
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   673
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   674
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   675
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   676
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   677
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   678
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   679
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   680
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   681
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   682
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   683
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   684
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   685
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   686
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   687
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   688
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   689
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   690
print_translation \<open>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   691
let
42287
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   692
  val All_binder = Mixfix.binder_name @{const_syntax All};
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   693
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
38786
e46e7a9cb622 formerly unnamed infix impliciation now named HOL.implies
haftmann
parents: 38715
diff changeset
   694
  val impl = @{const_syntax HOL.implies};
38795
848be46708dc formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents: 38786
diff changeset
   695
  val conj = @{const_syntax HOL.conj};
22916
haftmann
parents: 22886
diff changeset
   696
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   697
  val less_eq = @{const_syntax less_eq};
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   698
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   699
  val trans =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   700
   [((All_binder, impl, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   701
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   702
    ((All_binder, impl, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   703
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   704
    ((Ex_binder, conj, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   705
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   706
    ((Ex_binder, conj, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   707
    (@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   708
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   709
  fun matches_bound v t =
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   710
    (case t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   711
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   712
    | _ => false);
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   713
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
49660
de49d9b4d7bc more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents: 48891
diff changeset
   714
  fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   715
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   716
  fun tr' q = (q, fn _ =>
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   717
    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   718
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   719
        (case AList.lookup (op =) trans (q, c, d) of
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   720
          NONE => raise Match
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   721
        | SOME (l, g) =>
49660
de49d9b4d7bc more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents: 48891
diff changeset
   722
            if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
de49d9b4d7bc more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents: 48891
diff changeset
   723
            else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   724
            else raise Match)
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   725
      | _ => raise Match));
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   726
in [tr' All_binder, tr' Ex_binder] end
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   727
\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   728
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   729
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   730
subsection \<open>Transitivity reasoning\<close>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   731
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   732
context ord
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   733
begin
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   734
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   735
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   736
  by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   737
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   738
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   739
  by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   740
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   741
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   742
  by (rule subst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   743
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   744
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   745
  by (rule ssubst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   746
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   747
end
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   748
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   749
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   750
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   751
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   752
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   753
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   754
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   755
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   756
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   757
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   758
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   759
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   760
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   761
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   762
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   763
  also assume "b < c" hence "f b < f c" by (rule r)
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   764
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   765
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   766
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   767
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   768
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   769
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   770
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   771
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   772
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   773
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   774
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   775
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   776
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   777
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   778
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   779
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   780
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   781
  also assume "b < c" hence "f b < f c" by (rule r)
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   782
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   783
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   784
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   785
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   786
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   787
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   788
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   789
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   790
  also assume "f b <= c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   791
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   792
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   793
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   794
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   795
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   796
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   797
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   798
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   799
  also assume "b <= c" hence "f b <= f c" by (rule r)
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   800
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   801
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   802
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   803
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   804
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   805
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   806
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   807
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   808
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   809
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   810
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   811
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   812
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   813
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   814
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   815
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   816
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   817
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   818
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   819
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   820
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   821
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   822
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   823
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   824
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   825
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   826
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   827
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   828
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   829
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   830
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   831
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   832
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   833
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   834
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   835
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   836
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   837
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   838
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   839
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   841
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   842
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   843
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   844
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   845
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   846
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   847
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   848
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   849
  (!!x y. x < y ==> f x < f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   850
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   851
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   852
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   853
  also assume "b < c" hence "f b < f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   854
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   855
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   856
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   857
text \<open>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   858
  Note that this list of rules is in reverse order of priorities.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   859
\<close>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   860
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   861
lemmas [trans] =
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   862
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   863
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   864
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   865
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   866
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   867
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   868
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   869
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   870
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   871
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   872
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   873
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   874
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   875
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   876
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   877
  mp
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   878
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   879
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   880
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   881
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   882
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   883
lemmas (in preorder) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   884
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   885
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   886
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   887
  less_le_trans
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   888
  order_trans
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   889
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   890
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   891
  antisym
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   892
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   893
lemmas (in ord) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   894
  ord_le_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   895
  ord_eq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   896
  ord_less_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   897
  ord_eq_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   898
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   899
lemmas [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   900
  trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   901
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   902
lemmas order_trans_rules =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   903
  order_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   904
  order_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   905
  order_le_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   906
  order_le_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   907
  order_less_le_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   908
  order_less_le_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   909
  order_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   910
  order_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   911
  ord_le_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   912
  ord_eq_le_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   913
  ord_less_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   914
  ord_eq_less_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   915
  forw_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   916
  back_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   917
  rev_mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   918
  mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   919
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   920
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   921
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   922
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   923
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   924
  less_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   925
  order_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   926
  antisym
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   927
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   928
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   929
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   930
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   931
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   932
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   933
text \<open>These support proving chains of decreasing inequalities
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   934
    a >= b >= c ... in Isar proofs.\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   935
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   936
lemma xt1 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   937
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   938
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   939
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   940
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   941
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   942
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   943
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   944
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   945
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   946
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   947
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   948
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   949
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   950
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   951
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   952
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   953
  by auto
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   954
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   955
lemma xt2 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   956
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   957
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   958
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   959
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   960
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   961
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   962
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   963
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   964
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   965
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   966
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   967
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   968
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   969
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   970
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   971
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   972
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   973
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   974
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   975
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   976
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   977
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   978
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   979
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   980
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   981
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   982
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   983
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   984
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   985
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   986
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53216
diff changeset
   987
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   988
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   989
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   990
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   991
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   992
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   993
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   994
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   995
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   996
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   997
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   998
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   999
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1000
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1001
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1002
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1003
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1004
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1005
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1006
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1007
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1008
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1009
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1010
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1011
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1012
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1013
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1014
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1015
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1016
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1017
subsection \<open>Monotonicity\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1018
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1019
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1020
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1021
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1022
definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1023
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1024
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1025
lemma monoI [intro?]:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1026
  fixes f :: "'a \<Rightarrow> 'b::order"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1027
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1028
  unfolding mono_def by iprover
21216
1c8580913738 made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents: 21204
diff changeset
  1029
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1030
lemma monoD [dest?]:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1031
  fixes f :: "'a \<Rightarrow> 'b::order"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1032
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1033
  unfolding mono_def by iprover
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1034
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1035
lemma monoE:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1036
  fixes f :: "'a \<Rightarrow> 'b::order"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1037
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1038
  assumes "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1039
  obtains "f x \<le> f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1040
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1041
  from assms show "f x \<le> f y" by (simp add: mono_def)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1042
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1043
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1044
definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
56020
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1045
  "antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1046
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1047
lemma antimonoI [intro?]:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1048
  fixes f :: "'a \<Rightarrow> 'b::order"
56020
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1049
  shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1050
  unfolding antimono_def by iprover
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1051
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1052
lemma antimonoD [dest?]:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1053
  fixes f :: "'a \<Rightarrow> 'b::order"
56020
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1054
  shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1055
  unfolding antimono_def by iprover
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1056
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1057
lemma antimonoE:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1058
  fixes f :: "'a \<Rightarrow> 'b::order"
56020
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1059
  assumes "antimono f"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1060
  assumes "x \<le> y"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1061
  obtains "f x \<ge> f y"
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1062
proof
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1063
  from assms show "f x \<ge> f y" by (simp add: antimono_def)
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1064
qed
f92479477c52 introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents: 54868
diff changeset
  1065
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1066
definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1067
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1068
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1069
lemma strict_monoI [intro?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1070
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1071
  shows "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1072
  using assms unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1073
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1074
lemma strict_monoD [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1075
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1076
  unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1077
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1078
lemma strict_mono_mono [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1079
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1080
  shows "mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1081
proof (rule monoI)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1082
  fix x y
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1083
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1084
  show "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1085
  proof (cases "x = y")
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1086
    case True then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1087
  next
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1088
    case False with \<open>x \<le> y\<close> have "x < y" by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1089
    with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1090
    then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1091
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1092
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1093
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1094
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1095
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1096
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1097
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1098
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1099
lemma mono_invE:
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1100
  fixes f :: "'a \<Rightarrow> 'b::order"
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1101
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1102
  assumes "f x < f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1103
  obtains "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1104
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1105
  show "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1106
  proof (rule ccontr)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1107
    assume "\<not> x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1108
    then have "y \<le> x" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1109
    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1110
    with \<open>f x < f y\<close> show False by simp
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1111
  qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1112
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1113
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1114
lemma strict_mono_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1115
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1116
  shows "f x = f y \<longleftrightarrow> x = y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1117
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1118
  assume "f x = f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1119
  show "x = y" proof (cases x y rule: linorder_cases)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1120
    case less with assms strict_monoD have "f x < f y" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1121
    with \<open>f x = f y\<close> show ?thesis by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1122
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1123
    case equal then show ?thesis .
