src/HOL/Orderings.thy
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(*  Title:      HOL/Orderings.thy
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* Abstract orderings *}
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theory Orderings
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imports HOL
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keywords "print_orders" :: diag
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begin
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ML_file "~~/src/Provers/order.ML"
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ML_file "~~/src/Provers/quasi.ML"  (* FIXME unused? *)
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subsection {* Abstract ordering *}
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locale ordering =
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50)
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   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50)
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  assumes strict_iff_order: "a \<prec> b \<longleftrightarrow> a \<preceq> b \<and> a \<noteq> b"
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  assumes refl: "a \<preceq> a" -- {* not @{text iff}: makes problems due to multiple (dual) interpretations *}
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    and antisym: "a \<preceq> b \<Longrightarrow> b \<preceq> a \<Longrightarrow> a = b"
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    and trans: "a \<preceq> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<preceq> c"
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begin
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lemma strict_implies_order:
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  "a \<prec> b \<Longrightarrow> a \<preceq> b"
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  by (simp add: strict_iff_order)
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lemma strict_implies_not_eq:
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  "a \<prec> b \<Longrightarrow> a \<noteq> b"
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  by (simp add: strict_iff_order)
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lemma not_eq_order_implies_strict:
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  "a \<noteq> b \<Longrightarrow> a \<preceq> b \<Longrightarrow> a \<prec> b"
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  by (simp add: strict_iff_order)
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lemma order_iff_strict:
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  "a \<preceq> b \<longleftrightarrow> a \<prec> b \<or> a = b"
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  by (auto simp add: strict_iff_order refl)
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lemma irrefl: -- {* not @{text iff}: makes problems due to multiple (dual) interpretations *}
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  "\<not> a \<prec> a"
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  by (simp add: strict_iff_order)
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lemma asym:
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  "a \<prec> b \<Longrightarrow> b \<prec> a \<Longrightarrow> False"
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  by (auto simp add: strict_iff_order intro: antisym)
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lemma strict_trans1:
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  "a \<preceq> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c"
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  by (auto simp add: strict_iff_order intro: trans antisym)
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lemma strict_trans2:
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  "a \<prec> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<prec> c"
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  by (auto simp add: strict_iff_order intro: trans antisym)
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lemma strict_trans:
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  "a \<prec> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c"
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  by (auto intro: strict_trans1 strict_implies_order)
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end
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locale ordering_top = ordering +
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  fixes top :: "'a"
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  assumes extremum [simp]: "a \<preceq> top"
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begin
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lemma extremum_uniqueI:
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  "top \<preceq> a \<Longrightarrow> a = top"
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  by (rule antisym) auto
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lemma extremum_unique:
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  "top \<preceq> a \<longleftrightarrow> a = top"
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  by (auto intro: antisym)
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lemma extremum_strict [simp]:
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  "\<not> (top \<prec> a)"
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  using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
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lemma not_eq_extremum:
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  "a \<noteq> top \<longleftrightarrow> a \<prec> top"
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  by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
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end  
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subsection {* Syntactic orders *}
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class ord =
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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begin
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notation
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  less_eq  ("op <=") and
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  less_eq  ("(_/ <= _)" [51, 51] 50) and
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  less  ("op <") and
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  less  ("(_/ < _)"  [51, 51] 50)
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notation (xsymbols)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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notation (HTML output)
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  less_eq  ("op \<le>") and
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  less_eq  ("(_/ \<le> _)"  [51, 51] 50)
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abbreviation (input)
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  greater_eq  (infix ">=" 50) where
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  "x >= y \<equiv> y <= x"
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notation (input)
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  greater_eq  (infix "\<ge>" 50)
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abbreviation (input)
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  greater  (infix ">" 50) where
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  "x > y \<equiv> y < x"
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end
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subsection {* Quasi orders *}
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class preorder = ord +
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  assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
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  and order_refl [iff]: "x \<le> x"
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  and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
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begin
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text {* Reflexivity. *}
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
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    -- {* This form is useful with the classical reasoner. *}
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by (erule ssubst) (rule order_refl)
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lemma less_irrefl [iff]: "\<not> x < x"
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by (simp add: less_le_not_le)
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
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unfolding less_le_not_le by blast
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text {* Asymmetry. *}
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lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
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by (simp add: less_le_not_le)
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lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
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by (drule less_not_sym, erule contrapos_np) simp
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text {* Transitivity. *}
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lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
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by (auto simp add: less_le_not_le intro: order_trans) 
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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text {* Transitivity rules for calculational reasoning *}
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lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
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by (rule less_asym)
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text {* Dual order *}
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lemma dual_preorder:
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  "class.preorder (op \<ge>) (op >)"
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proof qed (auto simp add: less_le_not_le intro: order_trans)
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end
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subsection {* Partial orders *}
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class order = preorder +
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  assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
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begin
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lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
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  by (auto simp add: less_le_not_le intro: antisym)
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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 {* Reflexivity. *}
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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
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    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
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by (fact order.order_iff_strict)
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lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
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unfolding less_le by blast
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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by auto
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lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
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by auto
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text {* Transitivity rules for calculational reasoning *}
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
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by (fact order.not_eq_order_implies_strict)
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
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by (rule order.not_eq_order_implies_strict)
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text {* Asymmetry. *}
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lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
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by (blast intro: antisym)
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lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym)
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lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
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by (fact order.strict_implies_not_eq)
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text {* Least value operator *}
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definition (in ord)
