author | hoelzl |
Mon, 03 Dec 2012 18:19:08 +0100 | |
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parent 49769 | c7c2152322f2 |
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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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header {* Abstract orderings *} |
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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 {* Syntactic orders *} |
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class ord = |
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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begin |
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notation |
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less_eq ("op <=") and |
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less_eq ("(_/ <= _)" [51, 51] 50) and |
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less ("op <") and |
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less ("(_/ < _)" [51, 51] 50) |
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notation (xsymbols) |
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less_eq ("op \<le>") and |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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notation (HTML output) |
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less_eq ("op \<le>") and |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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abbreviation (input) |
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greater_eq (infix ">=" 50) where |
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"x >= y \<equiv> y <= x" |
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notation (input) |
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greater_eq (infix "\<ge>" 50) |
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abbreviation (input) |
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greater (infix ">" 50) where |
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"x > y \<equiv> y < x" |
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end |
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subsection {* Quasi orders *} |
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class preorder = ord + |
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assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)" |
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and order_refl [iff]: "x \<le> x" |
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and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
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begin |
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text {* Reflexivity. *} |
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lemma eq_refl: "x = y \<Longrightarrow> x \<le> y" |
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-- {* This form is useful with the classical reasoner. *} |
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by (erule ssubst) (rule order_refl) |
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lemma less_irrefl [iff]: "\<not> x < x" |
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by (simp add: less_le_not_le) |
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lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y" |
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unfolding less_le_not_le by blast |
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text {* Asymmetry. *} |
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lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)" |
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by (simp add: less_le_not_le) |
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lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P" |
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by (drule less_not_sym, erule contrapos_np) simp |
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text {* Transitivity. *} |
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lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z" |
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by (auto simp add: less_le_not_le intro: order_trans) |
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text {* Useful for simplification, but too risky to include by default. *} |
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lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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text {* Transitivity rules for calculational reasoning *} |
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lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P" |
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by (rule less_asym) |
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text {* Dual order *} |
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lemma dual_preorder: |
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"class.preorder (op \<ge>) (op >)" |
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proof qed (auto simp add: less_le_not_le intro: order_trans) |
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end |
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subsection {* Partial orders *} |
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class order = preorder + |
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assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
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begin |
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text {* Reflexivity. *} |
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lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" |
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by (auto simp add: less_le_not_le intro: antisym) |
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lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y" |
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-- {* NOT suitable for iff, since it can cause PROOF FAILED. *} |
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by (simp add: less_le) blast |
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lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y" |
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unfolding less_le by blast |
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text {* Useful for simplification, but too risky to include by default. *} |
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lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
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by auto |
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lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False" |
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by auto |
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text {* Transitivity rules for calculational reasoning *} |
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b" |
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by (simp add: less_le) |
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lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b" |
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by (simp add: less_le) |
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text {* Asymmetry. *} |
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lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x" |
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by (blast intro: antisym) |
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lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym) |
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lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y" |
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by (erule contrapos_pn, erule subst, rule less_irrefl) |
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text {* Least value operator *} |
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definition (in ord) |
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Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where |
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"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))" |
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lemma Least_equality: |
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assumes "P x" |
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and "\<And>y. P y \<Longrightarrow> x \<le> y" |
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shows "Least P = x" |
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unfolding Least_def by (rule the_equality) |
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(blast intro: assms antisym)+ |
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lemma LeastI2_order: |
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assumes "P x" |
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and "\<And>y. P y \<Longrightarrow> x \<le> y" |
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and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x" |
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shows "Q (Least P)" |
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unfolding Least_def by (rule theI2) |
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(blast intro: assms antisym)+ |
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text {* Dual order *} |
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lemma dual_order: |
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"class.order (op \<ge>) (op >)" |
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by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym) |
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end |
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subsection {* Linear (total) orders *} |
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class linorder = order + |
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assumes linear: "x \<le> y \<or> y \<le> x" |
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begin |
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lemma less_linear: "x < y \<or> x = y \<or> y < x" |
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unfolding less_le using less_le linear by blast |
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lemma le_less_linear: "x \<le> y \<or> y < x" |
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by (simp add: le_less less_linear) |
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lemma le_cases [case_names le ge]: |
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"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P" |
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using linear by blast |
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lemma linorder_cases [case_names less equal greater]: |
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"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P" |
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using less_linear by blast |
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lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x" |
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apply (simp add: less_le) |
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using linear apply (blast intro: antisym) |
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done |
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lemma not_less_iff_gr_or_eq: |
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"\<not>(x < y) \<longleftrightarrow> (x > y | x = y)" |
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apply(simp add:not_less le_less) |
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apply blast |
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done |
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lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x" |
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apply (simp add: less_le) |
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using linear apply (blast intro: antisym) |
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done |
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x" |
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by (cut_tac x = x and y = y in less_linear, auto) |
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R" |
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by (simp add: neq_iff) blast |
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lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym dest: not_less [THEN iffD1]) |
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lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
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by (blast intro: antisym dest: not_less [THEN iffD1]) |
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lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y" |
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by (blast intro: antisym dest: not_less [THEN iffD1]) |
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lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x" |
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unfolding not_less . |
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lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y" |
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unfolding not_less . |
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(*FIXME inappropriate name (or delete altogether)*) |
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lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y" |
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unfolding not_le . |
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text {* Dual order *} |
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lemma dual_linorder: |
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"class.linorder (op \<ge>) (op >)" |
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by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear) |
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text {* min/max *} |
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definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"min a b = (if a \<le> b then a else b)" |
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definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"max a b = (if a \<le> b then b else a)" |
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lemma min_le_iff_disj: |
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"min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z" |
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unfolding min_def using linear by (auto intro: order_trans) |
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lemma le_max_iff_disj: |
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"z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y" |
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unfolding max_def using linear by (auto intro: order_trans) |
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lemma min_less_iff_disj: |
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"min x y < z \<longleftrightarrow> x < z \<or> y < z" |
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unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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lemma less_max_iff_disj: |
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"z < max x y \<longleftrightarrow> z < x \<or> z < y" |
23212 | 287 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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288 |
|
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289 |
lemma min_less_iff_conj [simp]: |
25062 | 290 |
"z < min x y \<longleftrightarrow> z < x \<and> z < y" |
23212 | 291 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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|
292 |
|
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293 |
lemma max_less_iff_conj [simp]: |
25062 | 294 |
"max x y < z \<longleftrightarrow> x < z \<and> y < z" |
23212 | 295 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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|
296 |
|
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297 |
lemma split_min [no_atp]: |
25062 | 298 |
"P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)" |
23212 | 299 |
by (simp add: min_def) |
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300 |
|
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|
301 |
lemma split_max [no_atp]: |
25062 | 302 |
"P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)" |
23212 | 303 |
by (simp add: max_def) |
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304 |
|
21248 | 305 |
end |
306 |
||
23948 | 307 |
|
21083 | 308 |
subsection {* Reasoning tools setup *} |
309 |
||
21091 | 310 |
ML {* |
311 |
||
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312 |
signature ORDERS = |
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313 |
sig |
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314 |
val print_structures: Proof.context -> unit |
47432 | 315 |
val attrib_setup: theory -> theory |
32215 | 316 |
val order_tac: Proof.context -> thm list -> int -> tactic |
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317 |
end; |
21091 | 318 |
|
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319 |
structure Orders: ORDERS = |
21248 | 320 |
struct |
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|
321 |
|
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322 |
(** Theory and context data **) |
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323 |
|
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324 |
fun struct_eq ((s1: string, ts1), (s2, ts2)) = |
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325 |
(s1 = s2) andalso eq_list (op aconv) (ts1, ts2); |
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326 |
|
33519 | 327 |
structure Data = Generic_Data |
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328 |
( |
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329 |
type T = ((string * term list) * Order_Tac.less_arith) list; |
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330 |
(* Order structures: |
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331 |
identifier of the structure, list of operations and record of theorems |
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332 |
needed to set up the transitivity reasoner, |
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|
333 |
identifier and operations identify the structure uniquely. *) |
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334 |
val empty = []; |
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335 |
val extend = I; |
33519 | 336 |
fun merge data = AList.join struct_eq (K fst) data; |
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337 |
); |
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|
338 |
|
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339 |
fun print_structures ctxt = |
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340 |
let |
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341 |
val structs = Data.get (Context.Proof ctxt); |
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|
342 |
fun pretty_term t = Pretty.block |
24920 | 343 |
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1, |
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344 |
Pretty.str "::", Pretty.brk 1, |
24920 | 345 |
Pretty.quote (Syntax.pretty_typ ctxt (type_of t))]; |
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346 |
fun pretty_struct ((s, ts), _) = Pretty.block |
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347 |
[Pretty.str s, Pretty.str ":", Pretty.brk 1, |
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348 |
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; |
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349 |
in |
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350 |
Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs)) |
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351 |
end; |
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|
352 |
|
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|
353 |
|
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354 |
(** Method **) |
21091 | 355 |
|
32215 | 356 |
fun struct_tac ((s, [eq, le, less]), thms) ctxt prems = |
24641
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357 |
let |
30107
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|
358 |
fun decomp thy (@{const Trueprop} $ t) = |
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359 |
let |
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360 |
fun excluded t = |
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361 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
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362 |
let val T = type_of t |
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363 |
in |
32960
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|
364 |
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT |
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365 |
end; |
32960
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366 |
fun rel (bin_op $ t1 $ t2) = |
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367 |
if excluded t1 then NONE |
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368 |
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) |
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369 |
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) |
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370 |
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) |
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371 |
else NONE |
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372 |
| rel _ = NONE; |
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|
373 |
fun dec (Const (@{const_name Not}, _) $ t) = (case rel t |
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|
374 |
of NONE => NONE |
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|
375 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
24741
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|
376 |
| dec x = rel x; |
30107
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|
377 |
in dec t end |
f3b3b0e3d184
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|
378 |
| decomp thy _ = NONE; |
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|
379 |
in |
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|
380 |
case s of |
32215 | 381 |
"order" => Order_Tac.partial_tac decomp thms ctxt prems |
382 |
| "linorder" => Order_Tac.linear_tac decomp thms ctxt prems |
|
24641
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383 |
| _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.") |
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|
384 |
end |
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|
385 |
|
32215 | 386 |
fun order_tac ctxt prems = |
387 |
FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt))); |
|
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|
388 |
|
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|
389 |
|
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|
390 |
(** Attribute **) |
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|
391 |
|
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|
392 |
fun add_struct_thm s tag = |
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|
393 |
Thm.declaration_attribute |
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|
394 |
(fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm))); |
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|
395 |
fun del_struct s = |
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|
396 |
Thm.declaration_attribute |
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|
397 |
(fn _ => Data.map (AList.delete struct_eq s)); |
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|
398 |
|
30722
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|
399 |
val attrib_setup = |
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|
400 |
Attrib.setup @{binding order} |
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|
401 |
(Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --| |
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|
402 |
Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name -- |
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|
403 |
Scan.repeat Args.term |
623d4831c8cf
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|
404 |
>> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag |
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changeset
|
405 |
| ((NONE, n), ts) => del_struct (n, ts))) |
623d4831c8cf
simplified attribute and method setup: eliminating bottom-up styles makes it easier to keep things in one place, and also SML/NJ happy;
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|
406 |
"theorems controlling transitivity reasoner"; |
24641
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|
407 |
|
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|
408 |
|
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|
409 |
(** Diagnostic command **) |
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|
410 |
|
24867 | 411 |
val _ = |
46961
5c6955f487e5
outer syntax command definitions based on formal command_spec derived from theory header declarations;
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|
412 |
Outer_Syntax.improper_command @{command_spec "print_orders"} |
5c6955f487e5
outer syntax command definitions based on formal command_spec derived from theory header declarations;
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|
413 |
"print order structures available to transitivity reasoner" |
30806 | 414 |
(Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o |
415 |
Toplevel.keep (print_structures o Toplevel.context_of))); |
|
24641
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416 |
|
21091 | 417 |
end; |
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|
418 |
|
21091 | 419 |
*} |
420 |
||
47432 | 421 |
setup Orders.attrib_setup |
422 |
||
423 |
method_setup order = {* |
|
424 |
Scan.succeed (fn ctxt => SIMPLE_METHOD' (Orders.order_tac ctxt [])) |
|
425 |
*} "transitivity reasoner" |
|
24641
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|
426 |
|
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|
427 |
|
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|
428 |
text {* Declarations to set up transitivity reasoner of partial and linear orders. *} |
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|
429 |
|
25076 | 430 |
context order |
431 |
begin |
|
432 |
||
24641
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|
433 |
(* The type constraint on @{term op =} below is necessary since the operation |
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|
434 |
is not a parameter of the locale. *) |
25076 | 435 |
|
27689 | 436 |
declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] |
437 |
||
438 |
declare order_refl [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
439 |
||
440 |
declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
441 |
||
442 |
declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
443 |
||
444 |
declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
445 |
||
446 |
declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
447 |
||
448 |
declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
449 |
||
450 |
declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
451 |
||
452 |
declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
453 |
||
454 |
declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
455 |
||
456 |
declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
457 |
||
458 |
declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
459 |
||
460 |
declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
461 |
||
462 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
463 |
||
464 |
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
|
465 |
|
25076 | 466 |
end |
467 |
||
468 |
context linorder |
|
469 |
begin |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
470 |
|
27689 | 471 |
declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]] |
472 |
||
473 |
declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
474 |
||
475 |
declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
476 |
||
477 |
declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
478 |
||
479 |
declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
480 |
||
481 |
declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
482 |
||
483 |
declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
484 |
||
485 |
declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
486 |
||
487 |
declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
488 |
||
489 |
declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
25076 | 490 |
|
27689 | 491 |
declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
492 |
||
493 |
declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
494 |
||
495 |
declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
496 |
||
497 |
declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
498 |
||
499 |
declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
500 |
||
501 |
declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
502 |
||
503 |
declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
504 |
||
505 |
declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
506 |
||
507 |
declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] |
|
508 |
||
509 |
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
|
510 |
|
25076 | 511 |
end |
512 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
513 |
|
21083 | 514 |
setup {* |
515 |
let |
|
516 |
||
44058 | 517 |
fun prp t thm = Thm.prop_of thm = t; (* FIXME aconv!? *) |
15524 | 518 |
|
21083 | 519 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
43597 | 520 |
let val prems = Simplifier.prems_of ss; |
22916 | 521 |
val less = Const (@{const_name less}, T); |
21083 | 522 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
523 |
in case find_first (prp t) prems of |
|
524 |
NONE => |
|
525 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
526 |
in case find_first (prp t) prems of |
|
527 |
NONE => NONE |
|
24422 | 528 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})) |
21083 | 529 |
end |
24422 | 530 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv})) |
21083 | 531 |
end |
532 |
handle THM _ => NONE; |
|
15524 | 533 |
|
21083 | 534 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
43597 | 535 |
let val prems = Simplifier.prems_of ss; |
22916 | 536 |
val le = Const (@{const_name less_eq}, T); |
21083 | 537 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
538 |
in case find_first (prp t) prems of |
|
539 |
NONE => |
|
540 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
541 |
in case find_first (prp t) prems of |
|
542 |
NONE => NONE |
|
24422 | 543 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})) |
21083 | 544 |
end |
24422 | 545 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2})) |
21083 | 546 |
end |
547 |
handle THM _ => NONE; |
|
15524 | 548 |
|
21248 | 549 |
fun add_simprocs procs thy = |
42795
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
42287
diff
changeset
|
550 |
Simplifier.map_simpset_global (fn ss => ss |
21248 | 551 |
addsimprocs (map (fn (name, raw_ts, proc) => |
38715
6513ea67d95d
renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents:
38705
diff
changeset
|
552 |
Simplifier.simproc_global thy name raw_ts proc) procs)) thy; |
42795
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
42287
diff
changeset
|
553 |
|
26496
49ae9456eba9
purely functional setup of claset/simpset/clasimpset;
wenzelm
parents:
26324
diff
changeset
|
554 |
fun add_solver name tac = |
42795
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
42287
diff
changeset
|
555 |
Simplifier.map_simpset_global (fn ss => ss addSolver |
43597 | 556 |
mk_solver name (fn ss => tac (Simplifier.the_context ss) (Simplifier.prems_of ss))); |
21083 | 557 |
|
558 |
in |
|
21248 | 559 |
add_simprocs [ |
560 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
561 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
562 |
] |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
563 |
#> add_solver "Transitivity" Orders.order_tac |
21248 | 564 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
565 |
speed up, but the reasoning strength appears to be not higher (at least |
|
566 |
no breaking of additional proofs in the entire HOL distribution, as |
|
567 |
of 5 March 2004, was observed). *) |
|
21083 | 568 |
end |
569 |
*} |
|
15524 | 570 |
|
571 |
||
21083 | 572 |
subsection {* Bounded quantifiers *} |
573 |
||
574 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
575 |
"_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
|
576 |
"_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
|
577 |
"_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
|
578 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 579 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
580 |
"_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
|
581 |
"_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
|
582 |
"_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
|
583 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 584 |
|
585 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
586 |
"_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
|
587 |
"_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
|
588 |
"_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
|
589 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 590 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
591 |
"_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
|
592 |
"_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
|
593 |
"_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
|
594 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 595 |
|
596 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
597 |
"_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
|
598 |
"_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
|
599 |
"_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
|
600 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 601 |
|
602 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
603 |
"_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
|
604 |
"_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
|
605 |
"_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
|
606 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 607 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
608 |
"_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
|
609 |
"_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
|
610 |
"_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
|
611 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 612 |
|
613 |
translations |
|
614 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
615 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
616 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
617 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
618 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
619 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
620 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
621 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
622 |
||
623 |
print_translation {* |
|
624 |
let |
|
42287
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset
|
625 |
val All_binder = Mixfix.binder_name @{const_syntax All}; |
d98eb048a2e4
discontinued special treatment of structure Mixfix;
wenzelm
parents:
42284
diff
changeset
|
626 |
val Ex_binder = Mixfix.binder_name @{const_syntax Ex}; |
38786
e46e7a9cb622
formerly unnamed infix impliciation now named HOL.implies
haftmann
parents:
38715
diff
changeset
|
627 |
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
|
628 |
val conj = @{const_syntax HOL.conj}; |
22916 | 629 |
val less = @{const_syntax less}; |
630 |
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
|
631 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
632 |
val trans = |
35115 | 633 |
[((All_binder, impl, less), |
634 |
(@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})), |
|
635 |
((All_binder, impl, less_eq), |
|
636 |
(@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})), |
|
637 |
((Ex_binder, conj, less), |
|
638 |
(@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})), |
|
639 |
((Ex_binder, conj, less_eq), |
|
640 |
(@{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
|
641 |
|
35115 | 642 |
fun matches_bound v t = |
643 |
(case t of |
|
35364 | 644 |
Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v' |
35115 | 645 |
| _ => false); |
646 |
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
|
647 |
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
|
648 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
649 |
fun tr' q = (q, |
49660
de49d9b4d7bc
more explicit Syntax_Trans.mark_bound_abs/mark_bound_body: preserve type information for show_markup;
wenzelm
parents:
48891
diff
changeset
|
650 |
fn [Const (@{syntax_const "_bound"}, _) $ Free (v, T), |
35364 | 651 |
Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
35115 | 652 |
(case AList.lookup (op =) trans (q, c, d) of |
653 |
NONE => raise Match |
|
654 |
| 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
|
655 |
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
|
656 |
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P |
35115 | 657 |
else raise Match) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
658 |
| _ => raise Match); |
21524 | 659 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 660 |
*} |
661 |
||
662 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
663 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
664 |
|
25193 | 665 |
context ord |
666 |
begin |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
667 |
|
25193 | 668 |
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c" |
669 |
by (rule subst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
670 |
|
25193 | 671 |
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
672 |
by (rule ssubst) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
673 |
|
25193 | 674 |
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c" |
675 |
by (rule subst) |
|
676 |
||
677 |
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c" |
|
678 |
by (rule ssubst) |
|
679 |
||
680 |
end |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
681 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
682 |
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
|
683 |
(!!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
|
684 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
685 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
686 |
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
|
687 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
688 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
689 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
690 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
691 |
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
|
692 |
(!!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
|
693 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
694 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
695 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
696 |
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
|
697 |
finally (less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
698 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
699 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
700 |
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
|
701 |
(!!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
|
702 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
703 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
704 |
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
|
705 |
also assume "f b < c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
706 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
707 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
708 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
709 |
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
|
710 |
(!!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
|
711 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
712 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
713 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
714 |
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
|
715 |
finally (le_less_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
716 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
717 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
718 |
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
|
719 |
(!!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
|
720 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
721 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
722 |
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
|
723 |
also assume "f b <= c" |
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
724 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
725 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
726 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
727 |
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
|
728 |
(!!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
|
729 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
730 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
731 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
732 |
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
|
733 |
finally (less_le_trans) show ?thesis . |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
734 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
735 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
736 |
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
|
737 |
(!!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
|
738 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
739 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
740 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
741 |
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
|
742 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
743 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
744 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
745 |
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
|
746 |
(!!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
|
747 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
748 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
749 |
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
|
750 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
751 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
752 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
753 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
754 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
755 |
(!!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
|
756 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
757 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
758 |
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
|
759 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
760 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
761 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
762 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
763 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
764 |
(!!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
|
765 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
766 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
767 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
768 |
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
|
769 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
770 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
771 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
772 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
773 |
(!!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
|
774 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
775 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
776 |
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
|
777 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
778 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
779 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
780 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
781 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
782 |
(!!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
|
783 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
784 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
785 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
786 |
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
|
787 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
788 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
789 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
790 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
791 |
Note that this list of rules is in reverse order of priorities. |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
792 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
793 |
|
27682 | 794 |
lemmas [trans] = |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
795 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
796 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
797 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
798 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
799 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
800 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
801 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
802 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
803 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
804 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
805 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
806 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
807 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
808 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
809 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
810 |
mp |
27682 | 811 |
|
812 |
lemmas (in order) [trans] = |
|
813 |
neq_le_trans |
|
814 |
le_neq_trans |
|
815 |
||
816 |
lemmas (in preorder) [trans] = |
|
817 |
less_trans |
|
818 |
less_asym' |
|
819 |
le_less_trans |
|
820 |
less_le_trans |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
821 |
order_trans |
27682 | 822 |
|
823 |
lemmas (in order) [trans] = |
|
824 |
antisym |
|
825 |
||
826 |
lemmas (in ord) [trans] = |
|
827 |
ord_le_eq_trans |
|
828 |
ord_eq_le_trans |
|
829 |
ord_less_eq_trans |
|
830 |
ord_eq_less_trans |
|
831 |
||
832 |
lemmas [trans] = |
|
833 |
trans |
|
834 |
||
835 |
lemmas order_trans_rules = |
|
836 |
order_less_subst2 |
|
837 |
order_less_subst1 |
|
838 |
order_le_less_subst2 |
|
839 |
order_le_less_subst1 |
|
840 |
order_less_le_subst2 |
|
841 |
order_less_le_subst1 |
|
842 |
order_subst2 |
|
843 |
order_subst1 |
|
844 |
ord_le_eq_subst |
|
845 |
ord_eq_le_subst |
|
846 |
ord_less_eq_subst |
|
847 |
ord_eq_less_subst |
|
848 |
forw_subst |
|
849 |
back_subst |
|
850 |
rev_mp |
|
851 |
mp |
|
852 |
neq_le_trans |
|
853 |
le_neq_trans |
|
854 |
less_trans |
|
855 |
less_asym' |
|
856 |
le_less_trans |
|
857 |
less_le_trans |
|
858 |
order_trans |
|
859 |
antisym |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
860 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
862 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
|
21083 | 866 |
text {* These support proving chains of decreasing inequalities |
867 |
a >= b >= c ... in Isar proofs. *} |
|
868 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
869 |
lemma xt1 [no_atp]: |
21083 | 870 |
"a = b ==> b > c ==> a > c" |
871 |
"a > b ==> b = c ==> a > c" |
|
872 |
"a = b ==> b >= c ==> a >= c" |
|
873 |
"a >= b ==> b = c ==> a >= c" |
|
874 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
875 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
876 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
877 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
23417 | 878 |
"(a::'a::order) > b ==> b > a ==> P" |
21083 | 879 |
"(x::'a::order) > y ==> y > z ==> x > z" |
880 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
881 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
882 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
883 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
884 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
885 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
25076 | 886 |
by auto |
21083 | 887 |
|
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
888 |
lemma xt2 [no_atp]: |
21083 | 889 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
890 |
by (subgoal_tac "f b >= f c", force, force) |
|
891 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
892 |
lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
21083 | 893 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
894 |
by (subgoal_tac "f a >= f b", force, force) |
|
895 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
896 |
lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
21083 | 897 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
898 |
by (subgoal_tac "f b >= f c", force, force) |
|
899 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
900 |
lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
21083 | 901 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
902 |
by (subgoal_tac "f a > f b", force, force) |
|
903 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
904 |
lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==> |
21083 | 905 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
906 |
by (subgoal_tac "f b > f c", force, force) |
|
907 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
908 |
lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
21083 | 909 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
910 |
by (subgoal_tac "f a >= f b", force, force) |
|
911 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
912 |
lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
21083 | 913 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
914 |
by (subgoal_tac "f b > f c", force, force) |
|
915 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
916 |
lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
21083 | 917 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
918 |
by (subgoal_tac "f a > f b", force, force) |
|
919 |
||
45221
3eadb9b6a055
mark "xt..." rules as "no_atp", since they are easy consequences of other better named properties
blanchet
parents:
44921
diff
changeset
|
920 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 [no_atp] |
21083 | 921 |
|
922 |
(* |
|
923 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
924 |
for the wrong thing in an Isar proof. |
|
925 |
||
926 |
The extra transitivity rules can be used as follows: |
|
927 |
||
928 |
lemma "(a::'a::order) > z" |
|
929 |
proof - |
|
930 |
have "a >= b" (is "_ >= ?rhs") |
|
931 |
sorry |
|
932 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
933 |
sorry |
|
934 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
935 |
sorry |
|
936 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
937 |
sorry |
|
938 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
939 |
sorry |
|
940 |
also (xtrans) have "?rhs > z" |
|
941 |
sorry |
|
942 |
finally (xtrans) show ?thesis . |
|
943 |
qed |
|
944 |
||
945 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
946 |
leave out the "(xtrans)" above. |
|
947 |
*) |
|
948 |
||
23881 | 949 |
|
950 |
subsection {* Monotonicity, least value operator and min/max *} |
|
21083 | 951 |
|
25076 | 952 |
context order |
953 |
begin |
|
954 |
||
30298 | 955 |
definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where |
25076 | 956 |
"mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)" |
957 |
||
958 |
lemma monoI [intro?]: |
|
959 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" |
|
960 |
shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f" |
|
961 |
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
|
962 |
|
25076 | 963 |
lemma monoD [dest?]: |
964 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>order" |
|
965 |
shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
966 |
unfolding mono_def by iprover |
|
967 |
||
30298 | 968 |
definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where |
969 |
"strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)" |
|
970 |
||
971 |
lemma strict_monoI [intro?]: |
|
972 |
assumes "\<And>x y. x < y \<Longrightarrow> f x < f y" |
|
973 |
shows "strict_mono f" |
|
974 |
using assms unfolding strict_mono_def by auto |
|
975 |
||
976 |
lemma strict_monoD [dest?]: |
|
977 |
"strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y" |
|
978 |
unfolding strict_mono_def by auto |
|
979 |
||
980 |
lemma strict_mono_mono [dest?]: |
|
981 |
assumes "strict_mono f" |
|
982 |
shows "mono f" |
|
983 |
proof (rule monoI) |
|
984 |
fix x y |
|
985 |
assume "x \<le> y" |
|
986 |
show "f x \<le> f y" |
|
987 |
proof (cases "x = y") |
|
988 |
case True then show ?thesis by simp |
|
989 |
next |
|
990 |
case False with `x \<le> y` have "x < y" by simp |
|
991 |
with assms strict_monoD have "f x < f y" by auto |
|
992 |
then show ?thesis by simp |
|
993 |
qed |
|
994 |
qed |
|
995 |
||
25076 | 996 |
end |
997 |
||
998 |
context linorder |
|
999 |
begin |
|
1000 |
||
30298 | 1001 |
lemma strict_mono_eq: |
1002 |
assumes "strict_mono f" |
|
1003 |
shows "f x = f y \<longleftrightarrow> x = y" |
|
1004 |
proof |
|
1005 |
assume "f x = f y" |
|
1006 |
show "x = y" proof (cases x y rule: linorder_cases) |
|
1007 |
case less with assms strict_monoD have "f x < f y" by auto |
|
1008 |
with `f x = f y` show ?thesis by simp |
|
1009 |
next |
|
1010 |
case equal then show ?thesis . |
|
1011 |
next |
|
1012 |
case greater with assms strict_monoD have "f y < f x" by auto |
|
1013 |
with `f x = f y` show ?thesis by simp |
|
1014 |
qed |
|
1015 |
qed simp |
|
1016 |
||
1017 |
lemma strict_mono_less_eq: |
|
1018 |
assumes "strict_mono f" |
|
1019 |
shows "f x \<le> f y \<longleftrightarrow> x \<le> y" |
|
1020 |
proof |
|
1021 |
assume "x \<le> y" |
|
1022 |
with assms strict_mono_mono monoD show "f x \<le> f y" by auto |
|
1023 |
next |
|
1024 |
assume "f x \<le> f y" |
|
1025 |
show "x \<le> y" proof (rule ccontr) |
|
1026 |
assume "\<not> x \<le> y" then have "y < x" by simp |
|
1027 |
with assms strict_monoD have "f y < f x" by auto |
|
1028 |
with `f x \<le> f y` show False by simp |
|
1029 |
qed |
|
1030 |
qed |
|
1031 |
||
1032 |
lemma strict_mono_less: |
|
1033 |
assumes "strict_mono f" |
|
1034 |
shows "f x < f y \<longleftrightarrow> x < y" |
|
1035 |
using assms |
|
1036 |
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq) |
|
1037 |
||
25076 | 1038 |
lemma min_of_mono: |
1039 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder" |
|
25377 | 1040 |
shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)" |
25076 | 1041 |
by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym) |
1042 |
||
1043 |
lemma max_of_mono: |
|
1044 |
fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder" |
|
25377 | 1045 |
shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)" |
25076 | 1046 |
by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym) |
1047 |
||
1048 |
end |
|
21083 | 1049 |
|
45931 | 1050 |
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x" |
23212 | 1051 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1052 |
|
45931 | 1053 |
lemma max_absorb2: "x \<le> y ==> max x y = y" |
23212 | 1054 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1055 |
|
45931 | 1056 |
lemma min_absorb2: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> min x y = y" |
1057 |
by (simp add:min_def) |
|
45893 | 1058 |
|
45931 | 1059 |
lemma max_absorb1: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> max x y = x" |
45893 | 1060 |
by (simp add: max_def) |
1061 |
||
1062 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1063 |
|
43813
07f0650146f2
tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents:
43597
diff
changeset
|
1064 |
subsection {* (Unique) top and bottom elements *} |
28685 | 1065 |
|
43813
07f0650146f2
tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents:
43597
diff
changeset
|
1066 |
class bot = order + |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1067 |
fixes bot :: 'a ("\<bottom>") |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1068 |
assumes bot_least [simp]: "\<bottom> \<le> a" |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1069 |
begin |
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1070 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1071 |
lemma le_bot: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1072 |
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1073 |
by (auto intro: antisym) |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1074 |
|
43816 | 1075 |
lemma bot_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1076 |
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1077 |
by (auto intro: antisym) |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1078 |
|
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1079 |
lemma not_less_bot [simp]: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1080 |
"\<not> (a < \<bottom>)" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1081 |
using bot_least [of a] by (auto simp: le_less) |
43816 | 1082 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1083 |
lemma bot_less: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1084 |
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a" |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1085 |
by (auto simp add: less_le_not_le intro!: antisym) |
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1086 |
|
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1087 |
end |
41082 | 1088 |
|
43813
07f0650146f2
tightened specification of classes bot and top: uniqueness of top and bot elements
haftmann
parents:
43597
diff
changeset
|
1089 |
class top = order + |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1090 |
fixes top :: 'a ("\<top>") |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1091 |
assumes top_greatest [simp]: "a \<le> \<top>" |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1092 |
begin |
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1093 |
|
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1094 |
lemma top_le: |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1095 |
"\<top> \<le> a \<Longrightarrow> a = \<top>" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1096 |
by (rule antisym) auto |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1097 |
|
43816 | 1098 |
lemma top_unique: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1099 |
"\<top> \<le> a \<longleftrightarrow> a = \<top>" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1100 |
by (auto intro: antisym) |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1101 |
|
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1102 |
lemma not_top_less [simp]: "\<not> (\<top> < a)" |
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1103 |
using top_greatest [of a] by (auto simp: le_less) |
43816 | 1104 |
|
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1105 |
lemma less_top: |
43853
020ddc6a9508
consolidated bot and top classes, tuned notation
haftmann
parents:
43816
diff
changeset
|
1106 |
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>" |
43814
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1107 |
by (auto simp add: less_le_not_le intro!: antisym) |
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1108 |
|
58791b75cf1f
moved lemmas bot_less and less_top to classes bot and top respectively
haftmann
parents:
43813
diff
changeset
|
1109 |
end |
28685 | 1110 |
|
1111 |
||
27823 | 1112 |
subsection {* Dense orders *} |
1113 |
||
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
1114 |
class dense_linorder = linorder + |
27823 | 1115 |
assumes gt_ex: "\<exists>y. x < y" |
1116 |
and lt_ex: "\<exists>y. y < x" |
|
1117 |
and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)" |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1118 |
begin |
27823 | 1119 |
|
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1120 |
lemma dense_le: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1121 |
fixes y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1122 |
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
|
1123 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1124 |
proof (rule ccontr) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1125 |
assume "\<not> ?thesis" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1126 |
hence "z < y" by simp |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1127 |
from dense[OF this] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1128 |
obtain x where "x < y" and "z < x" by safe |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1129 |
moreover have "x \<le> z" using assms[OF `x < y`] . |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1130 |
ultimately show False by auto |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1131 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1132 |
|
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1133 |
lemma dense_le_bounded: |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1134 |
fixes x y z :: 'a |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1135 |
assumes "x < y" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1136 |
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
|
1137 |
shows "y \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1138 |
proof (rule dense_le) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1139 |
fix w assume "w < y" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1140 |
from dense[OF `x < y`] obtain u where "x < u" "u < y" by safe |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1141 |
from linear[of u w] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1142 |
show "w \<le> z" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1143 |
proof (rule disjE) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1144 |
assume "u \<le> w" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1145 |
from less_le_trans[OF `x < u` `u \<le> w`] `w < y` |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1146 |
show "w \<le> z" by (rule *) |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1147 |
next |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1148 |
assume "w \<le> u" |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1149 |
from `w \<le> u` *[OF `x < u` `u < y`] |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1150 |
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
|
1151 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1152 |
qed |
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1153 |
|
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35364
diff
changeset
|
1154 |
end |
27823 | 1155 |
|
1156 |
subsection {* Wellorders *} |
|
1157 |
||
1158 |
class wellorder = linorder + |
|
1159 |
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" |
|
1160 |
begin |
|
1161 |
||
1162 |
lemma wellorder_Least_lemma: |
|
1163 |
fixes k :: 'a |
|
1164 |
assumes "P k" |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1165 |
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k" |
27823 | 1166 |
proof - |
1167 |
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k" |
|
1168 |
using assms proof (induct k rule: less_induct) |
|
1169 |
case (less x) then have "P x" by simp |
|
1170 |
show ?case proof (rule classical) |
|
1171 |
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)" |
|
1172 |
have "\<And>y. P y \<Longrightarrow> x \<le> y" |
|
1173 |
proof (rule classical) |
|
1174 |
fix y |
|
38705 | 1175 |
assume "P y" and "\<not> x \<le> y" |
27823 | 1176 |
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
1177 |
by (auto simp add: not_le) |
|
1178 |
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y" |
|
1179 |
by auto |
|
1180 |
then show "x \<le> y" by auto |
|
1181 |
qed |
|
1182 |
with `P x` have Least: "(LEAST a. P a) = x" |
|
1183 |
by (rule Least_equality) |
|
1184 |
with `P x` show ?thesis by simp |
|
1185 |
qed |
|
1186 |
qed |
|
1187 |
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto |
|
1188 |
qed |
|
1189 |
||
1190 |
-- "The following 3 lemmas are due to Brian Huffman" |
|
1191 |
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)" |
|
1192 |
by (erule exE) (erule LeastI) |
|
1193 |
||
1194 |
lemma LeastI2: |
|
1195 |
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1196 |
by (blast intro: LeastI) |
|
1197 |
||
1198 |
lemma LeastI2_ex: |
|
1199 |
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)" |
|
1200 |
by (blast intro: LeastI_ex) |
|
1201 |
||
38705 | 1202 |
lemma LeastI2_wellorder: |
1203 |
assumes "P a" |
|
1204 |
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a" |
|
1205 |
shows "Q (Least P)" |
|
1206 |
proof (rule LeastI2_order) |
|
1207 |
show "P (Least P)" using `P a` by (rule LeastI) |
|
1208 |
next |
|
1209 |
fix y assume "P y" thus "Least P \<le> y" by (rule Least_le) |
|
1210 |
next |
|
1211 |
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2)) |
|
1212 |
qed |
|
1213 |
||
27823 | 1214 |
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k" |
1215 |
apply (simp (no_asm_use) add: not_le [symmetric]) |
|
1216 |
apply (erule contrapos_nn) |
|
1217 |
apply (erule Least_le) |
|
1218 |
done |
|
1219 |
||
38705 | 1220 |
end |
27823 | 1221 |
|
28685 | 1222 |
|
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
|
1223 |
subsection {* Order on @{typ bool} *} |
28685 | 1224 |
|
45262 | 1225 |
instantiation bool :: "{bot, top, linorder}" |
28685 | 1226 |
begin |
1227 |
||
1228 |
definition |
|
41080 | 1229 |
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q" |
28685 | 1230 |
|
1231 |
definition |
|
41080 | 1232 |
[simp]: "(P\<Colon>bool) < Q \<longleftrightarrow> \<not> P \<and> Q" |
28685 | 1233 |
|
1234 |
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
|
1235 |
[simp]: "\<bottom> \<longleftrightarrow> False" |
28685 | 1236 |
|
1237 |
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
|
1238 |
[simp]: "\<top> \<longleftrightarrow> True" |
28685 | 1239 |
|
1240 |
instance proof |
|
41080 | 1241 |
qed auto |
28685 | 1242 |
|
15524 | 1243 |
end |
28685 | 1244 |
|
1245 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
41080 | 1246 |
by simp |
28685 | 1247 |
|
1248 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
41080 | 1249 |
by simp |
28685 | 1250 |
|
1251 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
41080 | 1252 |
by simp |
28685 | 1253 |
|
1254 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
41080 | 1255 |
by simp |
32899 | 1256 |
|
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
|
1257 |
lemma bot_boolE: "\<bottom> \<Longrightarrow> P" |
41080 | 1258 |
by simp |
32899 | 1259 |
|
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
|
1260 |
lemma top_boolI: \<top> |
41080 | 1261 |
by simp |
28685 | 1262 |
|
1263 |
lemma [code]: |
|
1264 |
"False \<le> b \<longleftrightarrow> True" |
|
1265 |
"True \<le> b \<longleftrightarrow> b" |
|
1266 |
"False < b \<longleftrightarrow> b" |
|
1267 |
"True < b \<longleftrightarrow> False" |
|
41080 | 1268 |
by simp_all |
28685 | 1269 |
|
1270 |
||
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
|
1271 |
subsection {* Order on @{typ "_ \<Rightarrow> _"} *} |
28685 | 1272 |
|
1273 |
instantiation "fun" :: (type, ord) ord |
|
1274 |
begin |
|
1275 |
||
1276 |
definition |
|
37767 | 1277 |
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)" |
28685 | 1278 |
|
1279 |
definition |
|
41080 | 1280 |
"(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)" |
28685 | 1281 |
|
1282 |
instance .. |
|
1283 |
||
1284 |
end |
|
1285 |
||
1286 |
instance "fun" :: (type, preorder) preorder proof |
|
1287 |
qed (auto simp add: le_fun_def less_fun_def |
|
44921 | 1288 |
intro: order_trans antisym) |
28685 | 1289 |
|
1290 |
instance "fun" :: (type, order) order proof |
|
44921 | 1291 |
qed (auto simp add: le_fun_def intro: antisym) |
28685 | 1292 |
|
41082 | 1293 |
instantiation "fun" :: (type, bot) bot |
1294 |
begin |
|
1295 |
||
1296 |
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
|
1297 |
"\<bottom> = (\<lambda>x. \<bottom>)" |
41082 | 1298 |
|
49769 | 1299 |
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
|
1300 |
"\<bottom> x = \<bottom>" |
41082 | 1301 |
by (simp add: bot_fun_def) |
1302 |
||
1303 |
instance proof |
|
46884 | 1304 |
qed (simp add: le_fun_def) |
41082 | 1305 |
|
1306 |
end |
|
1307 |
||
28685 | 1308 |
instantiation "fun" :: (type, top) top |
1309 |
begin |
|
1310 |
||
1311 |
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
|
1312 |
[no_atp]: "\<top> = (\<lambda>x. \<top>)" |
28685 | 1313 |
|
49769 | 1314 |
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
|
1315 |
"\<top> x = \<top>" |
41080 | 1316 |
by (simp add: top_fun_def) |
1317 |
||
28685 | 1318 |
instance proof |
46884 | 1319 |
qed (simp add: le_fun_def) |
28685 | 1320 |
|
1321 |
end |
|
1322 |
||
1323 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
1324 |
unfolding le_fun_def by simp |
|
1325 |
||
1326 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
1327 |
unfolding le_fun_def by simp |
|
1328 |
||
1329 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
1330 |
unfolding le_fun_def by simp |
|
1331 |
||
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1332 |
|
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
|
1333 |
subsection {* Order on unary and binary predicates *} |
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
|
1334 |
|
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
|
1335 |
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
|
1336 |
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
|
1337 |
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
|
1338 |
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
|
1339 |
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
|
1340 |
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
|
1341 |
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
|
1342 |
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
|
1343 |
|
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
|
1344 |
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
|
1345 |
"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
|
1346 |
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
|
1347 |
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
|
1348 |
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
|
1349 |
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
|
1350 |
|
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
|
1351 |
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
|
1352 |
"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
|
1353 |
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
|
1354 |
|
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
|
1355 |
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
|
1356 |
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
|
1357 |
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
|
1358 |
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
|
1359 |
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
|
1360 |
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
|
1361 |
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
|
1362 |
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
|
1363 |
|
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
|
1364 |
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
|
1365 |
"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
|
1366 |
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
|
1367 |
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
|
1368 |
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
|
1369 |
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
|
1370 |
|
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
|
1371 |
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
|
1372 |
"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
|
1373 |
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
|
1374 |
|
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
|
1375 |
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
|
1376 |
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
|
1377 |
|
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
|
1378 |
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
|
1379 |
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
|
1380 |
|
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
|
1381 |
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
|
1382 |
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
|
1383 |
|
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
|
1384 |
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
|
1385 |
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
|
1386 |
|
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
|
1387 |
|
34250
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1388 |
subsection {* Name duplicates *} |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1389 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1390 |
lemmas order_eq_refl = preorder_class.eq_refl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1391 |
lemmas order_less_irrefl = preorder_class.less_irrefl |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1392 |
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
|
1393 |
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
|
1394 |
lemmas order_less_asym = preorder_class.less_asym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1395 |
lemmas order_less_trans = preorder_class.less_trans |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1396 |
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
|
1397 |
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
|
1398 |
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
|
1399 |
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
|
1400 |
lemmas order_less_asym' = preorder_class.less_asym' |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1401 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1402 |
lemmas order_less_le = order_class.less_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1403 |
lemmas order_le_less = order_class.le_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1404 |
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
|
1405 |
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
|
1406 |
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
|
1407 |
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
|
1408 |
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
|
1409 |
lemmas order_antisym = order_class.antisym |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1410 |
lemmas order_eq_iff = order_class.eq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1411 |
lemmas order_antisym_conv = order_class.antisym_conv |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1412 |
|
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1413 |
lemmas linorder_linear = linorder_class.linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1414 |
lemmas linorder_less_linear = linorder_class.less_linear |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1415 |
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
|
1416 |
lemmas linorder_le_cases = linorder_class.le_cases |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1417 |
lemmas linorder_not_less = linorder_class.not_less |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1418 |
lemmas linorder_not_le = linorder_class.not_le |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1419 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1420 |
lemmas linorder_neqE = linorder_class.neqE |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1421 |
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1422 |
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1423 |
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
3b619abaa67a
moved name duplicates to end of theory; reduced warning noise
haftmann
parents:
34065
diff
changeset
|
1424 |
|
28685 | 1425 |
end |