author | haftmann |
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parent 24741 | a53f5db5acbb |
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permissions | -rw-r--r-- |
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(* Title: HOL/Orderings.thy |
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ID: $Id$ |
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Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson |
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*) |
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header {* Syntactic and abstract orders *} |
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theory Orderings |
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imports Set Fun |
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uses |
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"~~/src/Provers/order.ML" |
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begin |
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subsection {* Partial orders *} |
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class order = ord + |
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assumes less_le: "x \<^loc>< y \<longleftrightarrow> x \<^loc>\<le> y \<and> x \<noteq> y" |
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and order_refl [iff]: "x \<^loc>\<le> x" |
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and order_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> z" |
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assumes antisym: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y" |
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begin |
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notation (input) |
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less_eq (infix "\<sqsubseteq>" 50) |
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and |
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less (infix "\<sqsubset>" 50) |
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text {* Reflexivity. *} |
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lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<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 \<^loc>< x" |
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by (simp add: less_le) |
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lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< 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 \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y" |
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unfolding less_le by blast |
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lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y" |
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unfolding less_le by blast |
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" |
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by (erule contrapos_pn, erule subst, rule less_irrefl) |
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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 \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
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by auto |
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lemma less_imp_not_eq2: "x \<^loc>< 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 \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b" |
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by (simp add: less_le) |
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lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b" |
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by (simp add: less_le) |
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text {* Asymmetry. *} |
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lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)" |
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by (simp add: less_le antisym) |
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lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P" |
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by (drule less_not_sym, erule contrapos_np) simp |
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lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x" |
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by (blast intro: antisym) |
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lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym) |
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y" |
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by (erule contrapos_pn, erule subst, rule less_irrefl) |
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text {* Transitivity. *} |
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lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z" |
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by (simp add: less_le) (blast intro: order_trans antisym) |
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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 \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< 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 \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P" |
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by (rule less_asym) |
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text {* Reverse order *} |
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lemma order_reverse: |
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"order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)" |
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by unfold_locales |
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(simp add: less_le, auto intro: antisym order_trans) |
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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 \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
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begin |
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lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x" |
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unfolding less_le using less_le linear by blast |
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lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< 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 \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<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 \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P" |
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using less_linear by blast |
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lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<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 \<^loc>< y) \<longleftrightarrow> (x \<^loc>> 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 \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< 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 \<^loc>< y \<or> y \<^loc>< 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 \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R" |
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by (simp add: neq_iff) blast |
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lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<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 \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< 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 \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y" |
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by (blast intro: antisym dest: not_less [THEN iffD1]) |
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text{*Replacing the old Nat.leI*} |
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lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x" |
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unfolding not_less . |
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lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< 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 \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y" |
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unfolding not_le . |
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text {* Reverse order *} |
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lemma linorder_reverse: |
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"linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)" |
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by unfold_locales |
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(simp add: less_le, auto intro: antisym order_trans simp add: linear) |
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text {* min/max *} |
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text {* for historic reasons, definitions are done in context ord *} |
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definition (in ord) |
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min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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[code unfold, code inline del]: "min a b = (if a \<^loc>\<le> b then a else b)" |
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definition (in ord) |