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1124
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1125
    case greater with assms strict_monoD have "f y < f x" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1126
    with \<open>f x = f y\<close> show ?thesis by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1127
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1128
qed simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1129
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1130
lemma strict_mono_less_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1131
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1132
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1133
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1134
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1135
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1136
next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1137
  assume "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1138
  show "x \<le> y" proof (rule ccontr)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1139
    assume "\<not> x \<le> y" then have "y < x" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1140
    with assms strict_monoD have "f y < f x" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1141
    with \<open>f x \<le> f y\<close> show False by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1142
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1143
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1144
  
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1145
lemma strict_mono_less:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1146
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1147
  shows "f x < f y \<longleftrightarrow> x < y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1148
  using assms
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1149
    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1150
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1151
end
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1152
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1153
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1154
subsection \<open>min and max -- fundamental\<close>
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1155
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1156
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1157
  "min a b = (if a \<le> b then a else b)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1158
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1159
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1160
  "max a b = (if a \<le> b then b else a)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1161
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1162
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1163
  by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1164
54857
5c05f7c5f8ae tuning and augmentation of min/max lemmas;
haftmann
parents: 54147
diff changeset
  1165
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1166
  by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1167
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1168
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1169
  by (simp add:min_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1170
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
  1171
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1172
  by (simp add: max_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1173
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1174
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1175
subsection \<open>(Unique) top and bottom elements\<close>
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1176
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1177
class bot =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1178
  fixes bot :: 'a ("\<bottom>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1179
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1180
class order_bot = order + bot +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1181
  assumes bot_least: "\<bottom> \<le> a"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1182
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1183
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1184
sublocale bot!: ordering_top greater_eq greater bot
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61076
diff changeset
  1185
  by standard (fact bot_least)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1186
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1187
lemma le_bot:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1188
  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1189
  by (fact bot.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1190
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1191
lemma bot_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1192
  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1193
  by (fact bot.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1194
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1195
lemma not_less_bot:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1196
  "\<not> a < \<bottom>"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1197
  by (fact bot.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1198
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1199
lemma bot_less:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1200
  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1201
  by (fact bot.not_eq_extremum)
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1202
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1203
end
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1204
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1205
class top =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1206
  fixes top :: 'a ("\<top>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1207
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1208
class order_top = order + top +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1209
  assumes top_greatest: "a \<le> \<top>"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1210
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1211
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1212
sublocale top!: ordering_top less_eq less top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61076
diff changeset
  1213
  by standard (fact top_greatest)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1214
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1215
lemma top_le:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1216
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1217
  by (fact top.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1218
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1219
lemma top_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1220
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1221
  by (fact top.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1222
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1223
lemma not_top_less:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1224
  "\<not> \<top> < a"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1225
  by (fact top.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1226
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1227
lemma less_top:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1228
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1229
  by (fact top.not_eq_extremum)
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1230
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1231
end
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1232
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1233
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1234
subsection \<open>Dense orders\<close>
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1235
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1236
class dense_order = order +
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1237
  assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1238
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1239
class dense_linorder = linorder + dense_order
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1240
begin
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1241
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1242
lemma dense_le:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1243
  fixes y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1244
  assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1245
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1246
proof (rule ccontr)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1247
  assume "\<not> ?thesis"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1248
  hence "z < y" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1249
  from dense[OF this]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1250
  obtain x where "x < y" and "z < x" by safe
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1251
  moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1252
  ultimately show False by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1253
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1254
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1255
lemma dense_le_bounded:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1256
  fixes x y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1257
  assumes "x < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1258
  assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1259
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1260
proof (rule dense_le)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1261
  fix w assume "w < y"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1262
  from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1263
  from linear[of u w]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1264
  show "w \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1265
  proof (rule disjE)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1266
    assume "u \<le> w"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1267
    from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1268
    show "w \<le> z" by (rule *)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1269
  next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1270
    assume "w \<le> u"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1271
    from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1272
    show "w \<le> z" by (rule order_trans)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1273
  qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1274
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1275
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1276
lemma dense_ge:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1277
  fixes y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1278
  assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1279
  shows "y \<le> z"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1280
proof (rule ccontr)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1281
  assume "\<not> ?thesis"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1282
  hence "z < y" by simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1283
  from dense[OF this]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1284
  obtain x where "x < y" and "z < x" by safe
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1285
  moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1286
  ultimately show False by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1287
qed
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1288
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1289
lemma dense_ge_bounded:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1290
  fixes x y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1291
  assumes "z < x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1292
  assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1293
  shows "y \<le> z"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1294
proof (rule dense_ge)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1295
  fix w assume "z < w"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1296
  from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1297
  from linear[of u w]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1298
  show "y \<le> w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1299
  proof (rule disjE)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1300
    assume "w \<le> u"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1301
    from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1302
    show "y \<le> w" by (rule *)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1303
  next
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1304
    assume "u \<le> w"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1305
    from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1306
    show "y \<le> w" by (rule order_trans)