src/HOL/Orderings.thy
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isabelle update_cartouches;
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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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notation (HTML output)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater_eq  (infix ">=" 50) where
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  "x >= y \<equiv> y <= x"
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notation (input)
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  greater_eq  (infix "\<ge>" 50)
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abbreviation (input)
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  greater  (infix ">" 50) where
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  "x > y \<equiv> y < x"
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end
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subsection \<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 default (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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parents: 23182
diff changeset
   327
apply (simp add: less_le)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   328
using linear apply (blast intro: antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   329
done
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   330
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   331
lemma not_less_iff_gr_or_eq:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   332
 "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   333
apply(simp add:not_less le_less)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   334
apply blast
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   335
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   336
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   337
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   338
apply (simp add: less_le)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   339
using linear apply (blast intro: antisym)
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   340
done
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   341
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   342
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   343
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
   344
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   345
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
   346
by (simp add: neq_iff) blast
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   347
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   348
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
   349
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   350
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   351
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
   352
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   353
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   354
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
   355
by (blast intro: antisym dest: not_less [THEN iffD1])
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   356
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   357
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   358
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   359
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   360
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   361
unfolding not_less .
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   362
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16743
diff changeset
   363
(*FIXME inappropriate name (or delete altogether)*)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24920
diff changeset
   364
lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
   365
unfolding not_le .
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   366
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   367
text \<open>Dual order\<close>
22916
haftmann
parents: 22886
diff changeset
   368
26014
00c2c3525bef dual orders and dual lattices
haftmann
parents: 25614
diff changeset
   369
lemma dual_linorder:
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   370
  "class.linorder (op \<ge>) (op >)"
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   371
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
22916
haftmann
parents: 22886
diff changeset
   372
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   373
end
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   374
23948
261bd4678076 using class target
haftmann
parents: 23881
diff changeset
   375
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   376
text \<open>Alternative introduction rule with bias towards strict order\<close>
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   377
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   378
lemma linorder_strictI:
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   379
  fixes less (infix "\<sqsubset>" 50)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   380
    and less_eq (infix "\<sqsubseteq>" 50)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   381
  assumes "class.order less_eq less"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   382
  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
   383
  shows "class.linorder less_eq less"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   384
proof -
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   385
  interpret order less_eq less
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   386
    by (fact \<open>class.order less_eq less\<close>)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   387
  show ?thesis
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   388
  proof
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   389
    fix a b
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   390
    show "a \<sqsubseteq> b \<or> b \<sqsubseteq> a"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   391
      using trichotomy by (auto simp add: le_less)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   392
  qed
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   393
qed
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   394
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56509
diff changeset
   395
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   396
subsection \<open>Reasoning tools setup\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   397
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   398
ML \<open>
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   399
signature ORDERS =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   400
sig
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   401
  val print_structures: Proof.context -> unit
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   402
  val order_tac: Proof.context -> thm list -> int -> tactic
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   403
  val add_struct: string * term list -> string -> attribute
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   404
  val del_struct: string * term list -> attribute
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   405
end;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   406
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   407
structure Orders: ORDERS =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   408
struct
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   409
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   410
(* context data *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   411
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   412
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   413
  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
   414
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   415
structure Data = Generic_Data
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   416
(
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   417
  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
   418
    (* Order structures:
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   419
       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
   420
       needed to set up the transitivity reasoner,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   421
       identifier and operations identify the structure uniquely. *)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   422
  val empty = [];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   423
  val extend = I;
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   424
  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
   425
);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   426
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   427
fun print_structures ctxt =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   428
  let
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   429
    val structs = Data.get (Context.Proof ctxt);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   430
    fun pretty_term t = Pretty.block
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   431
      [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
   432
        Pretty.str "::", Pretty.brk 1,
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   433
        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
   434
    fun pretty_struct ((s, ts), _) = Pretty.block
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   435
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   436
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   437
  in
51579
ec3b78ce0758 tuned message;
wenzelm
parents: 51546
diff changeset
   438
    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
   439
  end;
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   440
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   441
val _ =
59936
b8ffc3dc9e24 @{command_spec} is superseded by @{command_keyword};
wenzelm
parents: 59582
diff changeset
   442
  Outer_Syntax.command @{command_keyword print_orders}
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   443
    "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
   444
    (Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of)));
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   445
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   446
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   447
(* tactics *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   448
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   449
fun struct_tac ((s, ops), thms) ctxt facts =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   450
  let
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   451
    val [eq, le, less] = ops;
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   452
    fun decomp thy (@{const Trueprop} $ t) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   453
          let
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   454
            fun excluded t =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   455
              (* exclude numeric types: linear arithmetic subsumes transitivity *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   456
              let val T = type_of t
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   457
              in
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   458
                T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   459
              end;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   460
            fun rel (bin_op $ t1 $ t2) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   461
                  if excluded t1 then NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   462
                  else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   463
                  else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   464
                  else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   465
                  else NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   466
              | rel _ = NONE;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   467
            fun dec (Const (@{const_name Not}, _) $ t) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   468
                  (case rel t of NONE =>
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   469
                    NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   470
                  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   471
              | dec x = rel x;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   472
          in dec t end
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   473
      | decomp _ _ = NONE;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   474
  in
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   475
    (case s of
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   476
      "order" => Order_Tac.partial_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   477
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   478
    | _ => 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
   479
  end
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
fun order_tac ctxt facts =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   482
  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
   483
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   484
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   485
(* attributes *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   486
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   487
fun add_struct s tag =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   488
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   489
    (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
   490
fun del_struct s =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   491
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   492
    (fn _ => Data.map (AList.delete struct_eq s));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   493
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   494