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  Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
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  "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
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lemma Least_equality:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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  shows "Least P = x"
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unfolding Least_def by (rule the_equality)
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  (blast intro: assms antisym)+
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lemma LeastI2_order:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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    and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
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  shows "Q (Least P)"
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unfolding Least_def by (rule theI2)
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  (blast intro: assms antisym)+
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text {* Dual order *}
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lemma dual_order:
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  "class.order (op \<ge>) (op >)"
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym)
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end
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subsection {* Linear (total) orders *}
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class linorder = order +
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  assumes linear: "x \<le> y \<or> y \<le> x"
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begin
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lemma less_linear: "x < y \<or> x = y \<or> y < x"
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unfolding less_le using less_le linear by blast
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lemma le_less_linear: "x \<le> y \<or> y < x"
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by (simp add: le_less less_linear)
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lemma le_cases [case_names le ge]:
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  "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
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using linear by blast
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lemma linorder_cases [case_names less equal greater]:
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  "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
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using less_linear by blast
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lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma not_less_iff_gr_or_eq:
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 "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
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apply(simp add:not_less le_less)
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apply blast
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done
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lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
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by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
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unfolding not_less .
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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
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unfolding not_less .
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(*FIXME inappropriate name (or delete altogether)*)
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lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
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unfolding not_le .
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diff changeset
   333
22916
haftmann
parents: 22886
diff changeset
   334
26014
00c2c3525bef dual orders and dual lattices
haftmann
parents: 25614
diff changeset
   335
text {* Dual order *}
22916
haftmann
parents: 22886
diff changeset
   336
26014
00c2c3525bef dual orders and dual lattices
haftmann
parents: 25614
diff changeset
   337
lemma dual_linorder:
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   338
  "class.linorder (op \<ge>) (op >)"
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   339
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
22916
haftmann
parents: 22886
diff changeset
   340
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   341
end
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   342
23948
261bd4678076 using class target
haftmann
parents: 23881
diff changeset
   343
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   344
subsection {* Reasoning tools setup *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   345
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   346
ML {*
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   347
signature ORDERS =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   348
sig
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   349
  val print_structures: Proof.context -> unit
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   350
  val order_tac: Proof.context -> thm list -> int -> tactic
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   351
end;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   352
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   353
structure Orders: ORDERS =
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   354
struct
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   355
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   356
(* context data *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   357
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   358
fun struct_eq ((s1: string, ts1), (s2, ts2)) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   359
  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
   360
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   361
structure Data = Generic_Data
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   362
(
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   363
  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
   364
    (* Order structures:
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   365
       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
   366
       needed to set up the transitivity reasoner,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   367
       identifier and operations identify the structure uniquely. *)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   368
  val empty = [];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   369
  val extend = I;
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   370
  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
   371
);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   372
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   373
fun print_structures ctxt =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   374
  let
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   375
    val structs = Data.get (Context.Proof ctxt);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   376
    fun pretty_term t = Pretty.block
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   377
      [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
   378
        Pretty.str "::", Pretty.brk 1,
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   379
        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
   380
    fun pretty_struct ((s, ts), _) = Pretty.block
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   381
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   382
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   383
  in
51579
ec3b78ce0758 tuned message;
wenzelm
parents: 51546
diff changeset
   384
    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
   385
  end;
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   386
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   387
val _ =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   388
  Outer_Syntax.improper_command @{command_spec "print_orders"}
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   389
    "print order structures available to transitivity reasoner"
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   390
    (Scan.succeed (Toplevel.unknown_context o
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   391
      Toplevel.keep (print_structures o Toplevel.context_of)));
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   392
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   393
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   394
(* tactics *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   395
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   396
fun struct_tac ((s, ops), thms) ctxt facts =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   397
  let
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   398
    val [eq, le, less] = ops;
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   399
    fun decomp thy (@{const Trueprop} $ t) =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   400
          let
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   401
            fun excluded t =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   402
              (* exclude numeric types: linear arithmetic subsumes transitivity *)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   403
              let val T = type_of t
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   404
              in
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   405
                T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   406
              end;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   407
            fun rel (bin_op $ t1 $ t2) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   408
                  if excluded t1 then NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   409
                  else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   410
                  else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   411
                  else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   412
                  else NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   413
              | rel _ = NONE;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   414
            fun dec (Const (@{const_name Not}, _) $ t) =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   415
                  (case rel t of NONE =>
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   416
                    NONE
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   417
                  | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   418
              | dec x = rel x;
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   419
          in dec t end
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   420
      | decomp _ _ = NONE;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   421
  in
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   422
    (case s of
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   423
      "order" => Order_Tac.partial_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   424
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt facts
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   425
    | _ => 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
   426
  end
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   427
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   428
fun order_tac ctxt facts =
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   429
  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
   430
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   431
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   432
(* attributes *)
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   433
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   434
fun add_struct_thm s tag =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   435
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   436
    (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
   437
fun del_struct s =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   438
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   439
    (fn _ => Data.map (AList.delete struct_eq s));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   440
24867
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   441
val _ =
56508
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   442
  Theory.setup
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   443
    (Attrib.setup @{binding order}
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   444