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max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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[code unfold, code inline del]: "max a b = (if a \<^loc>\<le> b then b else a)" |
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lemma min_le_iff_disj: |
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"min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<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 \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<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 \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< 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 \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y" |
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unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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lemma min_less_iff_conj [simp]: |
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"z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y" |
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unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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lemma max_less_iff_conj [simp]: |
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"max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z" |
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unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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lemma split_min [noatp]: |
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"P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)" |
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by (simp add: min_def) |
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lemma split_max [noatp]: |
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"P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)" |
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by (simp add: max_def) |
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end |
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subsection {* Reasoning tools setup *} |
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ML {* |
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signature ORDERS = |
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sig |
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val print_structures: Proof.context -> unit |
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val setup: theory -> theory |
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val order_tac: thm list -> Proof.context -> int -> tactic |
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end; |
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structure Orders: ORDERS = |
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struct |
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(** Theory and context data **) |
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fun struct_eq ((s1: string, ts1), (s2, ts2)) = |
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(s1 = s2) andalso eq_list (op aconv) (ts1, ts2); |
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structure Data = GenericDataFun |
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( |
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type T = ((string * term list) * Order_Tac.less_arith) list; |
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(* Order structures: |
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identifier of the structure, list of operations and record of theorems |
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needed to set up the transitivity reasoner, |
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identifier and operations identify the structure uniquely. *) |
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val empty = []; |
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val extend = I; |
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fun merge _ = AList.join struct_eq (K fst); |
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); |
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fun print_structures ctxt = |
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let |
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val structs = Data.get (Context.Proof ctxt); |
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277 |
fun pretty_term t = Pretty.block |
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278 |
[Pretty.quote (ProofContext.pretty_term ctxt t), Pretty.brk 1, |
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|
279 |
Pretty.str "::", Pretty.brk 1, |
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|
280 |
Pretty.quote (ProofContext.pretty_typ ctxt (type_of t))]; |
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|
281 |
fun pretty_struct ((s, ts), _) = Pretty.block |
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|
282 |
[Pretty.str s, Pretty.str ":", Pretty.brk 1, |
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|
283 |
Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; |
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|
284 |
in |
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|
285 |
Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs)) |
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|
286 |
end; |
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|
287 |
|
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|
288 |
|
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|
289 |
(** Method **) |
21091 | 290 |
|
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|
291 |
fun struct_tac ((s, [eq, le, less]), thms) prems = |
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|
292 |
let |
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|
293 |
fun decomp thy (Trueprop $ t) = |
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|
294 |
let |
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|
295 |
fun excluded t = |
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|
296 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
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|
297 |
let val T = type_of t |
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|
298 |
in |
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|
299 |
T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT |
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|
300 |
end; |
24741
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|
301 |
fun rel (bin_op $ t1 $ t2) = |
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|
302 |
if excluded t1 then NONE |
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|
303 |
else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) |
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|
304 |
else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) |
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|
305 |
else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) |
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|
306 |
else NONE |
24741
a53f5db5acbb
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|
307 |
| rel _ = NONE; |
a53f5db5acbb
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changeset
|
308 |
fun dec (Const (@{const_name Not}, _) $ t) = (case rel t |
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Fixed setup of transitivity reasoner (function decomp).
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|
309 |
of NONE => NONE |
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Fixed setup of transitivity reasoner (function decomp).
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|
310 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
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Fixed setup of transitivity reasoner (function decomp).
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|
311 |
| dec x = rel x; |
24641
448edc627ee4
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|
312 |
in dec t end; |
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|
313 |
in |
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|
314 |
case s of |
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|
315 |
"order" => Order_Tac.partial_tac decomp thms prems |
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Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
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|
316 |
| "linorder" => Order_Tac.linear_tac decomp thms prems |
24641
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|
317 |
| _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.") |
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|
318 |
end |
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|
319 |
|
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|
320 |
fun order_tac prems ctxt = |
9a95634ab135
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changeset
|
321 |
FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt))); |
24641
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|
322 |
|
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Transitivity reasoner set up for locales order and linorder.
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changeset
|
323 |
|
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|
324 |
(** Attribute **) |
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changeset
|
325 |
|
448edc627ee4
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changeset
|
326 |
fun add_struct_thm s tag = |
448edc627ee4
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|
327 |
Thm.declaration_attribute |
448edc627ee4
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parents:
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changeset
|
328 |
(fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm))); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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|
329 |
fun del_struct s = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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diff
changeset
|
330 |
Thm.declaration_attribute |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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changeset
|
331 |
(fn _ => Data.map (AList.delete struct_eq s)); |
448edc627ee4
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changeset
|
332 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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changeset
|
333 |
val attribute = Attrib.syntax |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
334 |
(Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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changeset
|
335 |
Args.del >> K NONE) --| Args.colon (* FIXME || |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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changeset
|
336 |
Scan.succeed true *) ) -- Scan.lift Args.name -- |
448edc627ee4
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|
337 |
Scan.repeat Args.term |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
338 |
>> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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changeset
|
339 |
| ((NONE, n), ts) => del_struct (n, ts))); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
340 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
341 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
342 |
(** Diagnostic command **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
343 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
344 |
val print = Toplevel.unknown_context o |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
345 |
Toplevel.keep (Toplevel.node_case |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
346 |
(Context.cases (print_structures o ProofContext.init) print_structures) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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changeset
|
347 |
(print_structures o Proof.context_of)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
348 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
349 |
val printP = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
350 |
OuterSyntax.improper_command "print_orders" |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
351 |
"print order structures available to transitivity reasoner" OuterKeyword.diag |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
352 |
(Scan.succeed (Toplevel.no_timing o print)); |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
353 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
354 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
355 |
(** Setup **) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
356 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
357 |
val setup = let val _ = OuterSyntax.add_parsers [printP] in |
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
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diff
changeset
|
358 |
Method.add_methods [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), |
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
359 |
"normalisation of algebraic structure")] #> |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
360 |
Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
361 |
end; |
21091 | 362 |
|
363 |
end; |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
364 |
|
21091 | 365 |
*} |
366 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
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parents:
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diff
changeset
|
367 |
setup Orders.setup |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
368 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
369 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
370 |
text {* Declarations to set up transitivity reasoner of partial and linear orders. *} |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
371 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
372 |
(* The type constraint on @{term op =} below is necessary since the operation |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
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diff
changeset
|
373 |
is not a parameter of the locale. *) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
374 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
375 |
[order add less_reflE: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
376 |
less_irrefl [THEN notE] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
377 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
378 |
[order add le_refl: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
379 |
order_refl |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
380 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
381 |
[order add less_imp_le: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
382 |
less_imp_le |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
383 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
384 |
[order add eqI: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
385 |
antisym |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
386 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
387 |
[order add eqD1: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
388 |
eq_refl |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
389 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
390 |
[order add eqD2: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
391 |
sym [THEN eq_refl] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
392 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
393 |
[order add less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
394 |
less_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
395 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
396 |
[order add less_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
397 |
less_le_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
398 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
399 |
[order add le_less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
400 |
le_less_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
401 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
402 |
[order add le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
403 |
order_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
404 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
405 |
[order add le_neq_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
406 |
le_neq_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
407 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
408 |
[order add neq_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
409 |
neq_le_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
410 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
411 |
[order add less_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
412 |
less_imp_neq |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
413 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
414 |
[order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
415 |
eq_neq_eq_imp_neq |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
416 |
lemmas (in order) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
417 |
[order add not_sym: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
418 |
not_sym |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
419 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
420 |
lemmas (in linorder) [order del: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = _ |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
421 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
422 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
423 |
[order add less_reflE: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
424 |
less_irrefl [THEN notE] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
425 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
426 |
[order add le_refl: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
427 |
order_refl |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
428 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
429 |
[order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
430 |
less_imp_le |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
431 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
432 |
[order add not_lessI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
433 |
not_less [THEN iffD2] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
434 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
435 |
[order add not_leI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
436 |
not_le [THEN iffD2] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
437 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
438 |
[order add not_lessD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
439 |
not_less [THEN iffD1] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
440 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
441 |
[order add not_leD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
442 |
not_le [THEN iffD1] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
443 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
444 |
[order add eqI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
445 |
antisym |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
446 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
447 |
[order add eqD1: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
448 |
eq_refl |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
449 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
450 |
[order add eqD2: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
451 |
sym [THEN eq_refl] |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
452 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
453 |
[order add less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
454 |
less_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
455 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
456 |
[order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
457 |
less_le_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
458 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
459 |
[order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
460 |
le_less_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
461 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
462 |
[order add le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
463 |
order_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
464 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
465 |
[order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
466 |
le_neq_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
467 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
468 |
[order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
469 |
neq_le_trans |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
470 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
471 |
[order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
472 |
less_imp_neq |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
473 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
474 |
[order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
475 |
eq_neq_eq_imp_neq |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
476 |
lemmas (in linorder) |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
477 |
[order add not_sym: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
478 |
not_sym |
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
479 |
|
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
480 |
|
21083 | 481 |
setup {* |
482 |
let |
|
483 |
||
484 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 485 |
|
21083 | 486 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
487 |
let val prems = prems_of_ss ss; |
|
22916 | 488 |
val less = Const (@{const_name less}, T); |
21083 | 489 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
490 |
in case find_first (prp t) prems of |
|
491 |
NONE => |
|
492 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
493 |
in case find_first (prp t) prems of |
|
494 |
NONE => NONE |
|
24422 | 495 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})) |
21083 | 496 |
end |
24422 | 497 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv})) |
21083 | 498 |
end |
499 |
handle THM _ => NONE; |
|
15524 | 500 |
|
21083 | 501 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
502 |
let val prems = prems_of_ss ss; |
|
22916 | 503 |
val le = Const (@{const_name less_eq}, T); |
21083 | 504 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
505 |
in case find_first (prp t) prems of |
|
506 |
NONE => |
|
507 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
508 |
in case find_first (prp t) prems of |
|
509 |
NONE => NONE |
|
24422 | 510 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})) |
21083 | 511 |
end |
24422 | 512 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2})) |
21083 | 513 |
end |
514 |
handle THM _ => NONE; |
|
15524 | 515 |
|
21248 | 516 |
fun add_simprocs procs thy = |
517 |
(Simplifier.change_simpset_of thy (fn ss => ss |
|
518 |
addsimprocs (map (fn (name, raw_ts, proc) => |
|
519 |
Simplifier.simproc thy name raw_ts proc)) procs); thy); |
|
520 |
fun add_solver name tac thy = |
|
521 |
(Simplifier.change_simpset_of thy (fn ss => ss addSolver |
|
24704
9a95634ab135
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
ballarin
parents:
24641
diff
changeset
|
522 |
(mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy); |
21083 | 523 |
|
524 |
in |
|
21248 | 525 |
add_simprocs [ |
526 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
527 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
528 |
] |
|
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
529 |
#> add_solver "Transitivity" Orders.order_tac |
21248 | 530 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
531 |
speed up, but the reasoning strength appears to be not higher (at least |
|
532 |
no breaking of additional proofs in the entire HOL distribution, as |
|
533 |
of 5 March 2004, was observed). *) |
|
21083 | 534 |
end |
535 |
*} |
|
15524 | 536 |
|
537 |
||
24422 | 538 |
subsection {* Dense orders *} |
539 |
||
540 |
class dense_linear_order = linorder + |
|
541 |
assumes gt_ex: "\<exists>y. x \<sqsubset> y" |
|
542 |
and lt_ex: "\<exists>y. y \<sqsubset> x" |
|
543 |
and dense: "x \<sqsubset> y \<Longrightarrow> (\<exists>z. x \<sqsubset> z \<and> z \<sqsubset> y)" |
|
544 |
(*see further theory Dense_Linear_Order*) |
|
545 |
||
24641
448edc627ee4
Transitivity reasoner set up for locales order and linorder.
ballarin
parents:
24422
diff
changeset
|
546 |
|
24422 | 547 |
lemma interval_empty_iff: |
548 |
fixes x y z :: "'a\<Colon>dense_linear_order" |
|
549 |
shows "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z" |
|
550 |
by (auto dest: dense) |
|
551 |
||
552 |
subsection {* Name duplicates *} |
|
553 |
||
554 |
lemmas order_less_le = less_le |
|
555 |
lemmas order_eq_refl = order_class.eq_refl |
|
556 |
lemmas order_less_irrefl = order_class.less_irrefl |
|
557 |
lemmas order_le_less = order_class.le_less |
|
558 |
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
|
559 |
lemmas order_less_imp_le = order_class.less_imp_le |
|
560 |
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
|
561 |
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
|
562 |
lemmas order_neq_le_trans = order_class.neq_le_trans |
|
563 |
lemmas order_le_neq_trans = order_class.le_neq_trans |
|
564 |
||
565 |
lemmas order_antisym = antisym |
|
566 |
lemmas order_less_not_sym = order_class.less_not_sym |
|
567 |
lemmas order_less_asym = order_class.less_asym |
|
568 |
lemmas order_eq_iff = order_class.eq_iff |
|
569 |
lemmas order_antisym_conv = order_class.antisym_conv |
|
570 |
lemmas order_less_trans = order_class.less_trans |
|
571 |
lemmas order_le_less_trans = order_class.le_less_trans |
|
572 |
lemmas order_less_le_trans = order_class.less_le_trans |
|
573 |
lemmas order_less_imp_not_less = order_class.less_imp_not_less |
|
574 |
lemmas order_less_imp_triv = order_class.less_imp_triv |
|
575 |
lemmas order_less_asym' = order_class.less_asym' |
|
576 |
||
577 |
lemmas linorder_linear = linear |
|
578 |
lemmas linorder_less_linear = linorder_class.less_linear |
|
579 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
|
580 |
lemmas linorder_le_cases = linorder_class.le_cases |
|
581 |
lemmas linorder_not_less = linorder_class.not_less |
|
582 |
lemmas linorder_not_le = linorder_class.not_le |
|
583 |
lemmas linorder_neq_iff = linorder_class.neq_iff |
|
584 |
lemmas linorder_neqE = linorder_class.neqE |
|
585 |
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
|
586 |
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
|
587 |
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
|
588 |
||
589 |
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj |
|
590 |
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj |
|
591 |
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj |
|
592 |
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj |
|
593 |
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj |
|
594 |
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj |
|
595 |
lemmas split_min = linorder_class.split_min |
|
596 |
lemmas split_max = linorder_class.split_max |
|
597 |
||
598 |
||
21083 | 599 |
subsection {* Bounded quantifiers *} |
600 |
||
601 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
602 |
"_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
|
603 |
"_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
|
604 |
"_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
|
605 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 606 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
607 |
"_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
|
608 |
"_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
|
609 |
"_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
|
610 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 611 |
|
612 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
613 |
"_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
|
614 |
"_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
|
615 |
"_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
|
616 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 617 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
618 |
"_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
|
619 |
"_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
|
620 |
"_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
|
621 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 622 |
|
623 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
624 |
"_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
|
625 |
"_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
|
626 |
"_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
|
627 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 628 |
|
629 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
630 |
"_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
|
631 |
"_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
|
632 |
"_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
|
633 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 634 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
635 |
"_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
|
636 |
"_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
|
637 |
"_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
|
638 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 639 |
|
640 |
translations |
|
641 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
642 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
643 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
644 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
645 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
646 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
647 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
648 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
649 |
||
650 |
print_translation {* |
|
651 |
let |
|
22916 | 652 |
val All_binder = Syntax.binder_name @{const_syntax All}; |
653 |
val Ex_binder = Syntax.binder_name @{const_syntax Ex}; |
|
22377 | 654 |
val impl = @{const_syntax "op -->"}; |
655 |
val conj = @{const_syntax "op &"}; |
|
22916 | 656 |
val less = @{const_syntax less}; |
657 |
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
|
658 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
659 |
val trans = |
21524 | 660 |
[((All_binder, impl, less), ("_All_less", "_All_greater")), |
661 |
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
|
662 |
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), |
|
663 |
((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
664 |
|
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
665 |
fun matches_bound v t = |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
666 |
case t of (Const ("_bound", _) $ Free (v', _)) => (v = v') |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
667 |
| _ => false |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
668 |
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false) |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
669 |
fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
670 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
671 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
672 |
fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] => |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
673 |
(case AList.lookup (op =) trans (q, c, d) of |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
674 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
675 |
| SOME (l, g) => |
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
676 |
if matches_bound v t andalso not (contains_var v u) then mk v l u P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
677 |
else if matches_bound v u andalso not (contains_var v t) then mk v g t P |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
678 |
else raise Match) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
679 |
| _ => raise Match); |
21524 | 680 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 681 |
*} |
682 |
||
683 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
684 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
685 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
686 |
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" |
23212 | 687 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
688 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
689 |
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" |
23212 | 690 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
691 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
692 |
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" |
23212 | 693 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
694 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
695 |
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" |
23212 | 696 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
697 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
698 |
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
|
699 |
(!!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
|
700 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
701 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
702 |
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
|
703 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
704 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
705 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
706 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
707 |
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
|
708 |
(!!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
|
709 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
710 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
711 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
712 |
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
|
713 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
714 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
715 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
716 |
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
|
717 |
(!!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
|
718 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
719 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
720 |
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
|
721 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
722 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
723 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
724 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
725 |
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
|
726 |
(!!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
|
727 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
728 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
729 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
730 |
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
|
731 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
732 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
733 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
734 |
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
|
735 |
(!!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
|
736 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
737 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
738 |
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
|
739 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
740 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
741 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
742 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
743 |
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
|
744 |
(!!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
|
745 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
746 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
747 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
748 |
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
|
749 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
750 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
751 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
752 |
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
753 |
(!!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
|
754 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
755 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
756 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
757 |
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
|
758 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
759 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
760 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
761 |
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
762 |
(!!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
|
763 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
764 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
765 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
766 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
767 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
768 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
769 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
770 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
771 |
(!!x y. x <= y ==> f x <= f y) ==> f a <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
772 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
773 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
774 |
assume "a <= b" hence "f a <= f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
775 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
776 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
777 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
778 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
779 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
780 |
(!!x y. x <= y ==> f x <= f y) ==> a <= f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
781 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
782 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
783 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
784 |
also assume "b <= c" hence "f b <= f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
785 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
786 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
787 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
788 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
789 |
(!!x y. x < y ==> f x < f y) ==> f a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
790 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
791 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
792 |
assume "a < b" hence "f a < f b" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
793 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
794 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
795 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
796 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
797 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
798 |
(!!x y. x < y ==> f x < f y) ==> a < f c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
799 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
800 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
801 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
802 |
also assume "b < c" hence "f b < f c" by (rule r) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
803 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
804 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
805 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
806 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
807 |
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
|
808 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
809 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
810 |
lemmas order_trans_rules [trans] = |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
811 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
812 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
813 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
814 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
815 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
816 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
817 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
818 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
819 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
820 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
821 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
822 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
823 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
824 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
825 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
826 |
mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
827 |
order_neq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
828 |
order_le_neq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
829 |
order_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
830 |
order_less_asym' |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
831 |
order_le_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
832 |
order_less_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
833 |
order_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
834 |
order_antisym |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
835 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
836 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
837 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
838 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
839 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
840 |
|
21083 | 841 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
842 |
(* FIXME cleanup *) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
843 |
|
21083 | 844 |
text {* These support proving chains of decreasing inequalities |
845 |
a >= b >= c ... in Isar proofs. *} |
|
846 |
||
847 |
lemma xt1: |
|
848 |
"a = b ==> b > c ==> a > c" |
|
849 |
"a > b ==> b = c ==> a > c" |
|
850 |
"a = b ==> b >= c ==> a >= c" |
|
851 |
"a >= b ==> b = c ==> a >= c" |
|
852 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
853 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
854 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
855 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
23417 | 856 |
"(a::'a::order) > b ==> b > a ==> P" |
21083 | 857 |
"(x::'a::order) > y ==> y > z ==> x > z" |
858 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
859 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
860 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
861 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
862 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
863 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
864 |
by auto |
|
865 |
||
866 |
lemma xt2: |
|
867 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
868 |
by (subgoal_tac "f b >= f c", force, force) |
|
869 |
||
870 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
871 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
872 |
by (subgoal_tac "f a >= f b", force, force) |
|
873 |
||
874 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
875 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
876 |
by (subgoal_tac "f b >= f c", force, force) |
|
877 |
||
878 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
879 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
880 |
by (subgoal_tac "f a > f b", force, force) |
|
881 |
||
882 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
883 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
884 |
by (subgoal_tac "f b > f c", force, force) |
|
885 |
||
886 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
887 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
888 |
by (subgoal_tac "f a >= f b", force, force) |
|
889 |
||
890 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
891 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
892 |
by (subgoal_tac "f b > f c", force, force) |
|
893 |
||
894 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
895 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
896 |
by (subgoal_tac "f a > f b", force, force) |
|
897 |
||
898 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
899 |
||
900 |
(* |
|
901 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
902 |
for the wrong thing in an Isar proof. |
|
903 |
||
904 |
The extra transitivity rules can be used as follows: |
|
905 |
||
906 |
lemma "(a::'a::order) > z" |
|
907 |
proof - |
|
908 |
have "a >= b" (is "_ >= ?rhs") |
|
909 |
sorry |
|
910 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
911 |
sorry |
|
912 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
913 |
sorry |
|
914 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
915 |
sorry |
|
916 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
917 |
sorry |
|
918 |
also (xtrans) have "?rhs > z" |
|
919 |
sorry |
|
920 |
finally (xtrans) show ?thesis . |
|
921 |
qed |
|
922 |
||
923 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
924 |
leave out the "(xtrans)" above. |
|
925 |
*) |
|
926 |
||
21546 | 927 |
subsection {* Order on bool *} |
928 |
||
22886 | 929 |
instance bool :: order |
21546 | 930 |
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" |
931 |
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" |
|
22916 | 932 |
by intro_classes (auto simp add: le_bool_def less_bool_def) |
24422 | 933 |
lemmas [code func del] = le_bool_def less_bool_def |
21546 | 934 |
|
935 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
23212 | 936 |
by (simp add: le_bool_def) |
21546 | 937 |
|
938 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
23212 | 939 |
by (simp add: le_bool_def) |
21546 | 940 |
|
941 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
23212 | 942 |
by (simp add: le_bool_def) |
21546 | 943 |
|
944 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
23212 | 945 |
by (simp add: le_bool_def) |
21546 | 946 |
|
22348 | 947 |
lemma [code func]: |
948 |
"False \<le> b \<longleftrightarrow> True" |
|
949 |
"True \<le> b \<longleftrightarrow> b" |
|
950 |
"False < b \<longleftrightarrow> b" |
|
951 |
"True < b \<longleftrightarrow> False" |
|
952 |
unfolding le_bool_def less_bool_def by simp_all |
|
953 |
||
22424 | 954 |
|
23881 | 955 |
subsection {* Order on sets *} |
956 |
||
957 |
instance set :: (type) order |
|
958 |
by (intro_classes, |
|
959 |
(assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+) |
|
960 |
||
961 |
lemmas basic_trans_rules [trans] = |
|
962 |
order_trans_rules set_rev_mp set_mp |
|
963 |
||
964 |
||
965 |
subsection {* Order on functions *} |
|
966 |
||
967 |
instance "fun" :: (type, ord) ord |
|
968 |
le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x" |
|
969 |
less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" .. |
|
970 |
||
971 |
lemmas [code func del] = le_fun_def less_fun_def |
|
972 |
||
973 |
instance "fun" :: (type, order) order |
|
974 |
by default |
|
975 |
(auto simp add: le_fun_def less_fun_def expand_fun_eq |
|
976 |
intro: order_trans order_antisym) |
|
977 |
||
978 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
979 |
unfolding le_fun_def by simp |
|
980 |
||
981 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
982 |
unfolding le_fun_def by simp |
|
983 |
||
984 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
985 |
unfolding le_fun_def by simp |
|
986 |
||
987 |
text {* |
|
988 |
Handy introduction and elimination rules for @{text "\<le>"} |
|
989 |
on unary and binary predicates |
|
990 |
*} |
|
991 |
||
992 |
lemma predicate1I [Pure.intro!, intro!]: |
|
993 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
|
994 |
shows "P \<le> Q" |
|
995 |
apply (rule le_funI) |
|
996 |
apply (rule le_boolI) |
|
997 |
apply (rule PQ) |
|
998 |
apply assumption |
|
999 |
done |
|
1000 |
||
1001 |
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
|
1002 |
apply (erule le_funE) |
|
1003 |
apply (erule le_boolE) |
|
1004 |
apply assumption+ |
|
1005 |
done |
|
1006 |
||
1007 |
lemma predicate2I [Pure.intro!, intro!]: |
|
1008 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
|
1009 |
shows "P \<le> Q" |
|
1010 |
apply (rule le_funI)+ |
|
1011 |
apply (rule le_boolI) |
|
1012 |
apply (rule PQ) |
|
1013 |
apply assumption |
|
1014 |
done |
|
1015 |
||
1016 |
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
|
1017 |
apply (erule le_funE)+ |
|
1018 |
apply (erule le_boolE) |
|
1019 |
apply assumption+ |
|
1020 |
done |
|
1021 |
||
1022 |
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" |
|
1023 |
by (rule predicate1D) |
|
1024 |
||
1025 |
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" |
|
1026 |
by (rule predicate2D) |
|
1027 |
||
1028 |
||
1029 |
subsection {* Monotonicity, least value operator and min/max *} |
|
21083 | 1030 |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1031 |
locale mono = |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1032 |
fixes f |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1033 |
assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B" |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1034 |
|
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1035 |
lemmas monoI [intro?] = mono.intro |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
1036 |
and monoD [dest?] = mono.mono |
21083 | 1037 |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1038 |
lemma LeastI2_order: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1039 |
"[| P (x::'a::order); |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1040 |
!!y. P y ==> x <= y; |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1041 |
!!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1042 |
==> Q (Least P)" |
23212 | 1043 |
apply (unfold Least_def) |
1044 |
apply (rule theI2) |
|
1045 |
apply (blast intro: order_antisym)+ |
|
1046 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1047 |
|
23881 | 1048 |
lemma Least_mono: |
1049 |
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y |
|
1050 |
==> (LEAST y. y : f ` S) = f (LEAST x. x : S)" |
|
1051 |
-- {* Courtesy of Stephan Merz *} |
|
1052 |
apply clarify |
|
1053 |
apply (erule_tac P = "%x. x : S" in LeastI2_order, fast) |
|
1054 |
apply (rule LeastI2_order) |
|
1055 |
apply (auto elim: monoD intro!: order_antisym) |
|
1056 |
done |
|
1057 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1058 |
lemma Least_equality: |
23212 | 1059 |
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" |
1060 |
apply (simp add: Least_def) |
|
1061 |
apply (rule the_equality) |
|
1062 |
apply (auto intro!: order_antisym) |
|
1063 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1064 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1065 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
23212 | 1066 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1067 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1068 |
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" |
23212 | 1069 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1070 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1071 |
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" |
23212 | 1072 |
apply (simp add: min_def) |
1073 |
apply (blast intro: order_antisym) |
|
1074 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1075 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1076 |
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" |
23212 | 1077 |
apply (simp add: max_def) |
1078 |
apply (blast intro: order_antisym) |
|
1079 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1080 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1081 |
lemma min_of_mono: |
23212 | 1082 |
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)" |
1083 |
by (simp add: min_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1084 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1085 |
lemma max_of_mono: |
23212 | 1086 |
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)" |
1087 |
by (simp add: max_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
1088 |
|
22548 | 1089 |
|
1090 |
subsection {* legacy ML bindings *} |
|
21673 | 1091 |
|
1092 |
ML {* |
|
22548 | 1093 |
val monoI = @{thm monoI}; |
22886 | 1094 |
*} |
21673 | 1095 |
|
15524 | 1096 |
end |