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   495
\<close>
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   496
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   497
attribute_setup order = \<open>
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   498
  Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   499
    Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   500
    Scan.repeat Args.term
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   501
    >> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   502
         | ((NONE, n), ts) => Orders.del_struct (n, ts))
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   503
\<close> "theorems controlling transitivity reasoner"
58826
2ed2eaabe3df modernized setup;
wenzelm
parents: 57447
diff changeset
   504
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   505
method_setup order = \<open>
47432
e1576d13e933 more standard method setup;
wenzelm
parents: 46976
diff changeset
   506
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   507
\<close> "transitivity reasoner"
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   508
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   509
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   510
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
   511
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   512
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   513
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   514
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   515
(* 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
   516
   is not a parameter of the locale. *)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   517
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   518
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
   519
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   520
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   521
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   522
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   523
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   524
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   525
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   526
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   527
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   528
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   529
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   530
declare less_trans [order add 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 less_le_trans [order add less_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_less_trans [order add le_less_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 order_trans [order add 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 le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   539
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   540
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   541
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   542
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   543
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   544
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
   545
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   546
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
   547
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   548
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   549
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   550
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   551
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   552
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   553
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   554
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   555
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   556
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   557
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   558
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   559
declare less_imp_le [order add less_imp_le: 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 iffD2, order add not_lessI: 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 iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   564
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   565
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   566
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   567
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   568
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   569
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   570
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   571
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   572
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   573
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   574
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   575
declare less_trans [order add 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 less_le_trans [order add less_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_less_trans [order add le_less_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 order_trans [order add 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 le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   584
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   585
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   586
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   587
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   588
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   589
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
   590
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   591
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
   592
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   593
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   594
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   595
setup \<open>
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   596
  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   597
    mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt)))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   598
  (*Adding the transitivity reasoners also as safe solvers showed a slight
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   599
    speed up, but the reasoning strength appears to be not higher (at least
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   600
    no breaking of additional proofs in the entire HOL distribution, as
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   601
    of 5 March 2004, was observed).*)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   602
\<close>
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   603
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   604
ML \<open>
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   605
local
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   606
  fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   607
in
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   608
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   609
fun antisym_le_simproc ctxt ct =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59000
diff changeset
   610
  (case Thm.term_of ct of
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   611
    (le as Const (_, T)) $ r $ s =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   612
     (let
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   613
        val prems = Simplifier.prems_of ctxt;
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   614
        val less = Const (@{const_name less}, T);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   615
        val t = HOLogic.mk_Trueprop(le $ s $ r);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   616
      in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   617
        (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   618
          NONE =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   619
            let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   620
              (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   621
                NONE => NONE
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   622
              | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   623
             end
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   624
         | SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   625
      end handle THM _ => NONE)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   626
  | _ => NONE);
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   627
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   628
fun antisym_less_simproc ctxt ct =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59000
diff changeset
   629
  (case Thm.term_of ct of
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   630
    NotC $ ((less as Const(_,T)) $ r $ s) =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   631
     (let
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   632
       val prems = Simplifier.prems_of ctxt;
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   633
       val le = Const (@{const_name less_eq}, T);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   634
       val t = HOLogic.mk_Trueprop(le $ r $ s);
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   635
      in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   636
        (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   637
          NONE =>
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   638
            let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   639
              (case find_first (prp t) prems of
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   640
                NONE => NONE
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   641
              | SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   642
            end
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   643
        | SOME thm => SOME (mk_meta_eq (thm RS @{thm linorder_class.antisym_conv2})))
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   644
      end handle THM _ => NONE)
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   645
  | _ => NONE);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   646
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   647
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   648
\<close>
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   649
56509
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   650
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   651
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
e050d42dc21d modernized simproc_setup;
wenzelm
parents: 56508
diff changeset
   652
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   653
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   654
subsection \<open>Bounded quantifiers\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   655
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   656
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   657
  "_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
   658
  "_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
   659
  "_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
   660
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   661
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   662
  "_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
   663
  "_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
   664
  "_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
   665
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   666
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   667
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   668
  "_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
   669
  "_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
   670
  "_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
   671
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   672
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   673
  "_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
   674
  "_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
   675
  "_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
   676
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   677
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   678
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   679
  "_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
   680
  "_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
   681
  "_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
   682
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   683
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   684
syntax (HTML output)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   685
  "_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
   686
  "_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
   687
  "_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
   688
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   689
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   690
  "_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
   691
  "_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
   692
  "_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
   693
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   694
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   695
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   696
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   697
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   698
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   699
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   700
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   701
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   702
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   703
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   704
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   705
print_translation \<open>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   706
let
42287
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   707
  val All_binder = Mixfix.binder_name @{const_syntax All};
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   708
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
38786
e46e7a9cb622 formerly unnamed infix impliciation now named HOL.implies
haftmann
parents: 38715
diff changeset
   709
  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
   710
  val conj = @{const_syntax HOL.conj};
22916
haftmann
parents: 22886
diff changeset
   711
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   712
  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
   713
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   714
  val trans =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   715
   [((All_binder, impl, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   716
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   717
    ((All_binder, impl, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   718
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   719
    ((Ex_binder, conj, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   720
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   721
    ((Ex_binder, conj, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   722
    (@{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
   723
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   724
  fun matches_bound v t =
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   725
    (case t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   726
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   727
    | _ => false);
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   728
  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
   729
  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
   730
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   731
  fun tr' q = (q, fn _ =>
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   732
    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   733
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   734
        (case AList.lookup (op =) trans (q, c, d) of
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   735
          NONE => raise Match
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   736
        | 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
   737
            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
   738
            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
   739
            else raise Match)
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   740
      | _ => raise Match));
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   741
in [tr' All_binder, tr' Ex_binder] end
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   742
\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   743
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   744
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   745
subsection \<open>Transitivity reasoning\<close>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   746
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   747
context ord
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   748
begin
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   749
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   750
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   751
  by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   752
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   753
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   754
  by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   755
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   756
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   757
  by (rule subst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   758
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   759
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   760
  by (rule ssubst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   761
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   762
end
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   763
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   764
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
   765
  (!!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
   766
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   767
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   768
  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
   769
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   770
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   771
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   772
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   773
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
   774
  (!!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
   775
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   776
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   777
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   778
  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
   779
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   780
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   781
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   782
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
   783
  (!!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
   784
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   785
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   786
  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
   787
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   788
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   789
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   790
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   791
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
   792
  (!!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
   793
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   794
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   795
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   796
  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
   797
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   798
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   799
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   800
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
   801
  (!!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
   802
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   803
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   804
  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
   805
  also assume "f b <= c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   806
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   807
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   808
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   809
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
   810
  (!!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
   811
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   812
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   813
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   814
  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
   815
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   816
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   817
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   818
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
   819
  (!!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
   820
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   821
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   822
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   823
  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
   824
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   825
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   826
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   827
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
   828
  (!!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
   829
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   830
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   831
  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
   832
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   833
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   834
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   835
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   836
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   837
  (!!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
   838
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   839
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
  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
   841
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   842
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   843
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   844
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   845
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   846
  (!!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
   847
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   848
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   849
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   850
  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
   851
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   852
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   853
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   854
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   855
  (!!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
   856
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   857
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   858
  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
   859
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   860
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   861
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   862
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   863
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   864
  (!!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
   865
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   866
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   867
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   868
  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
   869
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   870
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   871
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   872
text \<open>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   873
  Note that this list of rules is in reverse order of priorities.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   874
\<close>
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   875
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   876
lemmas [trans] =
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   877
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   878
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   879
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   880
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   881
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   882
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   883
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   884
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   885
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   886
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   887
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   888
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   889
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   890
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   891
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   892
  mp
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   893
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   894
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   895
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   896
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   897
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   898
lemmas (in preorder) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   899
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   900
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   901
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   902
  less_le_trans
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   903
  order_trans
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   904
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   905
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   906
  antisym
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   907
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   908
lemmas (in ord) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   909
  ord_le_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   910
  ord_eq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   911
  ord_less_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   912
  ord_eq_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   913
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   914
lemmas [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   915
  trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   916
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   917
lemmas order_trans_rules =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   918
  order_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   919
  order_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   920
  order_le_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   921