      (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   445
        Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   446
        Scan.repeat Args.term
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   447
        >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   448
             | ((NONE, n), ts) => del_struct (n, ts)))
af08160c5a4c misc tuning;
wenzelm
parents: 56020
diff changeset
   449
      "theorems controlling transitivity reasoner");
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   450
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   451
end;
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   452
*}
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   453
47432
e1576d13e933 more standard method setup;
wenzelm
parents: 46976
diff changeset
   454
method_setup order = {*
e1576d13e933 more standard method setup;
wenzelm
parents: 46976
diff changeset
   455
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt []))
e1576d13e933 more standard method setup;
wenzelm
parents: 46976
diff changeset
   456
*} "transitivity reasoner"
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   457
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   458
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   459
text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   460
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   461
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   462
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   463
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   464
(* 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
   465
   is not a parameter of the locale. *)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   466
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   467
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
   468
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   469
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   470
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   471
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   472
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   473
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   474
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   475
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   476
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   477
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   478
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   479
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   480
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   481
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   482
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   483
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   484
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   485
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   486
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   487
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   488
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   489
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   490
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   491
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   492
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   493
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
   494
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   495
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
   496
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   497
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   498
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   499
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   500
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   501
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   502
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   503
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   504
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   505
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   506
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   507
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   508
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   509
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   510
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   511
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   512
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   513
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   514
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   515
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   516
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   517
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   518
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   519
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   520
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   521
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   522
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   523
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   524
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   525
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   526
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   527
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   528
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   529
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   530
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   531
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   532
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   533
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   534
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   535
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   536
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   537
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   538
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
   539
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   540
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
   541
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   542
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   543
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   544
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   545
setup {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   546
let
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   547
44058
ae85c5d64913 misc tuning -- eliminated old-fashioned rep_thm;
wenzelm
parents: 44025
diff changeset
   548
fun prp t thm = Thm.prop_of thm = t;  (* FIXME aconv!? *)
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   549
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   550
fun prove_antisym_le ctxt ((le as Const(_,T)) $ r $ s) =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   551
  let val prems = Simplifier.prems_of ctxt;
22916
haftmann
parents: 22886
diff changeset
   552
      val less = Const (@{const_name less}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   553
      val t = HOLogic.mk_Trueprop(le $ s $ r);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   554
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   555
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   556
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   557
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   558
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   559
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   560
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   561
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   562
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   563
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   564
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   565
fun prove_antisym_less ctxt (NotC $ ((less as Const(_,T)) $ r $ s)) =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   566
  let val prems = Simplifier.prems_of ctxt;
22916
haftmann
parents: 22886
diff changeset
   567
      val le = Const (@{const_name less_eq}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   568
      val t = HOLogic.mk_Trueprop(le $ r $ s);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   569
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   570
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   571
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   572
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   573
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   574
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   575
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   576
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   577
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   578
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   579
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   580
fun add_simprocs procs thy =
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   581
  map_theory_simpset (fn ctxt => ctxt
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   582
    addsimprocs (map (fn (name, raw_ts, proc) =>
38715
6513ea67d95d renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents: 38705
diff changeset
   583
      Simplifier.simproc_global thy name raw_ts proc) procs)) thy;
42795
66fcc9882784 clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents: 42287
diff changeset
   584
26496
49ae9456eba9 purely functional setup of claset/simpset/clasimpset;
wenzelm
parents: 26324
diff changeset
   585
fun add_solver name tac =
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   586
  map_theory_simpset (fn ctxt0 => ctxt0 addSolver
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51658
diff changeset
   587
    mk_solver name (fn ctxt => tac ctxt (Simplifier.prems_of ctxt)));
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   588
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   589
in
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   590
  add_simprocs [
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   591
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   592
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   593
     ]
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   594
  #> add_solver "Transitivity" Orders.order_tac
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   595
  (* Adding the transitivity reasoners also as safe solvers showed a slight
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   596
     speed up, but the reasoning strength appears to be not higher (at least
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   597
     no breaking of additional proofs in the entire HOL distribution, as
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   598
     of 5 March 2004, was observed). *)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   599
end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   600
*}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   601
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   602
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   603
subsection {* Bounded quantifiers *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   604
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   605
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   606
  "_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
   607
  "_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
   608
  "_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
   609
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   610
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   611
  "_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
   612
  "_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
   613
  "_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
   614
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   615
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   616
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   617
  "_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
   618
  "_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
   619
  "_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
   620
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   621
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   622
  "_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
   623
  "_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
   624
  "_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
   625
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   626
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   627
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   628
  "_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
   629
  "_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
   630
  "_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
   631
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   632
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   633
syntax (HTML output)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   634
  "_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
   635
  "_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
   636
  "_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
   637
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   638
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   639
  "_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
   640
  "_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
   641
  "_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
   642
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   643
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   644
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   645
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   646
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   647
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   648
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   649
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   650
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   651
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   652
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   653
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   654
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   655
let
42287
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   656
  val All_binder = Mixfix.binder_name @{const_syntax All};
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   657
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
38786
e46e7a9cb622 formerly unnamed infix impliciation now named HOL.implies
haftmann
parents: 38715
diff changeset
   658
  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
   659
  val conj = @{const_syntax HOL.conj};
22916
haftmann
parents: 22886
diff changeset
   660
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   661
  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
   662
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   663
  val trans =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   664
   [((All_binder, impl, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   665
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   666
    ((All_binder, impl, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   667
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   668
    ((Ex_binder, conj, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   669
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   670
    ((Ex_binder, conj, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   671
    (@{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
   672
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   673
  fun matches_bound v t =
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   674
    (case t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   675
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   676
    | _ => false);
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   677
  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
   678
  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
   679
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   680
  fun tr' q = (q, fn _ =>
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   681
    (fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T),
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   682
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   683
        (case AList.lookup (op =) trans (q, c, d) of
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   684
          NONE => raise Match
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   685
        | 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
   686
            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
   687
            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
   688
            else raise Match)
52143
36ffe23b25f8 syntax translations always depend on context;
wenzelm
parents: 51717
diff changeset
   689
      | _ => raise Match));
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   690
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   691
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   692
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   693
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   694
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   695
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   696
context ord
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   697
begin
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   698
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   699
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   700
  by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   701
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   702
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   703
  by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   704
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   705
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   706
  by (rule subst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   707
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   708
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   709
  by (rule ssubst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   710
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   711
end
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   712
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   713
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
   714
  (!!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
   715
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   716
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   717
  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
   718
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   719
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   720
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   721
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   722
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
   723
  (!!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
   724
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   725
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   726
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   727
  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
   728
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   729
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   730
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   731
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
   732
  (!!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
   733
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   734
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   735
  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
   736
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   737
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   738
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   739
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   740
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
   741
  (!!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
   742
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   743
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   744
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   745
  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
   746
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   747
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   748
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   749
lemma order_less_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
   750
  (!!x y. x < y ==> f x < f y) ==> f a < c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   751
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   752
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   753
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   754
  also assume "f b <= c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   755
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   756
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   757
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   758
lemma order_less_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
   759
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   760
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   761
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   762
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   763
  also assume "b <= c" hence "f b <= f c" by (rule r)
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   764
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   765
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   766
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   767
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   768
  (!!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
   769
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   770
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   771
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   772
  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
   773
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   774
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   775
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   776
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   777
  (!!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
   778
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   779
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   780
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   781
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   782
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   783
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   784
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   785
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   786
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   787
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   788
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   789
  assume "a <= b" hence "f a <= f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   790
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   791
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   792
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   793
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   794
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   795
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   796
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   797
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   798
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   799
  also assume "b <= c" hence "f b <= f c" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   800
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   801
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   802
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   803
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   804
  (!!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
   805
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   806
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   807
  assume "a < b" hence "f a < f b" by (rule r)
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   808
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   809
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   810
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   811
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   812
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   813
  (!!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
   814
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   815
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   816
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   817
  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
   818
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   819
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   820
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   821
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   822
  Note that this list of rules is in reverse order of priorities.
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   823
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   824
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   825
lemmas [trans] =
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   826
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   827
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   828
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   829
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   830
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   831
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   832
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   833
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   834
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   835
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   836
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   837
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   838
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   839
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   840
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   841
  mp
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   842
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   843
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   844
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   845
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   846
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   847
lemmas (in preorder) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   848
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   849
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   850
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   851
  less_le_trans
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   852
  order_trans
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   853
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   854
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   855
  antisym
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   856
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   857
lemmas (in ord) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   858
  ord_le_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   859
  ord_eq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   860
  ord_less_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   861
  ord_eq_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   862
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   863
lemmas [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   864
  trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   865
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   866
lemmas order_trans_rules =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   867
  order_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   868
  order_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   869
  order_le_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   870
  order_le_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   871
  order_less_le_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   872
  order_less_le_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   873
  order_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   874
  order_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   875
  ord_le_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   876
  ord_eq_le_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   877
  ord_less_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   878
  ord_eq_less_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   879
  forw_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   880
  back_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   881
  rev_mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   882
  mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   883
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   884
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   885
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   886
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   887
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   888
  less_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   889
  order_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   890
  antisym
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   891
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   892
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   893
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   894
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   895
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   896
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   897
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   898
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   899
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   900
lemma xt1 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   901
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   902
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   903
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   904
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   905
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   906
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   907
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   908
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   909
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   910
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   911
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   912
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   913
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   914
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   915
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   916
  "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
   917
  by auto
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   918
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   919
lemma xt2 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   920
  "(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
   921
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   922
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   923
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   924
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   925
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   926
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   927
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   928
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   929
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   930
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   931
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   932
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   933
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   934
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   935
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   936
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   937
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   938
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   939
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   940
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   941
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   942
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   943
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   944
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   945
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   946
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   947
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   948
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   949
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   950
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53216
diff changeset
   951
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   952
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   953
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   954
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   955
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   956
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   957
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   958
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   959
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   960
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   961
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   962
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   963
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   964
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   965
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   966
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   967
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   968
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   969
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   970
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   971
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   972
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   973
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   974
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   975
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   976
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   977
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   978
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   979
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   980
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
   981
subsection {* Monotonicity *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   982
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   983
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   984
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   985
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   986
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   987
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   988
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   989
lemma monoI [intro?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   990
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   991
  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
   992
  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
   993
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   994
lemma monoD [dest?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   995
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   996
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   997
  unfolding mono_def by iprover
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   998
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
   999
lemma monoE:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1000
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1001
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1002
  assumes "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1003
  obtains "f x \<le> f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1004
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1005
  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
  1006
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1007
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
  1008
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
  1009
  "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
  1010
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
  1011
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
  1012
  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
  1013
  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
  1014
  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
  1015
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
  1016
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
  1017
  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
  1018
  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
  1019
  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
  1020
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
  1021
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
  1022
  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
  1023
  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
  1024
  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
  1025
  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
  1026
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
  1027
  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
  1028
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
  1029
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1030
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1031
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1032
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1033
lemma strict_monoI [intro?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1034
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1035
  shows "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1036
  using assms unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1037
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1038
lemma strict_monoD [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1039
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1040
  unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1041
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1042
lemma strict_mono_mono [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1043
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1044
  shows "mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1045
proof (rule monoI)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1046
  fix x y
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1047
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1048
  show "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1049
  proof (cases "x = y")
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1050
    case True then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1051
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1052
    case False with `x \<le> y` have "x < y" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1053
    with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1054
    then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1055
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1056
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1057
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1058
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1059
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1060
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1061
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1062
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1063
lemma mono_invE:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1064
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1065
  assumes "mono f"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1066
  assumes "f x < f y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1067
  obtains "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1068
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1069
  show "x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1070
  proof (rule ccontr)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1071
    assume "\<not> x \<le> y"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1072
    then have "y \<le> x" by simp
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1073
    with `mono f` obtain "f y \<le> f x" by (rule monoE)
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1074
    with `f x < f y` show False by simp
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1075
  qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1076
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 49769
diff changeset
  1077
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1078
lemma strict_mono_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1079
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1080
  shows "f x = f y \<longleftrightarrow> x = y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1081
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1082
  assume "f x = f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1083
  show "x = y" proof (cases x y rule: linorder_cases)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1084
    case less with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1085
    with `f x = f y` show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1086
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1087
    case equal then show ?thesis .
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1088
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1089
    case greater with assms strict_monoD have "f y < f x" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1090
    with `f x = f y` show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1091
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1092
qed simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1093
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1094
lemma strict_mono_less_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1095
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1096
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1097
proof
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
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1100
next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1101
  assume "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1102
  show "x \<le> y" proof (rule ccontr)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1103
    assume "\<not> x \<le> y" then have "y < x" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1104
    with assms strict_monoD have "f y < f x" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1105
    with `f x \<le> f y` show False 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
  
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1109
lemma strict_mono_less:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1110
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1111
  shows "f x < f y \<longleftrightarrow> x < y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1112
  using assms
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1113
    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
  1114
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1115
end
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1116
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1117
54861
00d551179872 postponed min/max lemmas until abstract lattice is available
haftmann
parents: 54860
diff changeset
  1118
subsection {* min and max -- fundamental *}
54860
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1119
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1120
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1121
  "min a b = (if a \<le> b then a else b)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1122
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1123
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1124
  "max a b = (if a \<le> b then b else a)"
69b3e46d8fbe tuned structure of min/max lemmas
haftmann
parents: 54857
diff changeset
  1125
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1126
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
  1127
  by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1128
54857
5c05f7c5f8ae tuning and augmentation of min/max lemmas;
haftmann
parents: 54147
diff changeset
  1129
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
  1130
  by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1131
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1132
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
  1133
  by (simp add:min_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1134
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45893
diff changeset
  1135
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
  1136
  by (simp add: max_def)
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1137
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1138
43813
07f0650146f2 tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents: 43597
diff changeset
  1139
subsection {* (Unique) top and bottom elements *}
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1140
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1141
class bot =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1142
  fixes bot :: 'a ("\<bottom>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1143
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1144
class order_bot = order + bot +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1145
  assumes bot_least: "\<bottom> \<le> a"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1146
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1147
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1148
sublocale bot!: ordering_top greater_eq greater bot
51546
2e26df807dc7 more uniform style for interpretation and sublocale declarations
haftmann
parents: 51487
diff changeset
  1149
  by default (fact bot_least)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1150
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1151
lemma le_bot:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1152
  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1153
  by (fact bot.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1154
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1155
lemma bot_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1156
  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1157
  by (fact bot.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1158
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1159
lemma not_less_bot:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1160
  "\<not> a < \<bottom>"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1161
  by (fact bot.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1162
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1163
lemma bot_less:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1164
  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1165
  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
  1166
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1167
end
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1168
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1169
class top =
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1170
  fixes top :: 'a ("\<top>")
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1171
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52143
diff changeset
  1172
class order_top = order + top +
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1173
  assumes top_greatest: "a \<le> \<top>"
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1174
begin
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1175
54868
bab6cade3cc5 prefer target-style syntaxx for sublocale
haftmann
parents: 54861
diff changeset
  1176
sublocale top!: ordering_top less_eq less top
51546
2e26df807dc7 more uniform style for interpretation and sublocale declarations
haftmann
parents: 51487
diff changeset
  1177
  by default (fact top_greatest)
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1178
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1179
lemma top_le:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1180
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1181
  by (fact top.extremum_uniqueI)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1182
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1183
lemma top_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1184
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1185
  by (fact top.extremum_unique)
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1186
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1187
lemma not_top_less:
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1188
  "\<not> \<top> < a"
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1189
  by (fact top.extremum_strict)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1190
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1191
lemma less_top:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1192
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
51487
f4bfdee99304 locales for abstract orders
haftmann
parents: 51329
diff changeset
  1193
  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
  1194
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1195
end
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1196
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1197
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1198
subsection {* Dense orders *}
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1199
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1200
class dense_order = order +
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1201
  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
  1202
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1203
class dense_linorder = linorder + dense_order
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1204
begin
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1205
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1206
lemma dense_le:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1207
  fixes y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1208
  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
  1209
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1210
proof (rule ccontr)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1211
  assume "\<not> ?thesis"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1212
  hence "z < y" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1213
  from dense[OF this]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1214
  obtain x where "x < y" and "z < x" by safe
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1215
  moreover have "x \<le> z" using assms[OF `x < y`] .
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1216
  ultimately show False by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1217
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1218
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1219
lemma dense_le_bounded:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1220
  fixes x y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1221
  assumes "x < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1222
  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
  1223
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1224
proof (rule dense_le)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1225
  fix w assume "w < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1226
  from dense[OF `x < y`] obtain u where "x < u" "u < y" by safe
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1227
  from linear[of u w]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1228
  show "w \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1229
  proof (rule disjE)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1230
    assume "u \<le> w"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1231
    from less_le_trans[OF `x < u` `u \<le> w`] `w < y`
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1232
    show "w \<le> z" by (rule *)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1233
  next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1234
    assume "w \<le> u"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1235
    from `w \<le> u` *[OF `x < u` `u < y`]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1236
    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
  1237
  qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1238
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1239
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1240
lemma dense_ge:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1241
  fixes y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1242
  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
  1243
  shows "y \<le> z"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1244
proof (rule ccontr)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1245
  assume "\<not> ?thesis"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1246
  hence "z < y" by simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1247
  from dense[OF this]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1248
  obtain x where "x < y" and "z < x" by safe
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1249
  moreover have "y \<le> x" using assms[OF `z < x`] .
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1250
  ultimately show False by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1251
qed
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1252
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1253
lemma dense_ge_bounded:
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1254
  fixes x y z :: 'a
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1255
  assumes "z < x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1256
  assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1257
  shows "y \<le> z"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1258
proof (rule dense_ge)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1259
  fix w assume "z < w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1260
  from dense[OF `z < x`] obtain u where "z < u" "u < x" by safe
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1261
  from linear[of u w]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1262
  show "y \<le> w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1263
  proof (rule disjE)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1264
    assume "w \<le> u"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1265
    from `z < w` le_less_trans[OF `w \<le> u` `u < x`]
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1266
    show "y \<le> w" by (rule *)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1267
  next
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1268
    assume "u \<le> w"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1269
    from *[OF `z < u` `u < x`] `u \<le> w`
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1270
    show "y \<le> w" by (rule order_trans)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1271
  qed
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1272
qed
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1273
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1274
end
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1275
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1276
class no_top = order + 
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1277
  assumes gt_ex: "\<exists>y. x < y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1278
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1279
class no_bot = order + 
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1280
  assumes lt_ex: "\<exists>y. y < x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1281
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 53215
diff changeset
  1282
class unbounded_dense_linorder = dense_linorder + no_top + no_bot
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
  1283
51546
2e26df807dc7 more uniform style for interpretation and sublocale declarations
haftmann
parents: 51487
diff changeset
  1284
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1285
subsection {* Wellorders *}
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1286
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1287
class wellorder = linorder +
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1288
  assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1289
begin
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1290
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1291
lemma wellorder_Least_lemma:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1292
  fixes k :: 'a
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1293
  assumes "P k"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
  1294
  shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1295
proof -
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1296
  have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1297
  using assms proof (induct k rule: less_induct)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1298
    case (less x) then have "P x" by simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1299
    show ?case proof (rule classical)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1300
      assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1301
      have "\<And>y. P y \<Longrightarrow> x \<le> y"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1302
      proof (rule classical)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1303
        fix y
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1304
        assume "P y" and "\<not> x \<le> y"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1305
        with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1306
          by (auto simp add: not_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1307
        with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1308
          by auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1309
        then show "x \<le> y" by auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1310
      qed
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1311
      with `P x` have Least: "(LEAST a. P a) = x"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1312
        by (rule Least_equality)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: