author | paulson <lp15@cam.ac.uk> |
Thu, 10 Dec 2015 13:38:40 +0000 | |
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parent 61799 | 4cf66f21b764 |
child 61955 | e96292f32c3c |
permissions | -rw-r--r-- |
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(* Title: HOL/Orderings.thy |
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
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*) |
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section \<open>Abstract orderings\<close> |
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theory Orderings |
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distributed theory Algebras to theories Groups and Lattices
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imports HOL |
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declare command keywords via theory header, including strict checking outside Pure;
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keywords "print_orders" :: diag |
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begin |
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ML_file "~~/src/Provers/order.ML" |
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ML_file "~~/src/Provers/quasi.ML" (* FIXME unused? *) |
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subsection \<open>Abstract ordering\<close> |
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locale ordering = |
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<preceq>" 50) |
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<prec>" 50) |
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assumes strict_iff_order: "a \<prec> b \<longleftrightarrow> a \<preceq> b \<and> a \<noteq> b" |
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assumes refl: "a \<preceq> a" \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
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and antisym: "a \<preceq> b \<Longrightarrow> b \<preceq> a \<Longrightarrow> a = b" |
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and trans: "a \<preceq> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<preceq> c" |
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begin |
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lemma strict_implies_order: |
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"a \<prec> b \<Longrightarrow> a \<preceq> b" |
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by (simp add: strict_iff_order) |
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lemma strict_implies_not_eq: |
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"a \<prec> b \<Longrightarrow> a \<noteq> b" |
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by (simp add: strict_iff_order) |
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lemma not_eq_order_implies_strict: |
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"a \<noteq> b \<Longrightarrow> a \<preceq> b \<Longrightarrow> a \<prec> b" |
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by (simp add: strict_iff_order) |
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lemma order_iff_strict: |
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"a \<preceq> b \<longleftrightarrow> a \<prec> b \<or> a = b" |
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by (auto simp add: strict_iff_order refl) |
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lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close> |
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"\<not> a \<prec> a" |
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by (simp add: strict_iff_order) |
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lemma asym: |
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"a \<prec> b \<Longrightarrow> b \<prec> a \<Longrightarrow> False" |
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by (auto simp add: strict_iff_order intro: antisym) |
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lemma strict_trans1: |
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"a \<preceq> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c" |
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by (auto simp add: strict_iff_order intro: trans antisym) |
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lemma strict_trans2: |
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"a \<prec> b \<Longrightarrow> b \<preceq> c \<Longrightarrow> a \<prec> c" |
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by (auto simp add: strict_iff_order intro: trans antisym) |
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lemma strict_trans: |
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"a \<prec> b \<Longrightarrow> b \<prec> c \<Longrightarrow> a \<prec> c" |
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by (auto intro: strict_trans1 strict_implies_order) |
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end |
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locale ordering_top = ordering + |
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fixes top :: "'a" |
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assumes extremum [simp]: "a \<preceq> top" |
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begin |
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lemma extremum_uniqueI: |
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"top \<preceq> a \<Longrightarrow> a = top" |
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by (rule antisym) auto |
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lemma extremum_unique: |
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"top \<preceq> a \<longleftrightarrow> a = top" |
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by (auto intro: antisym) |
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lemma extremum_strict [simp]: |
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"\<not> (top \<prec> a)" |
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using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl) |
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lemma not_eq_extremum: |
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"a \<noteq> top \<longleftrightarrow> a \<prec> top" |
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by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum) |
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end |
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subsection \<open>Syntactic orders\<close> |
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class ord = |
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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begin |
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notation |
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less_eq ("op <=") and |
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less_eq ("(_/ <= _)" [51, 51] 50) and |
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less ("op <") and |
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less ("(_/ < _)" [51, 51] 50) |
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notation (xsymbols) |
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less_eq ("op \<le>") and |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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abbreviation (input) |
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greater_eq (infix ">=" 50) where |
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"x >= y \<equiv> y <= x" |
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notation (input) |
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greater_eq (infix "\<ge>" 50) |
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abbreviation (input) |
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greater (infix ">" 50) where |
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"x > y \<equiv> y < x" |
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end |
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subsection \<open>Quasi orders\<close> |
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class preorder = ord + |
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assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" |
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and order_refl [iff]: "x \<le> x" |
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and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
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begin |
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text \<open>Reflexivity.\<close> |
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" |
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\<comment> \<open>This form is useful with the classical reasoner.\<close> |
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by (erule ssubst) (rule order_refl) |
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lemma less_irrefl [iff]: "\<not> x < x" |
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by (simp add: less_le_not_le) |
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
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unfolding less_le_not_le by blast |
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text \<open>Asymmetry.\<close> |
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lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
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by (simp add: less_le_not_le) |
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lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" |
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by (drule less_not_sym, erule contrapos_np) simp |
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text \<open>Transitivity.\<close> |
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lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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text \<open>Useful for simplification, but too risky to include by default.\<close> |
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lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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text \<open>Transitivity rules for calculational reasoning\<close> |
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lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" |
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by (rule less_asym) |
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text \<open>Dual order\<close> |
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lemma dual_preorder: |
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"class.preorder (op \<ge>) (op >)" |
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proof qed (auto simp add: less_le_not_le intro: order_trans) |
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end |
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subsection \<open>Partial orders\<close> |
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class order = preorder + |
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assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
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begin |
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lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
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by (auto simp add: less_le_not_le intro: antisym) |
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sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater |
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by standard (auto intro: antisym order_trans simp add: less_le) |
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text \<open>Reflexivity.\<close> |
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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
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\<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close> |
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by (fact order.order_iff_strict) |
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lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
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unfolding less_le by blast |
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text \<open>Useful for simplification, but too risky to include by default.\<close> |
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lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
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by auto |
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lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
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by auto |
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text \<open>Transitivity rules for calculational reasoning\<close> |
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" |
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by (fact order.not_eq_order_implies_strict) |
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" |
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by (rule order.not_eq_order_implies_strict) |
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text \<open>Asymmetry.\<close> |
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lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" |
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by (blast intro: antisym) |
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lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym) |
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lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" |
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by (fact order.strict_implies_not_eq) |
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text \<open>Least value operator\<close> |
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definition (in ord) |
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Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where |
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"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" |
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lemma Least_equality: |
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assumes "P x" |
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and "\<And>y. P y \<Longrightarrow> x \<le> y" |
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shows "Least P = x" |
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unfolding Least_def by (rule the_equality) |
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(blast intro: assms antisym)+ |
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lemma LeastI2_order: |
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assumes "P x" |
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and "\<And>y. P y \<Longrightarrow> x \<le> y" |
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and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" |
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shows "Q (Least P)" |
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unfolding Least_def by (rule theI2) |
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(blast intro: assms antisym)+ |
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text \<open>Dual order\<close> |
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lemma dual_order: |
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"class.order (op \<ge>) (op >)" |
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym) |
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end |
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text \<open>Alternative introduction rule with bias towards strict order\<close> |
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lemma order_strictI: |
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fixes less (infix "\<sqsubset>" 50) |
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and less_eq (infix "\<sqsubseteq>" 50) |
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assumes less_eq_less: "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b" |
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assumes asym: "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a" |
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assumes irrefl: "\<And>a. \<not> a \<sqsubset> a" |
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assumes trans: "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> c" |
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shows "class.order less_eq less" |
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proof |
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fix a b |
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show "a \<sqsubset> b \<longleftrightarrow> a \<sqsubseteq> b \<and> \<not> b \<sqsubseteq> a" |
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by (auto simp add: less_eq_less asym irrefl) |
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next |
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fix a |
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show "a \<sqsubseteq> a" |
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by (auto simp add: less_eq_less) |
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next |
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fix a b c |
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assume "a \<sqsubseteq> b" and "b \<sqsubseteq> c" then show "a \<sqsubseteq> c" |
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by (auto simp add: less_eq_less intro: trans) |
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next |
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fix a b |
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assume "a \<sqsubseteq> b" and "b \<sqsubseteq> a" then show "a = b" |
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by (auto simp add: less_eq_less asym) |
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qed |
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subsection \<open>Linear (total) orders\<close> |
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class linorder = order + |
25207 | 300 |
assumes linear: "x \<le> y \<or> y \<le> x" |
21248 | 301 |
begin |
302 |
||
25062 | 303 |
lemma less_linear: "x < y \<or> x = y \<or> y < x" |
23212 | 304 |
unfolding less_le using less_le linear by blast |
21248 | 305 |
|
25062 | 306 |
lemma le_less_linear: "x \<le> y \<or> y < x" |
23212 | 307 |
by (simp add: le_less less_linear) |
21248 | 308 |
|
309 |
lemma le_cases [case_names le ge]: |
|
25062 | 310 |
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 311 |
using linear by blast |
21248 | 312 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
313 |
lemma (in linorder) le_cases3: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
314 |
"\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P; |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
315 |
\<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
316 |
by (blast intro: le_cases) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
317 |
|
22384
33a46e6c7f04
prefix of class interpretation not mandatory any longer
haftmann
parents:
22377
diff
changeset
|
318 |
lemma linorder_cases [case_names less equal greater]: |
25062 | 319 |
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" |
23212 | 320 |
using less_linear by blast |
21248 | 321 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
322 |
lemma linorder_wlog[case_names le sym]: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
323 |
"(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
324 |
by (cases rule: le_cases[of a b]) blast+ |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
56545
diff
changeset
|
325 |
|
25062 | 326 |
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" |
23212 | 327 |
apply (simp add: less_le) |
328 |
using linear apply (blast intro: antisym) |
|
329 |
done |
|
330 |
||
331 |
lemma not_less_iff_gr_or_eq: |
|
25062 | 332 |
"\<not>(x < y) \<longleftrightarrow> (x > y | x = y)" |
23212 | 333 |
apply(simp add:not_less le_less) |
334 |
apply blast |
|
335 |
done |
|
15524 | 336 |
|
25062 | 337 |
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" |
23212 | 338 |
apply (simp add: less_le) |
339 |
using linear apply (blast intro: antisym) |
|
340 |
done |
|
15524 | 341 |
|
25062 | 342 |
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" |
23212 | 343 |
by (cut_tac x = x and y = y in less_linear, auto) |
15524 | 344 |
|
25062 | 345 |
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" |
23212 | 346 |
by (simp add: neq_iff) blast |
15524 | 347 |
|
25062 | 348 |
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
23212 | 349 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 350 |
|
25062 | 351 |
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
23212 | 352 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 353 |
|
25062 | 354 |
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
23212 | 355 |
by (blast intro: antisym dest: not_less [THEN iffD1]) |
15524 | 356 |
|
25062 | 357 |
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" |
23212 | 358 |
unfolding not_less . |
16796 | 359 |
|
25062 | 360 |
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" |
23212 | 361 |
unfolding not_less . |
16796 | 362 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
363 |
lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y" |
23212 | 364 |
unfolding not_le . |
21248 | 365 |
|
60758 | 366 |
text \<open>Dual order\<close> |
22916 | 367 |
|
26014 | 368 |
lemma dual_linorder: |
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
35828
diff
changeset
|
369 |
"class.linorder (op \<ge>) (op >)" |
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
35828
diff
changeset
|
370 |
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear) |
22916 | 371 |
|
21248 | 372 |
end |
373 |
||
23948 | 374 |
|
60758 | 375 |
text \<open>Alternative introduction rule with bias towards strict order\<close> |
56545 | 376 |
|
377 |
lemma linorder_strictI: |
|
378 |
fixes less (infix "\<sqsubset>" 50) |
|
379 |
and less_eq (infix "\<sqsubseteq>" 50) |
|
380 |
assumes "class.order less_eq less" |
|
381 |
assumes trichotomy: "\<And>a b. a \<sqsubset> b \<or> a = b \<or> b \<sqsubset> a" |
|
382 |
shows "class.linorder less_eq less" |
|
383 |
proof - |
|
384 |
interpret order less_eq less |
|
60758 | 385 |
by (fact \<open>class.order less_eq less\<close>) |
56545 | 386 |
show ?thesis |
387 |
proof |
|
388 |
fix a b |
|
389 |
show "a \<sqsubseteq> b \<or> b \<sqsubseteq> a" |
|
390 |
using trichotomy by (auto simp add: le_less) |
|
391 |
qed |
|
392 |
qed |
|
393 |
||
394 |
||
60758 | 395 |
subsection \<open>Reasoning tools setup\<close> |
21083 | 396 |
|
60758 | 397 |
ML \<open> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
398 |
signature ORDERS = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
399 |
sig |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
400 |
val print_structures: Proof.context -> unit |
32215 | 401 |
val order_tac: Proof.context -> thm list -> int -> tactic |
58826 | 402 |
val add_struct: string * term list -> string -> attribute |
403 |
val del_struct: string * term list -> attribute |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
404 |
end; |
21091 | 405 |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
406 |
structure Orders: ORDERS = |
21248 | 407 |
struct |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
408 |
|
56508 | 409 |
(* context data *) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
410 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
411 |
fun struct_eq ((s1: string, ts1), (s2, ts2)) = |
56508 | 412 |
s1 = s2 andalso eq_list (op aconv) (ts1, ts2); |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
413 |
|
33519 | 414 |
structure Data = Generic_Data |
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 |
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
|
417 |
(* Order structures: |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
418 |
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
|
419 |
needed to set up the transitivity reasoner, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
420 |
identifier and operations identify the structure uniquely. *) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
421 |
val empty = []; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
422 |
val extend = I; |
33519 | 423 |
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
|
424 |
); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
425 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
426 |
fun print_structures ctxt = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
427 |
let |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
428 |
val structs = Data.get (Context.Proof ctxt); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
429 |
fun pretty_term t = Pretty.block |
24920 | 430 |
[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
|
431 |
Pretty.str "::", Pretty.brk 1, |
24920 | 432 |
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
|
433 |
fun pretty_struct ((s, ts), _) = Pretty.block |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
434 |
[Pretty.str s, Pretty.str ":", Pretty.brk 1, |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
435 |
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
436 |
in |
51579 | 437 |
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_struct structs)) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
438 |
end; |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
439 |
|
56508 | 440 |
val _ = |
59936
b8ffc3dc9e24
@{command_spec} is superseded by @{command_keyword};
wenzelm
parents:
59582
diff
changeset
|
441 |
Outer_Syntax.command @{command_keyword print_orders} |
56508 | 442 |
"print order structures available to transitivity reasoner" |
60097
d20ca79d50e4
discontinued pointless warnings: commands are only defined inside a theory context;
wenzelm
parents:
59936
diff
changeset
|
443 |
(Scan.succeed (Toplevel.keep (print_structures o Toplevel.context_of))); |
21091 | 444 |
|
56508 | 445 |
|
446 |
(* tactics *) |
|
447 |
||
448 |
fun struct_tac ((s, ops), thms) ctxt facts = |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
449 |
let |
56508 | 450 |
val [eq, le, less] = ops; |
30107
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29823
diff
changeset
|
451 |
fun decomp thy (@{const Trueprop} $ t) = |
56508 | 452 |
let |
453 |
fun excluded t = |
|
454 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
|
455 |
let val T = type_of t |
|
456 |
in |
|
457 |
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT |
|
458 |
end; |
|
459 |
fun rel (bin_op $ t1 $ t2) = |
|
460 |
if excluded t1 then NONE |
|
461 |
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) |
|
462 |
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) |
|
463 |
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) |
|
464 |
else NONE |
|
465 |
| rel _ = NONE; |
|
466 |
fun dec (Const (@{const_name Not}, _) $ t) = |
|
467 |
(case rel t of NONE => |
|
468 |
NONE |
|
469 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
470 |
| dec x = rel x; |
|
471 |
in dec t end |
|
472 |
| decomp _ _ = NONE; |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
473 |
in |
56508 | 474 |
(case s of |
475 |
"order" => Order_Tac.partial_tac decomp thms ctxt facts |
|
476 |
| "linorder" => Order_Tac.linear_tac decomp thms ctxt facts |
|
477 |
| _ => error ("Unknown order kind " ^ quote s ^ " encountered in transitivity reasoner")) |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
478 |
end |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
479 |
|
56508 | 480 |
fun order_tac ctxt facts = |
481 |
FIRST' (map (fn s => CHANGED o struct_tac s ctxt facts) (Data.get (Context.Proof ctxt))); |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
482 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
483 |
|
56508 | 484 |
(* attributes *) |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
485 |
|
58826 | 486 |
fun add_struct s tag = |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
487 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
488 |
(fn 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
|
489 |
fun del_struct s = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
490 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
491 |
(fn _ => Data.map (AList.delete struct_eq s)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
492 |
|
21091 | 493 |
end; |
60758 | 494 |
\<close> |
21091 | 495 |
|
60758 | 496 |
attribute_setup order = \<open> |
58826 | 497 |
Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --| |
498 |
Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name -- |
|
499 |
Scan.repeat Args.term |
|
500 |
>> (fn ((SOME tag, n), ts) => Orders.add_struct (n, ts) tag |
|
501 |
| ((NONE, n), ts) => Orders.del_struct (n, ts)) |
|
60758 | 502 |
\<close> "theorems controlling transitivity reasoner" |
58826 | 503 |
|
60758 | 504 |
method_setup order = \<open> |
47432 | 505 |
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt [])) |
60758 | 506 |
\<close> "transitivity reasoner" |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
507 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
508 |
|
60758 | 509 |
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close> |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
510 |
|
25076 | 511 |
context order |
512 |
begin |
|
513 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
514 |
(* 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
|
515 |
is not a parameter of the locale. *) |
25076 | 516 |
|
27689 | 517 |
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
518 |
|
27689 | 519 |
declare order_refl [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
520 |
|
27689 | 521 |
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
522 |
|
27689 | 523 |
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
524 |
||
525 |
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
526 |
||
527 |
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
528 |
||
529 |
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
530 |
|
27689 | 531 |
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
532 |
|
27689 | 533 |
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
534 |
||
535 |
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
536 |
||
537 |
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
538 |
||
539 |
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
540 |
||
541 |
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
542 |
||
543 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
544 |
||
545 |
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
|
546 |
|
25076 | 547 |
end |
548 |
||
549 |
context linorder |
|
550 |
begin |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
551 |
|
27689 | 552 |
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]] |
553 |
||
554 |
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
555 |
||
556 |
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
557 |
||
558 |
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
559 |
||
560 |
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
561 |
||
562 |
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
563 |
||
564 |
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
565 |
||
566 |
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
567 |
||
568 |
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
569 |
||
570 |
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
25076 | 571 |
|
27689 | 572 |
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
573 |
||
574 |
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
575 |
||
576 |
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
577 |
||
578 |
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
579 |
||
580 |
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
581 |
||
582 |
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
583 |
||
584 |
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
585 |
||
586 |
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
587 |
||
588 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
589 |
||
590 |
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
|
591 |
|
25076 | 592 |
end |
593 |
||
60758 | 594 |
setup \<open> |
56509 | 595 |
map_theory_simpset (fn ctxt0 => ctxt0 addSolver |
596 |
mk_solver "Transitivity" (fn ctxt => Orders.order_tac ctxt (Simplifier.prems_of ctxt))) |
|
597 |
(*Adding the transitivity reasoners also as safe solvers showed a slight |
|
598 |
speed up, but the reasoning strength appears to be not higher (at least |
|
599 |
no breaking of additional proofs in the entire HOL distribution, as |
|
600 |
of 5 March 2004, was observed).*) |
|
60758 | 601 |
\<close> |
15524 | 602 |
|
60758 | 603 |
ML \<open> |
56509 | 604 |
local |
605 |
fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *) |
|
606 |
in |
|
15524 | 607 |
|
56509 | 608 |
fun antisym_le_simproc ctxt ct = |
59582 | 609 |
(case Thm.term_of ct of |
56509 | 610 |
(le as Const (_, T)) $ r $ s => |
611 |
(let |
|
612 |
val prems = Simplifier.prems_of ctxt; |
|
613 |
val less = Const (@{const_name less}, T); |
|
614 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
|
615 |
in |
|
616 |
(case find_first (prp t) prems of |
|
617 |
NONE => |
|
618 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in |
|
619 |
(case find_first (prp t) prems of |
|
620 |
NONE => NONE |
|
621 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))) |
|
622 |
end |
|
623 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv}))) |
|
624 |
end handle THM _ => NONE) |
|
625 |
| _ => NONE); |
|
15524 | 626 |
|
56509 | 627 |
fun antisym_less_simproc ctxt ct = |
59582 | 628 |
(case Thm.term_of ct of |
56509 | 629 |
NotC $ ((less as Const(_,T)) $ r $ s) => |
630 |
(let |
|
631 |
val prems = Simplifier.prems_of ctxt; |
|
632 |
val le = Const (@{const_name less_eq}, T); |
|
633 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
|
634 |
in |
|
635 |
(case find_first (prp t) prems of |
|
636 |
NONE => |
|
637 |
let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in |
|
638 |
(case find_first (prp t) prems of |
|
639 |
NONE => NONE |
|
640 |
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))) |
|
641 |
end |
|
642 |
| SOME thm => SOME (mk_meta_eq (thm RS @{thm linorder_class.antisym_conv2}))) |
|
643 |
end handle THM _ => NONE) |
|
644 |
| _ => NONE); |
|
21083 | 645 |
|
56509 | 646 |
end; |
60758 | 647 |
\<close> |
15524 | 648 |
|
56509 | 649 |
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc" |
650 |
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc" |
|
651 |
||
15524 | 652 |
|
60758 | 653 |
subsection \<open>Bounded quantifiers\<close> |
21083 | 654 |
|
655 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
656 |
"_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
|
657 |
"_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
|
658 |
"_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
|
659 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 660 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
661 |
"_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
|
662 |
"_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
|
663 |
"_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
|
664 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 665 |
|
666 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
667 |
"_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
|
668 |
"_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
|
669 |
"_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
|
670 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 671 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
672 |
"_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
|
673 |
"_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
|
674 |
"_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
|
675 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 676 |
|
677 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
678 |
"_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
|
679 |
"_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
|
680 |
"_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
|
681 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 682 |
|
683 |
translations |
|
684 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
685 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
686 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
687 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
688 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
689 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
690 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
691 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
692 |
||
60758 | 693 |
print_translation \<open> |
21083 | 694 |
let |
42287
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset
|
695 |
val All_binder = Mixfix.binder_name @{const_syntax All}; |
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset
|
696 |
val Ex_binder = Mixfix.binder_name @{const_syntax Ex}; |
38786
e46e7a9cb622
formerly unnamed infix impliciation now named HOL.implies
haftmann
parents:
38715
diff
changeset
|
697 |
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
|
698 |
val conj = @{const_syntax HOL.conj}; |
22916 | 699 |
val less = @{const_syntax less}; |
700 |
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
|
701 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
702 |
val trans = |
35115 | 703 |
[((All_binder, impl, less), |
704 |
(@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})), |
|
705 |
((All_binder, impl, less_eq), |
|
706 |
(@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})), |
|
707 |
((Ex_binder, conj, less), |
|
708 |
(@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})), |
|
709 |
((Ex_binder, conj, less_eq), |
|
710 |
(@{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
|
711 |
|
35115 | 712 |
fun matches_bound v t = |
713 |
(case t of |
|
35364 | 714 |
Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v' |
35115 | 715 |
| _ => false); |
716 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false); |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
717 |
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
718 |
|
52143 | 719 |
fun tr' q = (q, fn _ => |
720 |
(fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T), |
|
35364 | 721 |
Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
35115 | 722 |
(case AList.lookup (op =) trans (q, c, d) of |
723 |
NONE => raise Match |
|
724 |
| SOME (l, g) => |
|
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
725 |
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P |
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
726 |
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P |
35115 | 727 |
else raise Match) |
52143 | 728 |
| _ => raise Match)); |
21524 | 729 |
in [tr' All_binder, tr' Ex_binder] end |
60758 | 730 |
\<close> |
21083 | 731 |
|
732 |
||
60758 | 733 |
subsection \<open>Transitivity reasoning\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
734 |
|
25193 | 735 |
context ord |
736 |
begin |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
737 |
|
25193 | 738 |
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" |
739 |
by (rule subst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
740 |
|
25193 | 741 |
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
742 |
by (rule ssubst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
743 |
|
25193 | 744 |
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" |
745 |
by (rule subst) |
|
746 |
||
747 |
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" |
|
748 |
by (rule ssubst) |
|
749 |
||
750 |
end |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
751 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
752 |
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
|
753 |
(!!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
|
754 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
755 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
756 |
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
|
757 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
758 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
759 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
760 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
761 |
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
|
762 |
(!!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
|
763 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
764 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
765 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
766 |
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
|
767 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
768 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
769 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
770 |
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
|
771 |
(!!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
|
772 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
773 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
774 |
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
|
775 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
776 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
777 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
778 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
779 |
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
|
780 |
(!!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
|
781 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
782 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
783 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
784 |
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
|
785 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
786 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
787 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
788 |
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
|
789 |
(!!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
|
790 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
791 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
792 |
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
|
793 |
also assume "f b <= c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
794 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
795 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
796 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
797 |
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
|
798 |
(!!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
|
799 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
800 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
801 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
802 |
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
|
803 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
804 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
805 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
806 |
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
|
807 |
(!!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
|
808 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
809 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
810 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
811 |
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
|
812 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
813 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
814 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
815 |
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
|
816 |
(!!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
|
817 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
818 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
819 |
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
|
820 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
821 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
822 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
823 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
824 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
825 |
(!!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
|
826 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
827 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
828 |
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
|
829 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
830 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
831 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
832 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
833 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
834 |
(!!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
|
835 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
836 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
837 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
838 |
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
|
839 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
840 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
841 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
842 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
843 |
(!!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
|
844 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
845 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
846 |
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
|
847 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
848 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
849 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
850 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
851 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
852 |
(!!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
|
853 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
854 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
855 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
856 |
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
|
857 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
858 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
859 |
|
60758 | 860 |
text \<open> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
Note that this list of rules is in reverse order of priorities. |
60758 | 862 |
\<close> |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
|
27682 | 864 |
lemmas [trans] = |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
866 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
867 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
868 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
869 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
870 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
871 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
872 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
873 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
874 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
875 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
876 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
877 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
878 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
879 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
880 |
mp |
27682 | 881 |
|
882 |
lemmas (in order) [trans] = |
|
883 |
neq_le_trans |
|
884 |
le_neq_trans |
|
885 |
||
886 |
lemmas (in preorder) [trans] = |
|
887 |
less_trans |
|
888 |
less_asym' |
|
889 |
le_less_trans |
|
890 |
less_le_trans |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
891 |
order_trans |
27682 | 892 |
|
893 |
lemmas (in order) [trans] = |
|
894 |
antisym |
|
895 |
||
896 |
lemmas (in ord) [trans] = |
|
897 |
ord_le_eq_trans |
|
898 |
ord_eq_le_trans |
|
899 |
ord_less_eq_trans |
|
900 |
ord_eq_less_trans |
|
901 |
||
902 |
lemmas [trans] = |
|
903 |
trans |
|
904 |
||
905 |
lemmas order_trans_rules = |
|
906 |
order_less_subst2 |
|
907 |
order_less_subst1 |
|
908 |
order_le_less_subst2 |
|
909 |
order_le_less_subst1 |
|
910 |
order_less_le_subst2 |
|
911 |
order_less_le_subst1 |
|
912 |
order_subst2 |
|
913 |
order_subst1 |
|
914 |
ord_le_eq_subst |
|
915 |
ord_eq_le_subst |
|
916 |
ord_less_eq_subst |
|
917 |
ord_eq_less_subst |
|
918 |
forw_subst |
|
919 |
back_subst |
|
920 |
rev_mp |
|
921 |
mp |
|
922 |
neq_le_trans |
|
923 |
le_neq_trans |
|
924 |
less_trans |
|
925 |
less_asym' |
|
926 |
le_less_trans |
|
927 |
less_le_trans |
|
928 |
order_trans |
|
929 |
antisym |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
930 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
931 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
932 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
933 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
934 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
935 |
|
60758 | 936 |
text \<open>These support proving chains of decreasing inequalities |
937 |
a >= b >= c ... in Isar proofs.\<close> |
|
21083 | 938 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
939 |
lemma xt1 [no_atp]: |
21083 | 940 |
"a = b ==> b > c ==> a > c" |
941 |
"a > b ==> b = c ==> a > c" |
|
942 |
"a = b ==> b >= c ==> a >= c" |
|
943 |
"a >= b ==> b = c ==> a >= c" |
|
944 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
945 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
946 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
947 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
23417 | 948 |
"(a::'a::order) > b ==> b > a ==> P" |
21083 | 949 |
"(x::'a::order) > y ==> y > z ==> x > z" |
950 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
951 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
952 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
21083 | 953 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
954 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
955 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
25076 | 956 |
by auto |
21083 | 957 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
958 |
lemma xt2 [no_atp]: |
21083 | 959 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
960 |
by (subgoal_tac "f b >= f c", force, force) |
|
961 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
962 |
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
21083 | 963 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
964 |
by (subgoal_tac "f a >= f b", force, force) |
|
965 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
966 |
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
21083 | 967 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
968 |
by (subgoal_tac "f b >= f c", force, force) |
|
969 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
970 |
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
21083 | 971 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
972 |
by (subgoal_tac "f a > f b", force, force) |
|
973 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
974 |
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==> |
21083 | 975 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
976 |
by (subgoal_tac "f b > f c", force, force) |
|
977 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
978 |
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
21083 | 979 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
980 |
by (subgoal_tac "f a >= f b", force, force) |
|
981 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
982 |
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
21083 | 983 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
984 |
by (subgoal_tac "f b > f c", force, force) |
|
985 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
986 |
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
21083 | 987 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
988 |
by (subgoal_tac "f a > f b", force, force) |
|
989 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53216
diff
changeset
|
990 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
21083 | 991 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
992 |
(* |
21083 | 993 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
994 |
for the wrong thing in an Isar proof. |
|
995 |
||
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
996 |
The extra transitivity rules can be used as follows: |
21083 | 997 |
|
998 |
lemma "(a::'a::order) > z" |
|
999 |
proof - |
|
1000 |
have "a >= b" (is "_ >= ?rhs") |
|
1001 |
sorry |
|
1002 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
1003 |
sorry |
|
1004 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
1005 |
sorry |
|
1006 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
1007 |
sorry |
|
1008 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
1009 |
sorry |
|
1010 |
also (xtrans) have "?rhs > z" |
|
1011 |
sorry |
|
1012 |
finally (xtrans) show ?thesis . |
|
1013 |
qed |
|
1014 |
||
1015 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
1016 |
leave out the "(xtrans)" above. |
|
1017 |
*) |
|
1018 |
||
23881 | 1019 |
|
60758 | 1020 |
subsection \<open>Monotonicity\<close> |
21083 | 1021 |
|
25076 | 1022 |
context order |
1023 |
begin |
|
1024 |
||
61076 | 1025 |
definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
25076 | 1026 |
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)" |
1027 |
||
1028 |
lemma monoI [intro?]: |
|
61076 | 1029 |
fixes f :: "'a \<Rightarrow> 'b::order" |
25076 | 1030 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f" |
1031 |
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
|
1032 |
|
25076 | 1033 |
lemma monoD [dest?]: |
61076 | 1034 |
fixes f :: "'a \<Rightarrow> 'b::order" |
25076 | 1035 |
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
1036 |
unfolding mono_def by iprover |
|
1037 |
||
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1038 |
lemma monoE: |
61076 | 1039 |
fixes f :: "'a \<Rightarrow> 'b::order" |
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1040 |
assumes "mono f" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1041 |
assumes "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1042 |
obtains "f x \<le> f y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1043 |
proof |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1044 |
from assms show "f x \<le> f y" by (simp add: mono_def) |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1045 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1046 |
|
61076 | 1047 |
definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1048 |
"antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1049 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1050 |
lemma antimonoI [intro?]: |
61076 | 1051 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1052 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1053 |
unfolding antimono_def by iprover |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1054 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1055 |
lemma antimonoD [dest?]: |
61076 | 1056 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1057 |
shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1058 |
unfolding antimono_def by iprover |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1059 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1060 |
lemma antimonoE: |
61076 | 1061 |
fixes f :: "'a \<Rightarrow> 'b::order" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1062 |
assumes "antimono f" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1063 |
assumes "x \<le> y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1064 |
obtains "f x \<ge> f y" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1065 |
proof |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1066 |
from assms show "f x \<ge> f y" by (simp add: antimono_def) |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1067 |
qed |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
54868
diff
changeset
|
1068 |
|
61076 | 1069 |
definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where |
30298 | 1070 |
"strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)" |
1071 |
||
1072 |
lemma strict_monoI [intro?]: |
|
1073 |
assumes "\<And>x y. x < y \<Longrightarrow> f x < f y" |
|
1074 |
shows "strict_mono f" |
|
1075 |
using assms unfolding strict_mono_def by auto |
|
1076 |
||
1077 |
lemma strict_monoD [dest?]: |
|
1078 |
"strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y" |
|
1079 |
unfolding strict_mono_def by auto |
|
1080 |
||
1081 |
lemma strict_mono_mono [dest?]: |
|
1082 |
assumes "strict_mono f" |
|
1083 |
shows "mono f" |
|
1084 |
proof (rule monoI) |
|
1085 |
fix x y |
|
1086 |
assume "x \<le> y" |
|
1087 |
show "f x \<le> f y" |
|
1088 |
proof (cases "x = y") |
|
1089 |
case True then show ?thesis by simp |
|
1090 |
next |
|
60758 | 1091 |
case False with \<open>x \<le> y\<close> have "x < y" by simp |
30298 | 1092 |
with assms strict_monoD have "f x < f y" by auto |
1093 |
then show ?thesis by simp |
|
1094 |
qed |
|
1095 |
qed |
|
1096 |
||
25076 | 1097 |
end |
1098 |
||
1099 |
context linorder |
|
1100 |
begin |
|
1101 |
||
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1102 |
lemma mono_invE: |
61076 | 1103 |
fixes f :: "'a \<Rightarrow> 'b::order" |
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1104 |
assumes "mono f" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1105 |
assumes "f x < f y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1106 |
obtains "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1107 |
proof |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1108 |
show "x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1109 |
proof (rule ccontr) |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1110 |
assume "\<not> x \<le> y" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1111 |
then have "y \<le> x" by simp |
60758 | 1112 |
with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE) |
1113 |
with \<open>f x < f y\<close> show False by simp |
|
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1114 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1115 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
49769
diff
changeset
|
1116 |
|
30298 | 1117 |
lemma strict_mono_eq: |
1118 |
assumes "strict_mono f" |
|
1119 |
shows "f x = f y \<longleftrightarrow> x = y" |
|
1120 |
proof |
|
1121 |
assume "f x = f y" |
|
1122 |
show "x = y" proof (cases x y rule: linorder_cases) |
|
1123 |
case less with assms strict_monoD have "f x < f y" by auto |
|
60758 | 1124 |
with \<open>f x = f y\<close> show ?thesis by simp |
30298 | 1125 |
next |
1126 |
case equal then show ?thesis . |
|
1127 |
next |
|
1128 |
case greater with assms strict_monoD have "f y < f x" by auto |
|
60758 | 1129 |
with \<open>f x = f y\<close> show ?thesis by simp |
30298 | 1130 |
qed |
1131 |
qed simp |
|
1132 |
||
1133 |
lemma strict_mono_less_eq: |
|
1134 |
assumes "strict_mono f" |
|
1135 |
shows "f x \<le> f y \<longleftrightarrow> x \<le> y" |
|
1136 |
proof |
|
1137 |
assume "x \<le> y" |
|
1138 |
with assms strict_mono_mono monoD show "f x \<le> f y" by auto |
|
1139 |
next |
|
1140 |
assume "f x \<le> f y" |
|
1141 |
show "x \<le> y" proof (rule ccontr) |
|
1142 |
assume "\<not> x \<le> y" then have "y < x" by simp |
|
1143 |
with assms strict_monoD have "f y < f x" by auto |
|
60758 | 1144 |
with \<open>f x \<le> f y\<close> show False by simp |
30298 | 1145 |
qed |
1146 |
qed |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1147 |
|
30298 | 1148 |
lemma strict_mono_less: |
1149 |
assumes "strict_mono f" |
|
1150 |
shows "f x < f y \<longleftrightarrow> x < y" |
|
1151 |
using assms |
|
1152 |
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq) |
|
1153 |
||
54860 | 1154 |
end |
1155 |
||
1156 |
||
60758 | 1157 |
subsection \<open>min and max -- fundamental\<close> |
54860 | 1158 |
|
1159 |
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1160 |
"min a b = (if a \<le> b then a else b)" |
|
1161 |
||
1162 |
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
|
1163 |
"max a b = (if a \<le> b then b else a)" |
|
1164 |
||
45931 | 1165 |
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1166 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1167 |
|
54857 | 1168 |
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1169 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1170 |
|
61076 | 1171 |
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1172 |
by (simp add:min_def) |
45893 | 1173 |
|
61076 | 1174 |
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x" |
54861
00d551179872
postponed min/max lemmas until abstract lattice is available
haftmann
parents:
54860
diff
changeset
|
1175 |
by (simp add: max_def) |
45893 | 1176 |
|
61630 | 1177 |
lemma max_min_same [simp]: |
1178 |
fixes x y :: "'a :: linorder" |
|
1179 |
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y" |
|
1180 |
by(auto simp add: max_def min_def) |
|
45893 | 1181 |
|
60758 | 1182 |
subsection \<open>(Unique) top and bottom elements\<close> |
28685 | 1183 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1184 |
class bot = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1185 |
fixes bot :: 'a ("\<bottom>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1186 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1187 |
class order_bot = order + bot + |
51487 | 1188 |
assumes bot_least: "\<bottom> \<le> a" |
54868 | 1189 |
begin |
51487 | 1190 |
|
61605 | 1191 |
sublocale bot: ordering_top greater_eq greater bot |
61169 | 1192 |
by standard (fact bot_least) |
51487 | 1193 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1194 |
lemma le_bot: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1195 |
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>" |
51487 | 1196 |
by (fact bot.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1197 |
|
43816 | 1198 |
lemma bot_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1199 |
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>" |
51487 | 1200 |
by (fact bot.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1201 |
|
51487 | 1202 |
lemma not_less_bot: |
1203 |
"\<not> a < \<bottom>" |
|
1204 |
by (fact bot.extremum_strict) |
|
43816 | 1205 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1206 |
lemma bot_less: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1207 |
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a" |
51487 | 1208 |
by (fact bot.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1209 |
|
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1210 |
end |
41082 | 1211 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1212 |
class top = |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1213 |
fixes top :: 'a ("\<top>") |
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1214 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1215 |
class order_top = order + top + |
51487 | 1216 |
assumes top_greatest: "a \<le> \<top>" |
54868 | 1217 |
begin |
51487 | 1218 |
|
61605 | 1219 |
sublocale top: ordering_top less_eq less top |
61169 | 1220 |
by standard (fact top_greatest) |
51487 | 1221 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1222 |
lemma top_le: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1223 |
"\<top> \<le> a \<Longrightarrow> a = \<top>" |
51487 | 1224 |
by (fact top.extremum_uniqueI) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1225 |
|
43816 | 1226 |
lemma top_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1227 |
"\<top> \<le> a \<longleftrightarrow> a = \<top>" |
51487 | 1228 |
by (fact top.extremum_unique) |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1229 |
|
51487 | 1230 |
lemma not_top_less: |
1231 |
"\<not> \<top> < a" |
|
1232 |
by (fact top.extremum_strict) |
|
43816 | 1233 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1234 |
lemma less_top: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1235 |
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>" |
51487 | 1236 |
by (fact top.not_eq_extremum) |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1237 |
|
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1238 |
end |
28685 | 1239 |
|
1240 |
||
60758 | 1241 |
subsection \<open>Dense orders\<close> |
27823 | 1242 |
|
53216 | 1243 |
class dense_order = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1244 |
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1245 |
|
53216 | 1246 |
class dense_linorder = linorder + dense_order |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1247 |
begin |
27823 | 1248 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1249 |
lemma dense_le: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1250 |
fixes y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1251 |
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
|
1252 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1253 |
proof (rule ccontr) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1254 |
assume "\<not> ?thesis" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1255 |
hence "z < y" by simp |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1256 |
from dense[OF this] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1257 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1258 |
moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] . |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1259 |
ultimately show False by auto |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1260 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1261 |
|
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1262 |
lemma dense_le_bounded: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1263 |
fixes x y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1264 |
assumes "x < y" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1265 |
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
|
1266 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1267 |
proof (rule dense_le) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1268 |
fix w assume "w < y" |
60758 | 1269 |
from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1270 |
from linear[of u w] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1271 |
show "w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1272 |
proof (rule disjE) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1273 |
assume "u \<le> w" |
60758 | 1274 |
from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close> |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1275 |
show "w \<le> z" by (rule *) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1276 |
next |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1277 |
assume "w \<le> u" |
60758 | 1278 |
from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>] |
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1279 |
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
|
1280 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1281 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1282 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1283 |
lemma dense_ge: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1284 |
fixes y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1285 |
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1286 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1287 |
proof (rule ccontr) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1288 |
assume "\<not> ?thesis" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1289 |
hence "z < y" by simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1290 |
from dense[OF this] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1291 |
obtain x where "x < y" and "z < x" by safe |
60758 | 1292 |
moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] . |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1293 |
ultimately show False by auto |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1294 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1295 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1296 |
lemma dense_ge_bounded: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1297 |
fixes x y z :: 'a |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1298 |
assumes "z < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1299 |
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1300 |
shows "y \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1301 |
proof (rule dense_ge) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1302 |
fix w assume "z < w" |
60758 | 1303 |
from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1304 |
from linear[of u w] |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1305 |
show "y \<le> w" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1306 |
proof (rule disjE) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1307 |
assume "w \<le> u" |
60758 | 1308 |
from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>] |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1309 |
show "y \<le> w" by (rule *) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1310 |
next |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1311 |
assume "u \<le> w" |
60758 | 1312 |
from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close> |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1313 |
show "y \<le> w" by (rule order_trans) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1314 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1315 |
qed |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1316 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1317 |
end |
27823 | 1318 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1319 |
class no_top = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1320 |
assumes gt_ex: "\<exists>y. x < y" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1321 |
|
61824
dcbe9f756ae0
not_leE -> not_le_imp_less and other tidying
paulson <lp15@cam.ac.uk>
parents:
61799
diff
changeset
|
1322 |
class no_bot = order + |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1323 |
assumes lt_ex: "\<exists>y. y < x" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1324 |
|
53216 | 1325 |
class unbounded_dense_linorder = dense_linorder + no_top + no_bot |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
1326 |
|
51546
2e26df807dc7
more uniform style for interpretation and sublocale declarations
haftmann
parents:
51487
diff
changeset
|
1327 |
|
60758 | 1328 |
subsection \<open>Wellorders\<close> |
27823 | 1329 |
|
1330 |
class wellorder = linorder + |
|
1331 |
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" |
|
1332 |
begin |
|
1333 |
||
1334 |
lemma wellorder_Least_lemma: |
|
1335 |
fixes k :: 'a |
|
1336 |
assumes "P k" |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1337 |
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k" |
27823 | 1338 |
proof - |
1339 |
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" |
|
1340 |
using assms proof (induct k rule: less_induct) |
|
1341 |
case (less x) then have "P x" by simp |
|
1342 |
show ?case proof (rule classical) |
|
1343 |
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" |
|
1344 |
have "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
1345 |
proof (rule classical) |
|
1346 |
fix y |
|
38705 | 1347 |
assume "P y" and "\<not> x \<le> y" |
27823 | 1348 |
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
1349 |
by (auto simp add: not_le) |
|
1350 |
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1351 |
by auto |
|
1352 |
then show "x \<le> y" by auto |
|
1353 |
qed |
|
60758 | 1354 |
with \<open>P x\<close> have Least: "(LEAST a. P a) = x" |
27823 | 1355 |
by (rule Least_equality) |
60758 | 1356 |
with \<open>P x\<close> show ?thesis by simp |
27823 | 1357 |
qed |
1358 |
qed |
|
1359 |
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto |
|
1360 |
qed |
|
1361 |
||
61799 | 1362 |
\<comment> "The following 3 lemmas are due to Brian Huffman" |
27823 | 1363 |
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)" |
1364 |
by (erule exE) (erule LeastI) |
|
1365 |
||
1366 |
lemma LeastI2: |
|
1367 |
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1368 |
by (blast intro: LeastI) |
|
1369 |
||
1370 |
lemma LeastI2_ex: |
|
1371 |
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1372 |
by (blast intro: LeastI_ex) |
|
1373 |
||
38705 | 1374 |
lemma LeastI2_wellorder: |
1375 |
assumes "P a" |
|
1376 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
|
1377 |
shows "Q (Least P)" |
|
1378 |
proof (rule LeastI2_order) |
|
60758 | 1379 |
show "P (Least P)" using \<open>P a\<close> by (rule LeastI) |
38705 | 1380 |
next |
1381 |
fix y assume "P y" thus "Least P \<le> y" by (rule Least_le) |
|
1382 |
next |
|
1383 |
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2)) |
|
1384 |
qed |
|
1385 |
||
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1386 |
lemma LeastI2_wellorder_ex: |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1387 |
assumes "\<exists>x. P x" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1388 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1389 |
shows "Q (Least P)" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1390 |
using assms by clarify (blast intro!: LeastI2_wellorder) |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1391 |
|
27823 | 1392 |
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k" |
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61630
diff
changeset
|
1393 |
apply (simp add: not_le [symmetric]) |
27823 | 1394 |
apply (erule contrapos_nn) |
1395 |
apply (erule Least_le) |
|
1396 |
done |
|
1397 |
||
38705 | 1398 |
end |
27823 | 1399 |
|
28685 | 1400 |
|
60758 | 1401 |
subsection \<open>Order on @{typ bool}\<close> |
28685 | 1402 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1403 |
instantiation bool :: "{order_bot, order_top, linorder}" |
28685 | 1404 |
begin |
1405 |
||
1406 |
definition |
|
41080 | 1407 |
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q" |
28685 | 1408 |
|
1409 |
definition |
|
61076 | 1410 |
[simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q" |
28685 | 1411 |
|
1412 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1413 |
[simp]: "\<bottom> \<longleftrightarrow> False" |
28685 | 1414 |
|
1415 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1416 |
[simp]: "\<top> \<longleftrightarrow> True" |
28685 | 1417 |
|
1418 |
instance proof |
|
41080 | 1419 |
qed auto |
28685 | 1420 |
|
15524 | 1421 |
end |
28685 | 1422 |
|
1423 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
41080 | 1424 |
by simp |
28685 | 1425 |
|
1426 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
41080 | 1427 |
by simp |
28685 | 1428 |
|
1429 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
41080 | 1430 |
by simp |
28685 | 1431 |
|
1432 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
41080 | 1433 |
by simp |
32899 | 1434 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1435 |
lemma bot_boolE: "\<bottom> \<Longrightarrow> P" |
41080 | 1436 |
by simp |
32899 | 1437 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1438 |
lemma top_boolI: \<top> |
41080 | 1439 |
by simp |
28685 | 1440 |
|
1441 |
lemma [code]: |
|
1442 |
"False \<le> b \<longleftrightarrow> True" |
|
1443 |
"True \<le> b \<longleftrightarrow> b" |
|
1444 |
"False < b \<longleftrightarrow> b" |
|
1445 |
"True < b \<longleftrightarrow> False" |
|
41080 | 1446 |
by simp_all |
28685 | 1447 |
|
1448 |
||
60758 | 1449 |
subsection \<open>Order on @{typ "_ \<Rightarrow> _"}\<close> |
28685 | 1450 |
|
1451 |
instantiation "fun" :: (type, ord) ord |
|
1452 |
begin |
|
1453 |
||
1454 |
definition |
|
37767 | 1455 |
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)" |
28685 | 1456 |
|
1457 |
definition |
|
61076 | 1458 |
"(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)" |
28685 | 1459 |
|
1460 |
instance .. |
|
1461 |
||
1462 |
end |
|
1463 |
||
1464 |
instance "fun" :: (type, preorder) preorder proof |
|
1465 |
qed (auto simp add: le_fun_def less_fun_def |
|
44921 | 1466 |
intro: order_trans antisym) |
28685 | 1467 |
|
1468 |
instance "fun" :: (type, order) order proof |
|
44921 | 1469 |
qed (auto simp add: le_fun_def intro: antisym) |
28685 | 1470 |
|
41082 | 1471 |
instantiation "fun" :: (type, bot) bot |
1472 |
begin |
|
1473 |
||
1474 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1475 |
"\<bottom> = (\<lambda>x. \<bottom>)" |
41082 | 1476 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1477 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1478 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1479 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1480 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1481 |
instantiation "fun" :: (type, order_bot) order_bot |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1482 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1483 |
|
49769 | 1484 |
lemma bot_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1485 |
"\<bottom> x = \<bottom>" |
41082 | 1486 |
by (simp add: bot_fun_def) |
1487 |
||
1488 |
instance proof |
|
46884 | 1489 |
qed (simp add: le_fun_def) |
41082 | 1490 |
|
1491 |
end |
|
1492 |
||
28685 | 1493 |
instantiation "fun" :: (type, top) top |
1494 |
begin |
|
1495 |
||
1496 |
definition |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1497 |
[no_atp]: "\<top> = (\<lambda>x. \<top>)" |
28685 | 1498 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1499 |
instance .. |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1500 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1501 |
end |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1502 |
|
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1503 |
instantiation "fun" :: (type, order_top) order_top |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1504 |
begin |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52143
diff
changeset
|
1505 |
|
49769 | 1506 |
lemma top_apply [simp, code]: |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1507 |
"\<top> x = \<top>" |
41080 | 1508 |
by (simp add: top_fun_def) |
1509 |
||
28685 | 1510 |
instance proof |
46884 | 1511 |
qed (simp add: le_fun_def) |
28685 | 1512 |
|
1513 |
end |
|
1514 |
||
1515 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
1516 |
unfolding le_fun_def by simp |
|
1517 |
||
1518 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
1519 |
unfolding le_fun_def by simp |
|
1520 |
||
1521 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
54860 | 1522 |
by (rule le_funE) |
28685 | 1523 |
|
59000 | 1524 |
lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))" |
1525 |
unfolding mono_def le_fun_def by auto |
|
1526 |
||
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1527 |
|
60758 | 1528 |
subsection \<open>Order on unary and binary predicates\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1529 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1530 |
lemma predicate1I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1531 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1532 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1533 |
apply (rule le_funI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1534 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1535 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1536 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1537 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1538 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1539 |
lemma predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1540 |
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1541 |
apply (erule le_funE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1542 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1543 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1544 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1545 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1546 |
lemma rev_predicate1D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1547 |
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1548 |
by (rule predicate1D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1549 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1550 |
lemma predicate2I: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1551 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1552 |
shows "P \<le> Q" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1553 |
apply (rule le_funI)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1554 |
apply (rule le_boolI) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1555 |
apply (rule PQ) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1556 |
apply assumption |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1557 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1558 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1559 |
lemma predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1560 |
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1561 |
apply (erule le_funE)+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1562 |
apply (erule le_boolE) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1563 |
apply assumption+ |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1564 |
done |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1565 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1566 |
lemma rev_predicate2D: |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1567 |
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1568 |
by (rule predicate2D) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1569 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1570 |
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1571 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1572 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1573 |
lemma bot2E: "\<bottom> x y \<Longrightarrow> P" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1574 |
by (simp add: bot_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1575 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1576 |
lemma top1I: "\<top> x" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1577 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1578 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1579 |
lemma top2I: "\<top> x y" |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1580 |
by (simp add: top_fun_def) |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1581 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1582 |
|
60758 | 1583 |
subsection \<open>Name duplicates\<close> |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1584 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1585 |
lemmas order_eq_refl = preorder_class.eq_refl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1586 |
lemmas order_less_irrefl = preorder_class.less_irrefl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1587 |
lemmas order_less_imp_le = preorder_class.less_imp_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1588 |
lemmas order_less_not_sym = preorder_class.less_not_sym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1589 |
lemmas order_less_asym = preorder_class.less_asym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1590 |
lemmas order_less_trans = preorder_class.less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1591 |
lemmas order_le_less_trans = preorder_class.le_less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1592 |
lemmas order_less_le_trans = preorder_class.less_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1593 |
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1594 |
lemmas order_less_imp_triv = preorder_class.less_imp_triv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1595 |
lemmas order_less_asym' = preorder_class.less_asym' |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1596 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1597 |
lemmas order_less_le = order_class.less_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1598 |
lemmas order_le_less = order_class.le_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1599 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1600 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1601 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1602 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1603 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1604 |
lemmas order_antisym = order_class.antisym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1605 |
lemmas order_eq_iff = order_class.eq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1606 |
lemmas order_antisym_conv = order_class.antisym_conv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1607 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1608 |
lemmas linorder_linear = linorder_class.linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1609 |
lemmas linorder_less_linear = linorder_class.less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1610 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1611 |
lemmas linorder_le_cases = linorder_class.le_cases |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1612 |
lemmas linorder_not_less = linorder_class.not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1613 |
lemmas linorder_not_le = linorder_class.not_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1614 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1615 |
lemmas linorder_neqE = linorder_class.neqE |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1616 |
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1617 |
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1618 |
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1619 |
|
28685 | 1620 |
end |