  order_le_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   922
  order_less_le_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   923
  order_less_le_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   924
  order_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   925
  order_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   926
  ord_le_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   927
  ord_eq_le_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   928
  ord_less_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   929
  ord_eq_less_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   930
  forw_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   931
  back_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   932
  rev_mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   933
  mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   934
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   935
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   936
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   937
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   938
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   939
  less_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   940
  order_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   941
  antisym
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   942
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   943
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   944
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   945
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   946
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   947
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   948
text \<open>These support proving chains of decreasing inequalities
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
   949
    a >= b >= c ... in Isar proofs.\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   950
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   951
lemma xt1 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   952
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   953
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   954
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   955
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   956
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   957
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   958
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   959
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   960
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   961
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   962
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   963
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   964
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   965
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   966
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   967
  "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
   968
  by auto
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   969
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   970
lemma xt2 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   971
  "(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
   972
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   973
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   974
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   975
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   976
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   977
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   978
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   979
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   980
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   981
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   982
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   983
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   984
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   985
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   986
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   987
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   988
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   989
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   990
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   991
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   992
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   993
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   994
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   995
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   996
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   997
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   998
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   999
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1000
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1001
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53216
diff changeset
  1002
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1003
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1004
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1005
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1006
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1007
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1008
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1009
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1010
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1011
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1012
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1013
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1014
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1015
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1016
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1017
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1018
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1019
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1020
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1021
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1022
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1023
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1024
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1025
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1026
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1027
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1028
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1029
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1030
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
  1031
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1032
subsection \<open>Monotonicity\<close>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1033
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1034
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1035
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1036
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1037
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1038
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1039
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1040
lemma monoI [intro?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1041
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1042
  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
  1043
  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
  1044
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1045
lemma monoD [dest?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1046
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1047
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1048
  unfolding mono_def by iprover
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1049
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1050
lemma monoE:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1051
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1052
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1053
  assumes "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1054
  obtains "f x \<le> f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1055
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1056
  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
  1057
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1058
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
definition antimono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
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
  "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
  1061
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
lemma antimonoI [intro?]:
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
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
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
  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
  1065
  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
  1066
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
  1067
lemma antimonoD [dest?]:
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
  1068
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
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
  1069
  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
  1070
  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
  1071
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
  1072
lemma antimonoE:
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
  1073
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
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
  1074
  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
  1075
  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
  1076
  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
  1077
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
  1078
  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
  1079
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
  1080
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1081
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1082
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1083
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1084
lemma strict_monoI [intro?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1085
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1086
  shows "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1087
  using assms unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1088
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1089
lemma strict_monoD [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1090
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1091
  unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1092
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1093
lemma strict_mono_mono [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1094
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1095
  shows "mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1096
proof (rule monoI)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1097
  fix x y
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1098
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1099
  show "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1100
  proof (cases "x = y")
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1101
    case True then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1102
  next
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1103
    case False with \<open>x \<le> y\<close> have "x < y" by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1104
    with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1105
    then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1106
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1107
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1108
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1109
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1110
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1111
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1112
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1113
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1114
lemma mono_invE:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1115
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1116
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1117
  assumes "f x < f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1118
  obtains "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1119
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1120
  show "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1121
  proof (rule ccontr)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1122
    assume "\<not> x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1123
    then have "y \<le> x" by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1124
    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1125
    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
  1126
  qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1127
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1128
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1129
lemma strict_mono_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1130
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1131
  shows "f x = f y \<longleftrightarrow> x = y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1132
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1133
  assume "f x = f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1134
  show "x = y" proof (cases x y rule: linorder_cases)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1135
    case less with assms strict_monoD have "f x < f y" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1136
    with \<open>f x = f y\<close> show ?thesis by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1137
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1138
    case equal then show ?thesis .
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1139
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1140
    case greater 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 = f y\<close> show ?thesis 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 simp
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_eq:
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 \<le> f y \<longleftrightarrow> x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1148
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1149
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1150
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1151
next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1152
  assume "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1153
  show "x \<le> y" proof (rule ccontr)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1154
    assume "\<not> x \<le> y" then have "y < x" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1155
    with assms strict_monoD have "f y < f x" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1156
    with \<open>f x \<le> f y\<close> show False by simp
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1157
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1158
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1159
  
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1160
lemma strict_mono_less:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1161
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1162
  shows "f x < f y \<longleftrightarrow> x < y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1163
  using assms
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1164
    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
  1165
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1166
end
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1167
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1168
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1169
subsection \<open>min and max -- fundamental\<close>
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1170
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1171
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1172
  "min a b = (if a \<le> b then a else b)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1173
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1174
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1175
  "max a b = (if a \<le> b then b else a)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1176
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1177
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
  1178
  by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1179
54857
5c05f7c5f8ae tuning and augmentation of min/max lemmas;
haftmann
parents: 54147
diff changeset
  1180
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
  1181
  by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1182
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1183
lemma min_absorb2: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> min x y = y"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1184
  by (simp add:min_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1185
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1186
lemma max_absorb1: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> max x y = x"
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1187
  by (simp add: max_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1188
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1189
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1190
subsection \<open>(Unique) top and bottom elements\<close>
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1191
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1192
class bot =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1193
  fixes bot :: 'a ("\<bottom>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1194
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1195
class order_bot = order + bot +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1196
  assumes bot_least: "\<bottom> \<le> a"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1197
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1198
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1199
sublocale bot!: ordering_top greater_eq greater bot
51546
2e26df807dc7 more uniform style for interpretation and sublocale declarations
haftmann
parents: 51487
diff changeset
  1200
  by default (fact bot_least)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1201
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1202
lemma le_bot:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1203
  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1204
  by (fact bot.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1205
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1206
lemma bot_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1207
  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1208
  by (fact bot.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1209
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1210
lemma not_less_bot:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1211
  "\<not> a < \<bottom>"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1212
  by (fact bot.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1213
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1214
lemma bot_less:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1215
  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1216
  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
  1217
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1218
end
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1219
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1220
class top =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1221
  fixes top :: 'a ("\<top>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1222
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1223
class order_top = order + top +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1224
  assumes top_greatest: "a \<le> \<top>"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1225
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1226
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1227
sublocale top!: ordering_top less_eq less top
51546
2e26df807dc7 more uniform style for interpretation and sublocale declarations
haftmann
parents: 51487
diff changeset
  1228
  by default (fact top_greatest)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1229
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1230
lemma top_le:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1231
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1232
  by (fact top.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1233
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1234
lemma top_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1235
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1236
  by (fact top.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1237
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1238
lemma not_top_less:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1239
  "\<not> \<top> < a"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1240
  by (fact top.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1241
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1242
lemma less_top:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1243
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1244
  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
  1245
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1246
end
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1247
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1248
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1249
subsection \<open>Dense orders\<close>
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1250
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1251
class dense_order = order +
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1252
  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
  1253
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1254
class dense_linorder = linorder + dense_order
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1255
begin
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1256
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1257
lemma dense_le:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1258
  fixes y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1259
  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
  1260
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1261
proof (rule ccontr)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1262
  assume "\<not> ?thesis"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1263
  hence "z < y" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1264
  from dense[OF this]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1265
  obtain x where "x < y" and "z < x" by safe
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1266
  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
  1267
  ultimately show False by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1268
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1269
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1270
lemma dense_le_bounded:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1271
  fixes x y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1272
  assumes "x < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1273
  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
  1274
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1275
proof (rule dense_le)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1276
  fix w assume "w < y"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1277
  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
  1278
  from linear[of u w]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1279
  show "w \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1280
  proof (rule disjE)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1281
    assume "u \<le> w"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1282
    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
  1283
    show "w \<le> z" by (rule *)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1284
  next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1285
    assume "w \<le> u"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1286
    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
  1287
    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
  1288
  qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1289
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1290
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1291
lemma dense_ge:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1292
  fixes y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1293
  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
  1294
  shows "y \<le> z"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1295
proof (rule ccontr)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1296
  assume "\<not> ?thesis"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1297
  hence "z < y" by simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1298
  from dense[OF this]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1299
  obtain x where "x < y" and "z < x" by safe
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60097
diff changeset
  1300
  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
  1301
  ultimately show False by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1302
qed
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1303
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1304
lemma dense_ge_bounded:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1305
  fixes x y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot