src/HOL/Orderings.thy
author noschinl
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add complementary lemmas for {min,max}_least
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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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uses
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  "~~/src/Provers/order.ML"
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  "~~/src/Provers/quasi.ML"  (* FIXME unused? *)
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begin
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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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cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
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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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text {* Reflexivity. *}
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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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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 (simp add: less_le) blast
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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 (simp add: less_le)
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
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by (simp add: less_le)
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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 (erule contrapos_pn, erule subst, rule less_irrefl)
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text {* Least value operator *}
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definition (in ord)
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  Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
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  "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
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lemma Least_equality:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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  shows "Least P = x"
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unfolding Least_def by (rule the_equality)
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  (blast intro: assms antisym)+
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lemma LeastI2_order:
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  assumes "P x"
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    and "\<And>y. P y \<Longrightarrow> x \<le> y"
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    and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
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  shows "Q (Least P)"
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unfolding Least_def by (rule theI2)
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  (blast intro: assms antisym)+
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text {* Dual order *}
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lemma dual_order:
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  "class.order (op \<ge>) (op >)"
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym)
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end
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subsection {* Linear (total) orders *}
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class linorder = order +
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  assumes linear: "x \<le> y \<or> y \<le> x"
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begin
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lemma less_linear: "x < y \<or> x = y \<or> y < x"
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unfolding less_le using less_le linear by blast
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lemma le_less_linear: "x \<le> y \<or> y < x"
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by (simp add: le_less less_linear)
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lemma le_cases [case_names le ge]:
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  "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
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using linear by blast
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lemma linorder_cases [case_names less equal greater]:
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  "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
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using less_linear by blast
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lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma not_less_iff_gr_or_eq:
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 "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
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apply(simp add:not_less le_less)
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apply blast
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done
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lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
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by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
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unfolding not_less .
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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
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unfolding not_less .
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(*FIXME inappropriate name (or delete altogether)*)
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lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
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unfolding not_le .
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text {* Dual order *}
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lemma dual_linorder:
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  "class.linorder (op \<ge>) (op >)"
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by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
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text {* min/max *}
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definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  "min a b = (if a \<le> b then a else b)"
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definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
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  "max a b = (if a \<le> b then b else a)"
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lemma min_le_iff_disj:
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  "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
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unfolding min_def using linear by (auto intro: order_trans)
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lemma le_max_iff_disj:
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  "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
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unfolding max_def using linear by (auto intro: order_trans)
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lemma min_less_iff_disj:
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  "min x y < z \<longleftrightarrow> x < z \<or> y < z"
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unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma less_max_iff_disj:
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  "z < max x y \<longleftrightarrow> z < x \<or> z < y"
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unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma min_less_iff_conj [simp]:
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  "z < min x y \<longleftrightarrow> z < x \<and> z < y"
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unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma max_less_iff_conj [simp]:
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  "max x y < z \<longleftrightarrow> x < z \<and> y < z"
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unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma split_min [no_atp]:
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  "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
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by (simp add: min_def)
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lemma split_max [no_atp]:
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  "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
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by (simp add: max_def)
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end
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subsection {* Reasoning tools setup *}
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ML {*
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signature ORDERS =
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sig
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  val print_structures: Proof.context -> unit
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  val setup: theory -> theory
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  val order_tac: Proof.context -> thm list -> int -> tactic
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end;
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structure Orders: ORDERS =
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struct
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(** Theory and context data **)
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fun struct_eq ((s1: string, ts1), (s2, ts2)) =
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   324
  (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   325
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   326
structure Data = Generic_Data
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   327
(
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   328
  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
   329
    (* Order structures:
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   330
       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
   331
       needed to set up the transitivity reasoner,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   332
       identifier and operations identify the structure uniquely. *)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   333
  val empty = [];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   334
  val extend = I;
33519
e31a85f92ce9 adapted Generic_Data, Proof_Data;
wenzelm
parents: 32960
diff changeset
   335
  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
   336
);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   337
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   338
fun print_structures ctxt =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   339
  let
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   340
    val structs = Data.get (Context.Proof ctxt);
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   341
    fun pretty_term t = Pretty.block
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   342
      [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
   343
        Pretty.str "::", Pretty.brk 1,
24920
2a45e400fdad generic Syntax.pretty/string_of operations;
wenzelm
parents: 24867
diff changeset
   344
        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
   345
    fun pretty_struct ((s, ts), _) = Pretty.block
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   346
      [Pretty.str s, Pretty.str ":", Pretty.brk 1,
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   347
       Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   348
  in
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   349
    Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   350
  end;
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   351
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   352
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   353
(** Method **)
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   354
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   355
fun struct_tac ((s, [eq, le, less]), thms) ctxt prems =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   356
  let
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   357
    fun decomp thy (@{const Trueprop} $ t) =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   358
      let
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   359
        fun excluded t =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   360
          (* exclude numeric types: linear arithmetic subsumes transitivity *)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   361
          let val T = type_of t
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   362
          in
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   363
            T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   364
          end;
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   365
        fun rel (bin_op $ t1 $ t2) =
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   366
              if excluded t1 then NONE
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   367
              else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   368
              else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   369
              else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   370
              else NONE
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   371
          | rel _ = NONE;
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   372
        fun dec (Const (@{const_name Not}, _) $ t) = (case rel t
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   373
              of NONE => NONE
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32899
diff changeset
   374
               | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
24741
a53f5db5acbb Fixed setup of transitivity reasoner (function decomp).
ballarin
parents: 24704
diff changeset
   375
          | dec x = rel x;
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   376
      in dec t end
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29823
diff changeset
   377
      | decomp thy _ = NONE;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   378
  in
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   379
    case s of
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   380
      "order" => Order_Tac.partial_tac decomp thms ctxt prems
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   381
    | "linorder" => Order_Tac.linear_tac decomp thms ctxt prems
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   382
    | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   383
  end
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   384
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   385
fun order_tac ctxt prems =
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   386
  FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt)));
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   387
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   388
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   389
(** Attribute **)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   390
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   391
fun add_struct_thm s tag =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   392
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   393
    (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
   394
fun del_struct s =
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   395
  Thm.declaration_attribute
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   396
    (fn _ => Data.map (AList.delete struct_eq s));
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   397
30722
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   398
val attrib_setup =
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   399
  Attrib.setup @{binding order}
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   400
    (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   401
      Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   402
      Scan.repeat Args.term
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   403
      >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   404
           | ((NONE, n), ts) => del_struct (n, ts)))
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   405
    "theorems controlling transitivity reasoner";
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   406
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   407
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   408
(** Diagnostic command **)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   409
24867
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   410
val _ =
36960
01594f816e3a prefer structure Keyword, Parse, Parse_Spec, Outer_Syntax;
wenzelm
parents: 36635
diff changeset
   411
  Outer_Syntax.improper_command "print_orders"
01594f816e3a prefer structure Keyword, Parse, Parse_Spec, Outer_Syntax;
wenzelm
parents: 36635
diff changeset
   412
    "print order structures available to transitivity reasoner" Keyword.diag
30806
342c73345237 simplified 'print_orders' command;
wenzelm
parents: 30722
diff changeset
   413
    (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o
342c73345237 simplified 'print_orders' command;
wenzelm
parents: 30722
diff changeset
   414
        Toplevel.keep (print_structures o Toplevel.context_of)));
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   415
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   416
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   417
(** Setup **)
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   418
24867
e5b55d7be9bb simplified interfaces for outer syntax;
wenzelm
parents: 24748
diff changeset
   419
val setup =
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31998
diff changeset
   420
  Method.setup @{binding order} (Scan.succeed (fn ctxt => SIMPLE_METHOD' (order_tac ctxt [])))
30722
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   421
    "transitivity reasoner" #>
623d4831c8cf simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
wenzelm
parents: 30528
diff changeset
   422
  attrib_setup;
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   423
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   424
end;
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   425
21091
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   426
*}
5061e3e56484 cleanup in ML setup code
haftmann
parents: 21083
diff changeset
   427
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   428
setup Orders.setup
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   429
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
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
   432
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   433
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   434
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   435
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   436
(* 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
   437
   is not a parameter of the locale. *)
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   438
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   439
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
   440
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   441
declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   442
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   443
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   444
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   445
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   446
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   447
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   448
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   449
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   450
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   451
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   452
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   453
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   454
  
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   455
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   456
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   457
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   458
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   459
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   460
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   461
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   462
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   463
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   464
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   465
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
   466
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   467
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
   468
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   469
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   470
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   471
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   472
begin
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   473
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   474
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   475
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   476
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   477
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   478
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   479
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   480
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   481
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   482
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   483
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   484
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   485
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   486
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   487
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   488
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   489
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   490
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   491
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   492
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   493
27689
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   494
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   495
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   496
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   497
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   498
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   499
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   500
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   501
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   502
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   503
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   504
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   505
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   506
declare neq_le_trans [order add neq_le_trans: 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_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   509
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   510
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
   511
268a7d02cf7a declare
haftmann
parents: 27682
diff changeset
   512
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
   513
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   514
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   515
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   516
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   517
setup {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   518
let
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   519
44058
ae85c5d64913 misc tuning -- eliminated old-fashioned rep_thm;
wenzelm
parents: 44025
diff changeset
   520
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
   521
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   522
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
43597
b4a093e755db tuned signature;
wenzelm
parents: 43596
diff changeset
   523
  let val prems = Simplifier.prems_of ss;
22916
haftmann
parents: 22886
diff changeset
   524
      val less = Const (@{const_name less}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   525
      val t = HOLogic.mk_Trueprop(le $ s $ r);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   526
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   527
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   528
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   529
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   530
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   531
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   532
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   533
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   534
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   535
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   536
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   537
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
43597
b4a093e755db tuned signature;
wenzelm
parents: 43596
diff changeset
   538
  let val prems = Simplifier.prems_of ss;
22916
haftmann
parents: 22886
diff changeset
   539
      val le = Const (@{const_name less_eq}, T);
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   540
      val t = HOLogic.mk_Trueprop(le $ r $ s);
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   541
  in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   542
       NONE =>
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   543
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   544
         in case find_first (prp t) prems of
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   545
              NONE => NONE
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   546
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   547
         end
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   548
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   549
  end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   550
  handle THM _ => NONE;
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   551
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   552
fun add_simprocs procs thy =
42795
66fcc9882784 clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents: 42287
diff changeset
   553
  Simplifier.map_simpset_global (fn ss => ss
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   554
    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
   555
      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
   556
26496
49ae9456eba9 purely functional setup of claset/simpset/clasimpset;
wenzelm
parents: 26324
diff changeset
   557
fun add_solver name tac =
42795
66fcc9882784 clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents: 42287
diff changeset
   558
  Simplifier.map_simpset_global (fn ss => ss addSolver
43597
b4a093e755db tuned signature;
wenzelm
parents: 43596
diff changeset
   559
    mk_solver name (fn ss => tac (Simplifier.the_context ss) (Simplifier.prems_of ss)));
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   560
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   561
in
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   562
  add_simprocs [
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   563
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   564
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   565
     ]
24641
448edc627ee4 Transitivity reasoner set up for locales order and linorder.
ballarin
parents: 24422
diff changeset
   566
  #> add_solver "Transitivity" Orders.order_tac
21248
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   567
  (* Adding the transitivity reasoners also as safe solvers showed a slight
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   568
     speed up, but the reasoning strength appears to be not higher (at least
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   569
     no breaking of additional proofs in the entire HOL distribution, as
3fd22b0939ff abstract ordering theories
haftmann
parents: 21216
diff changeset
   570
     of 5 March 2004, was observed). *)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   571
end
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   572
*}
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   573
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
   574
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   575
subsection {* Bounded quantifiers *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   576
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   577
syntax
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   578
  "_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
   579
  "_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
   580
  "_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
   581
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   582
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   583
  "_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
   584
  "_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
   585
  "_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
   586
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   587
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   588
syntax (xsymbols)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   589
  "_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
   590
  "_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
   591
  "_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
   592
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   593
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   594
  "_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
   595
  "_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
   596
  "_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
   597
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   598
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   599
syntax (HOL)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   600
  "_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
   601
  "_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
   602
  "_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
   603
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   604
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   605
syntax (HTML output)
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"    ("(3\<forall>_<_./ _)"  [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"    ("(3\<exists>_<_./ _)"  [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"    ("(3\<forall>_\<le>_./ _)" [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"    ("(3\<exists>_\<le>_./ _)" [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"    ("(3\<forall>_>_./ _)"  [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"    ("(3\<exists>_>_./ _)"  [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"    ("(3\<forall>_\<ge>_./ _)" [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"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   615
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   616
translations
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   617
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   618
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   619
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   620
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   621
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   622
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   623
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   624
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   625
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   626
print_translation {*
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   627
let
42287
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   628
  val All_binder = Mixfix.binder_name @{const_syntax All};
d98eb048a2e4 discontinued special treatment of structure Mixfix;
wenzelm
parents: 42284
diff changeset
   629
  val Ex_binder = Mixfix.binder_name @{const_syntax Ex};
38786
e46e7a9cb622 formerly unnamed infix impliciation now named HOL.implies
haftmann
parents: 38715
diff changeset
   630
  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
   631
  val conj = @{const_syntax HOL.conj};
22916
haftmann
parents: 22886
diff changeset
   632
  val less = @{const_syntax less};
haftmann
parents: 22886
diff changeset
   633
  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
   634
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   635
  val trans =
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   636
   [((All_binder, impl, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   637
    (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   638
    ((All_binder, impl, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   639
    (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   640
    ((Ex_binder, conj, less),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   641
    (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   642
    ((Ex_binder, conj, less_eq),
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   643
    (@{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
   644
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   645
  fun matches_bound v t =
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   646
    (case t of
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   647
      Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   648
    | _ => false);
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   649
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41082
diff changeset
   650
  fun mk v c n P = Syntax.const c $ Syntax_Trans.mark_bound v $ n $ P;
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   651
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   652
  fun tr' q = (q,
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   653
    fn [Const (@{syntax_const "_bound"}, _) $ Free (v, _),
b8c62d60195c more antiquotations;
wenzelm
parents: 35301
diff changeset
   654
        Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   655
        (case AList.lookup (op =) trans (q, c, d) of
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   656
          NONE => raise Match
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   657
        | SOME (l, g) =>
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   658
            if matches_bound v t andalso not (contains_var v u) then mk v l u P
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   659
            else if matches_bound v u andalso not (contains_var v t) then mk v g t P
446c5063e4fd modernized translations;
wenzelm
parents: 35092
diff changeset
   660
            else raise Match)
21180
f27f12bcafb8 fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents: 21091
diff changeset
   661
     | _ => raise Match);
21524
7843e2fd14a9 updated (binder) syntax/notation;
wenzelm
parents: 21412
diff changeset
   662
in [tr' All_binder, tr' Ex_binder] end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   663
*}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   664
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   665
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   666
subsection {* Transitivity reasoning *}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   667
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   668
context ord
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   669
begin
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   670
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   671
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   672
  by (rule subst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   673
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   674
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   675
  by (rule ssubst)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   676
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   677
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   678
  by (rule subst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   679
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   680
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   681
  by (rule ssubst)
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   682
e2e1a4b00de3 various localizations
haftmann
parents: 25103
diff changeset
   683
end
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   684
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   685
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
   686
  (!!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
   687
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   688
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   689
  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
   690
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   691
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   692
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   693
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   694
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
   695
  (!!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
   696
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   697
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   698
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   699
  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
   700
  finally (less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   701
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   702
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   703
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
   704
  (!!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
   705
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   706
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   707
  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
   708
  also assume "f b < c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   709
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   710
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   711
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   712
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
   713
  (!!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
   714
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   715
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   716
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   717
  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
   718
  finally (le_less_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   719
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   720
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   721
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
   722
  (!!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
   723
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   724
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   725
  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
   726
  also assume "f b <= c"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
   727
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   728
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   729
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   730
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
   731
  (!!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
   732
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   733
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   734
  assume "a < f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   735
  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
   736
  finally (less_le_trans) show ?thesis .
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   737
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   738
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   739
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
   740
  (!!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
   741
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   742
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   743
  assume "a <= f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   744
  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
   745
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   746
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   747
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   748
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
   749
  (!!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
   750
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   751
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   752
  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
   753
  also assume "f b <= c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   754
  finally (order_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   755
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   756
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   757
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   758
  (!!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
   759
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   760
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   761
  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
   762
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   763
  finally (ord_le_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   764
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   765
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   766
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   767
  (!!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
   768
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   769
  assume r: "!!x y. x <= y ==> f x <= f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   770
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   771
  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
   772
  finally (ord_eq_le_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   773
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   774
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   775
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   776
  (!!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
   777
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   778
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   779
  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
   780
  also assume "f b = c"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   781
  finally (ord_less_eq_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   782
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   783
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   784
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   785
  (!!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
   786
proof -
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   787
  assume r: "!!x y. x < y ==> f x < f y"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   788
  assume "a = f b"
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   789
  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
   790
  finally (ord_eq_less_trans) show ?thesis .
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   791
qed
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   792
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   793
text {*
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   794
  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
   795
*}
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   796
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   797
lemmas [trans] =
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   798
  order_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   799
  order_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   800
  order_le_less_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   801
  order_le_less_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   802
  order_less_le_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   803
  order_less_le_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   804
  order_subst2
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   805
  order_subst1
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   806
  ord_le_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   807
  ord_eq_le_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   808
  ord_less_eq_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   809
  ord_eq_less_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   810
  forw_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   811
  back_subst
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   812
  rev_mp
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   813
  mp
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   814
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   815
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   816
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   817
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   818
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   819
lemmas (in preorder) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   820
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   821
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   822
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   823
  less_le_trans
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   824
  order_trans
27682
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   825
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   826
lemmas (in order) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   827
  antisym
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   828
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   829
lemmas (in ord) [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   830
  ord_le_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   831
  ord_eq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   832
  ord_less_eq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   833
  ord_eq_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   834
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   835
lemmas [trans] =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   836
  trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   837
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   838
lemmas order_trans_rules =
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   839
  order_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   840
  order_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   841
  order_le_less_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   842
  order_le_less_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   843
  order_less_le_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   844
  order_less_le_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   845
  order_subst2
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   846
  order_subst1
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   847
  ord_le_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   848
  ord_eq_le_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   849
  ord_less_eq_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   850
  ord_eq_less_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   851
  forw_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   852
  back_subst
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   853
  rev_mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   854
  mp
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   855
  neq_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   856
  le_neq_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   857
  less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   858
  less_asym'
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   859
  le_less_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   860
  less_le_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   861
  order_trans
25aceefd4786 added class preorder
haftmann
parents: 27299
diff changeset
   862
  antisym
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   863
  ord_le_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   864
  ord_eq_le_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   865
  ord_less_eq_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   866
  ord_eq_less_trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   867
  trans
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
   868
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   869
text {* These support proving chains of decreasing inequalities
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   870
    a >= b >= c ... in Isar proofs. *}
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   871
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   872
lemma xt1 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   873
  "a = b ==> b > c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   874
  "a > b ==> b = c ==> a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   875
  "a = b ==> b >= c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   876
  "a >= b ==> b = c ==> a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   877
  "(x::'a::order) >= y ==> y >= x ==> x = y"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   878
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   879
  "(x::'a::order) > y ==> y >= z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   880
  "(x::'a::order) >= y ==> y > z ==> x > z"
23417
wenzelm
parents: 23263
diff changeset
   881
  "(a::'a::order) > b ==> b > a ==> P"
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   882
  "(x::'a::order) > y ==> y > z ==> x > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   883
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   884
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   885
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   886
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   887
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   888
  "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
   889
  by auto
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   890
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   891
lemma xt2 [no_atp]:
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   892
  "(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
   893
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   894
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   895
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   896
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   897
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   898
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   899
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   900
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   901
by (subgoal_tac "f b >= f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   902
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   903
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   904
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   905
by (subgoal_tac "f a > f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   906
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   907
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   908
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   909
by (subgoal_tac "f b > f c", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   910
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   911
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   912
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   913
by (subgoal_tac "f a >= f b", force, force)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   914
45221
3eadb9b6a055 mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents: 44921
diff changeset
   915
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   916
    (!!x y. x > y ==> f x > f y) ==> a > f c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   917
by (subgoal_tac "f b > f c", force, force)
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 xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   920
    (!!x y. x > y ==> f x > f y) ==> f a > c"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   921
by (subgoal_tac "f a > f b", 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
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 [no_atp]
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   924
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   925
(* 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   926
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   927
  for the wrong thing in an Isar proof.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   928
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   929
  The extra transitivity rules can be used as follows: 
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   930
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   931
lemma "(a::'a::order) > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   932
proof -
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   933
  have "a >= b" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   934
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   935
  also have "?rhs >= c" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   936
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   937
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   938
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   939
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   940
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   941
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   942
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   943
  also (xtrans) have "?rhs > z"
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   944
    sorry
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   945
  finally (xtrans) show ?thesis .
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   946
qed
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   947
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   948
  Alternatively, one can use "declare xtrans [trans]" and then
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   949
  leave out the "(xtrans)" above.
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   950
*)
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   951
23881
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   952
851c74f1bb69 moved class ord from Orderings.thy to HOL.thy
haftmann
parents: 23417
diff changeset
   953
subsection {* Monotonicity, least value operator and min/max *}
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
   954
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   955
context order
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   956
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   957
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   958
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   959
  "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   960
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   961
lemma monoI [intro?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   962
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   963
  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
   964
  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
   965
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   966
lemma monoD [dest?]:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   967
  fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   968
  shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   969
  unfolding mono_def by iprover
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   970
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   971
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   972
  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   973
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   974
lemma strict_monoI [intro?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   975
  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   976
  shows "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   977
  using assms unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   978
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   979
lemma strict_monoD [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   980
  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   981
  unfolding strict_mono_def by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   982
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   983
lemma strict_mono_mono [dest?]:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   984
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   985
  shows "mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   986
proof (rule monoI)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   987
  fix x y
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   988
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   989
  show "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   990
  proof (cases "x = y")
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   991
    case True then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   992
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   993
    case False with `x \<le> y` have "x < y" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   994
    with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   995
    then show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   996
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   997
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
   998
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   999
end
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1000
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1001
context linorder
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1002
begin
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1003
30298
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1004
lemma strict_mono_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1005
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1006
  shows "f x = f y \<longleftrightarrow> x = y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1007
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1008
  assume "f x = f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1009
  show "x = y" proof (cases x y rule: linorder_cases)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1010
    case less with assms strict_monoD have "f x < f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1011
    with `f x = f y` show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1012
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1013
    case equal then show ?thesis .
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1014
  next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1015
    case greater with assms strict_monoD have "f y < f x" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1016
    with `f x = f y` show ?thesis by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1017
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1018
qed simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1019
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1020
lemma strict_mono_less_eq:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1021
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1022
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1023
proof
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1024
  assume "x \<le> y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1025
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1026
next
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1027
  assume "f x \<le> f y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1028
  show "x \<le> y" proof (rule ccontr)
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1029
    assume "\<not> x \<le> y" then have "y < x" by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1030
    with assms strict_monoD have "f y < f x" by auto
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1031
    with `f x \<le> f y` show False by simp
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1032
  qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1033
qed
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1034
  
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1035
lemma strict_mono_less:
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1036
  assumes "strict_mono f"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1037
  shows "f x < f y \<longleftrightarrow> x < y"
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1038
  using assms
abefe1dfadbb added strict_mono predicate
haftmann
parents: 30107
diff changeset
  1039
    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
  1040
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1041
lemma min_of_mono:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1042
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
25377
dcde128c84a2 Orderings.min/max: no need to qualify consts;
wenzelm
parents: 25207
diff changeset
  1043
  shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1044
  by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1045
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1046
lemma max_of_mono:
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1047
  fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
25377
dcde128c84a2 Orderings.min/max: no need to qualify consts;
wenzelm
parents: 25207
diff changeset
  1048
  shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1049
  by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1050
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
  1051
end
21083
a1de02f047d0 cleaned up
haftmann
parents: 21044
diff changeset
  1052
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1053
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1054
by (simp add: min_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1055
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1056
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
23212
82881b1ae9c6 tuned list comprehension, added lemma
nipkow
parents: 23182
diff changeset
  1057
by (simp add: max_def)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1058
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1059
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1060
by (simp add: min_def) (blast intro: antisym)
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1061
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1062
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
45893
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1063
by (simp add: max_def) (blast intro: antisym)
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1064
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1065
lemma min_greatestL: "(\<And>x::'a::order. x \<le> greatest) \<Longrightarrow> min greatest x = x"
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1066
by (simp add: min_def) (blast intro: antisym)
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1067
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1068
lemma max_greatestL: "(\<And>x::'a::order. x \<le> greatest) \<Longrightarrow> max greatest x = greatest"
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1069
by (simp add: max_def) (blast intro: antisym)
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1070
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1071
lemma min_greatestR: "(\<And>x. x \<le> greatest) \<Longrightarrow> min x greatest = x"
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1072
by (simp add: min_def)
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1073
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1074
lemma max_greatestR: "(\<And>x. x \<le> greatest) \<Longrightarrow> max x greatest = greatest"
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1075
by (simp add: max_def)
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1076
e7dbb27c1308 add complementary lemmas for {min,max}_least
noschinl
parents: 45262
diff changeset
  1077
21383
17e6275e13f5 added transitivity rules, reworking of min/max lemmas
haftmann
parents: 21329
diff changeset
  1078
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1079
43813
07f0650146f2 tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents: 43597
diff changeset
  1080
subsection {* (Unique) top and bottom elements *}
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1081
43813
07f0650146f2 tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents: 43597
diff changeset
  1082
class bot = order +
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1083
  fixes bot :: 'a ("\<bottom>")
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1084
  assumes bot_least [simp]: "\<bottom> \<le> a"
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1085
begin
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1086
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1087
lemma le_bot:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1088
  "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1089
  by (auto intro: antisym)
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1090
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1091
lemma bot_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1092
  "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1093
  by (auto intro: antisym)
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1094
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1095
lemma not_less_bot [simp]:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1096
  "\<not> (a < \<bottom>)"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1097
  using bot_least [of a] by (auto simp: le_less)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1098
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1099
lemma bot_less:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1100
  "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1101
  by (auto simp add: less_le_not_le intro!: antisym)
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1102
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1103
end
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1104
43813
07f0650146f2 tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents: 43597
diff changeset
  1105
class top = order +
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1106
  fixes top :: 'a ("\<top>")
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1107
  assumes top_greatest [simp]: "a \<le> \<top>"
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1108
begin
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1109
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1110
lemma top_le:
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1111
  "\<top> \<le> a \<Longrightarrow> a = \<top>"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1112
  by (rule antisym) auto
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1113
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1114
lemma top_unique:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1115
  "\<top> \<le> a \<longleftrightarrow> a = \<top>"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1116
  by (auto intro: antisym)
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1117
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1118
lemma not_top_less [simp]: "\<not> (\<top> < a)"
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1119
  using top_greatest [of a] by (auto simp: le_less)
43816
05ab37be94ed uniqueness lemmas for bot and top
haftmann
parents: 43814
diff changeset
  1120
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1121
lemma less_top:
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1122
  "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
43814
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1123
  by (auto simp add: less_le_not_le intro!: antisym)
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1124
58791b75cf1f moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents: 43813
diff changeset
  1125
end
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1126
43853
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1127
no_notation
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1128
  bot ("\<bottom>") and
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1129
  top ("\<top>")
020ddc6a9508 consolidated bot and top classes, tuned notation
haftmann
parents: 43816
diff changeset
  1130
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1131
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1132
subsection {* Dense orders *}
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1133
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
  1134
class dense_linorder = linorder + 
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1135
  assumes gt_ex: "\<exists>y. x < y" 
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1136
  and lt_ex: "\<exists>y. y < x"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1137
  and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1138
begin
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1139
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1140
lemma dense_le:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1141
  fixes y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1142
  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
  1143
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1144
proof (rule ccontr)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1145
  assume "\<not> ?thesis"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1146
  hence "z < y" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1147
  from dense[OF this]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1148
  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
  1149
  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
  1150
  ultimately show False by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1151
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1152
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1153
lemma dense_le_bounded:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1154
  fixes x y z :: 'a
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1155
  assumes "x < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1156
  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
  1157
  shows "y \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1158
proof (rule dense_le)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1159
  fix w assume "w < y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1160
  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
  1161
  from linear[of u w]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1162
  show "w \<le> z"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1163
  proof (rule disjE)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1164
    assume "u \<le> w"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1165
    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
  1166
    show "w \<le> z" by (rule *)
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1167
  next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1168
    assume "w \<le> u"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1169
    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
  1170
    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
  1171
  qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1172
qed
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1173
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35364
diff changeset
  1174
end
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1175
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1176
subsection {* Wellorders *}
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1177
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1178
class wellorder = linorder +
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1179
  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
  1180
begin
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1181
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1182
lemma wellorder_Least_lemma:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1183
  fixes k :: 'a
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1184
  assumes "P k"
34250
3b619abaa67a moved name duplicates to end of theory; reduced warning noise
haftmann
parents: 34065
diff changeset
  1185
  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
  1186
proof -
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1187
  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
  1188
  using assms proof (induct k rule: less_induct)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1189
    case (less x) then have "P x" by simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1190
    show ?case proof (rule classical)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1191
      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
  1192
      have "\<And>y. P y \<Longrightarrow> x \<le> y"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1193
      proof (rule classical)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1194
        fix y
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1195
        assume "P y" and "\<not> x \<le> y"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1196
        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
  1197
          by (auto simp add: not_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1198
        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
  1199
          by auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1200
        then show "x \<le> y" by auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1201
      qed
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1202
      with `P x` have Least: "(LEAST a. P a) = x"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1203
        by (rule Least_equality)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1204
      with `P x` show ?thesis by simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1205
    qed
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1206
  qed
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1207
  then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1208
qed
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1209
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1210
-- "The following 3 lemmas are due to Brian Huffman"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1211
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1212
  by (erule exE) (erule LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1213
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1214
lemma LeastI2:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1215
  "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1216
  by (blast intro: LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1217
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1218
lemma LeastI2_ex:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1219
  "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1220
  by (blast intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1221
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1222
lemma LeastI2_wellorder:
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1223
  assumes "P a"
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1224
  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1225
  shows "Q (Least P)"
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1226
proof (rule LeastI2_order)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1227
  show "P (Least P)" using `P a` by (rule LeastI)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1228
next
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1229
  fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1230
next
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1231
  fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1232
qed
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1233
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1234
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1235
apply (simp (no_asm_use) add: not_le [symmetric])
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1236
apply (erule contrapos_nn)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1237
apply (erule Least_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1238
done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1239
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38650
diff changeset
  1240
end
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27689
diff changeset
  1241
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1242
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1243
subsection {* Order on bool *}
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1244
45262
b0cea4362430 instance bool :: linorder
huffman
parents: 45261
diff changeset
  1245
instantiation bool :: "{bot, top, linorder}"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1246
begin
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1247
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1248
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1249
  le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1250
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1251
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1252
  [simp]: "(P\<Colon>bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1253
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1254
definition
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1255
  [simp]: "bot \<longleftrightarrow> False"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1256
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1257
definition
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1258
  [simp]: "top \<longleftrightarrow> True"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1259
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1260
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1261
qed auto
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1262
15524
2ef571f80a55 Moved oderings from HOL into the new Orderings.thy
nipkow
parents:
diff changeset
  1263
end
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1264
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1265
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1266
  by simp
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1267
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1268
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1269
  by simp
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1270
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1271
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1272
  by simp
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1273
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1274
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1275
  by simp
32899
c913cc831630 tuned order of lemmas
haftmann
parents: 32887
diff changeset
  1276
c913cc831630 tuned order of lemmas
haftmann
parents: 32887
diff changeset
  1277
lemma bot_boolE: "bot \<Longrightarrow> P"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1278
  by simp
32899
c913cc831630 tuned order of lemmas
haftmann
parents: 32887
diff changeset
  1279
c913cc831630 tuned order of lemmas
haftmann
parents: 32887
diff changeset
  1280
lemma top_boolI: top
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1281
  by simp
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1282
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1283
lemma [code]:
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1284
  "False \<le> b \<longleftrightarrow> True"
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1285
  "True \<le> b \<longleftrightarrow> b"
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1286
  "False < b \<longleftrightarrow> b"
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1287
  "True < b \<longleftrightarrow> False"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1288
  by simp_all
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1289
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1290
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1291
subsection {* Order on functions *}
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1292
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1293
instantiation "fun" :: (type, ord) ord
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1294
begin
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1295
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1296
definition
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36960
diff changeset
  1297
  le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1298
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1299
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1300
  "(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1301
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1302
instance ..
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1303
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1304
end
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1305
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1306
instance "fun" :: (type, preorder) preorder proof
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1307
qed (auto simp add: le_fun_def less_fun_def
44921
58eef4843641 tuned proofs
huffman
parents: 44058
diff changeset
  1308
  intro: order_trans antisym)
28685
275122631271 new classes "top" and "bot"
haftmann
parents: 28562
diff changeset
  1309
275122631271 new classes "top" and "bot"