author | haftmann |
Fri, 20 Jul 2007 14:28:25 +0200 | |
changeset 23881 | 851c74f1bb69 |
parent 23417 | 42c1a89b45c1 |
child 23948 | 261bd4678076 |
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/quasi.ML"*) |
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"~~/src/Provers/order.ML" |
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begin |
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subsection {* Partial orders *} |
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|
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class order = ord + |
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assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" |
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and order_refl [iff]: "x \<sqsubseteq> x" |
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and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z" |
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assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" |
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begin |
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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"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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"max a b = (if a \<^loc>\<le> b then b else a)" |
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lemmas (in -) min_def [code func, code unfold, code inline del] = min_def [folded ord_class.min] |
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lemmas (in -) max_def [code func, code unfold, code inline del] = max_def [folded ord_class.max] |
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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: |
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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: |
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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 {* Name duplicates -- including min/max interpretation *} |
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lemmas order_less_le = less_le |
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lemmas order_eq_refl = order_class.eq_refl |
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lemmas order_less_irrefl = order_class.less_irrefl |
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lemmas order_le_less = order_class.le_less |
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lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq |
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lemmas order_less_imp_le = order_class.less_imp_le |
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lemmas order_less_imp_not_eq = order_class.less_imp_not_eq |
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lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 |
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lemmas order_neq_le_trans = order_class.neq_le_trans |
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lemmas order_le_neq_trans = order_class.le_neq_trans |
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lemmas order_antisym = antisym |
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lemmas order_less_not_sym = order_class.less_not_sym |
256 |
lemmas order_less_asym = order_class.less_asym |
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lemmas order_eq_iff = order_class.eq_iff |
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lemmas order_antisym_conv = order_class.antisym_conv |
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lemmas order_less_trans = order_class.less_trans |
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lemmas order_le_less_trans = order_class.le_less_trans |
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lemmas order_less_le_trans = order_class.less_le_trans |
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lemmas order_less_imp_not_less = order_class.less_imp_not_less |
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lemmas order_less_imp_triv = order_class.less_imp_triv |
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lemmas order_less_asym' = order_class.less_asym' |
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lemmas linorder_linear = linear |
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lemmas linorder_less_linear = linorder_class.less_linear |
268 |
lemmas linorder_le_less_linear = linorder_class.le_less_linear |
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lemmas linorder_le_cases = linorder_class.le_cases |
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lemmas linorder_not_less = linorder_class.not_less |
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lemmas linorder_not_le = linorder_class.not_le |
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lemmas linorder_neq_iff = linorder_class.neq_iff |
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lemmas linorder_neqE = linorder_class.neqE |
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lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 |
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lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 |
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lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 |
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lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [folded ord_class.min] |
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lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [folded ord_class.max] |
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lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [folded ord_class.min] |
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lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [folded ord_class.max] |
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lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [folded ord_class.min] |
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lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [folded ord_class.max] |
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lemmas split_min = linorder_class.split_min [folded ord_class.min] |
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lemmas split_max = linorder_class.split_max [folded ord_class.max] |
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subsection {* Reasoning tools setup *} |
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ML {* |
291 |
local |
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fun decomp_gen sort thy (Trueprop $ t) = |
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let |
295 |
fun of_sort t = |
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let |
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val T = type_of t |
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in |
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(* exclude numeric types: linear arithmetic subsumes transitivity *) |
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T <> HOLogic.natT andalso T <> HOLogic.intT |
301 |
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) |
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end; |
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fun dec (Const (@{const_name Not}, _) $ t) = (case dec t |
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of NONE => NONE |
305 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
22916 | 306 |
| dec (Const (@{const_name "op ="}, _) $ t1 $ t2) = |
21248 | 307 |
if of_sort t1 |
308 |
then SOME (t1, "=", t2) |
|
309 |
else NONE |
|
23881 | 310 |
| dec (Const (@{const_name HOL.less_eq}, _) $ t1 $ t2) = |
21248 | 311 |
if of_sort t1 |
312 |
then SOME (t1, "<=", t2) |
|
313 |
else NONE |
|
23881 | 314 |
| dec (Const (@{const_name HOL.less}, _) $ t1 $ t2) = |
21248 | 315 |
if of_sort t1 |
316 |
then SOME (t1, "<", t2) |
|
317 |
else NONE |
|
318 |
| dec _ = NONE; |
|
21091 | 319 |
in dec t end; |
320 |
||
321 |
in |
|
322 |
||
22841 | 323 |
(* sorry - there is no preorder class |
21248 | 324 |
structure Quasi_Tac = Quasi_Tac_Fun ( |
325 |
struct |
|
326 |
val le_trans = thm "order_trans"; |
|
327 |
val le_refl = thm "order_refl"; |
|
328 |
val eqD1 = thm "order_eq_refl"; |
|
329 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
330 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
331 |
val less_imp_le = thm "order_less_imp_le"; |
|
332 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
333 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
334 |
val less_imp_neq = thm "less_imp_neq"; |
|
22738 | 335 |
val decomp_trans = decomp_gen ["Orderings.preorder"]; |
336 |
val decomp_quasi = decomp_gen ["Orderings.preorder"]; |
|
22841 | 337 |
end);*) |
21091 | 338 |
|
339 |
structure Order_Tac = Order_Tac_Fun ( |
|
21248 | 340 |
struct |
341 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
342 |
val le_refl = thm "order_refl"; |
|
343 |
val less_imp_le = thm "order_less_imp_le"; |
|
344 |
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; |
|
345 |
val not_leI = thm "linorder_not_le" RS thm "iffD2"; |
|
346 |
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; |
|
347 |
val not_leD = thm "linorder_not_le" RS thm "iffD1"; |
|
348 |
val eqI = thm "order_antisym"; |
|
349 |
val eqD1 = thm "order_eq_refl"; |
|
350 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
351 |
val less_trans = thm "order_less_trans"; |
|
352 |
val less_le_trans = thm "order_less_le_trans"; |
|
353 |
val le_less_trans = thm "order_le_less_trans"; |
|
354 |
val le_trans = thm "order_trans"; |
|
355 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
356 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
357 |
val less_imp_neq = thm "less_imp_neq"; |
|
358 |
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; |
|
359 |
val not_sym = thm "not_sym"; |
|
360 |
val decomp_part = decomp_gen ["Orderings.order"]; |
|
361 |
val decomp_lin = decomp_gen ["Orderings.linorder"]; |
|
362 |
end); |
|
21091 | 363 |
|
364 |
end; |
|
365 |
*} |
|
366 |
||
21083 | 367 |
setup {* |
368 |
let |
|
369 |
||
370 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 371 |
|
21083 | 372 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
373 |
let val prems = prems_of_ss ss; |
|
22916 | 374 |
val less = Const (@{const_name less}, T); |
21083 | 375 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
376 |
in case find_first (prp t) prems of |
|
377 |
NONE => |
|
378 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
379 |
in case find_first (prp t) prems of |
|
380 |
NONE => NONE |
|
22738 | 381 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1})) |
21083 | 382 |
end |
22738 | 383 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv})) |
21083 | 384 |
end |
385 |
handle THM _ => NONE; |
|
15524 | 386 |
|
21083 | 387 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
388 |
let val prems = prems_of_ss ss; |
|
22916 | 389 |
val le = Const (@{const_name less_eq}, T); |
21083 | 390 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
391 |
in case find_first (prp t) prems of |
|
392 |
NONE => |
|
393 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
394 |
in case find_first (prp t) prems of |
|
395 |
NONE => NONE |
|
22738 | 396 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3})) |
21083 | 397 |
end |
22738 | 398 |
| SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2})) |
21083 | 399 |
end |
400 |
handle THM _ => NONE; |
|
15524 | 401 |
|
21248 | 402 |
fun add_simprocs procs thy = |
403 |
(Simplifier.change_simpset_of thy (fn ss => ss |
|
404 |
addsimprocs (map (fn (name, raw_ts, proc) => |
|
405 |
Simplifier.simproc thy name raw_ts proc)) procs); thy); |
|
406 |
fun add_solver name tac thy = |
|
407 |
(Simplifier.change_simpset_of thy (fn ss => ss addSolver |
|
408 |
(mk_solver name (K tac))); thy); |
|
21083 | 409 |
|
410 |
in |
|
21248 | 411 |
add_simprocs [ |
412 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
413 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
414 |
] |
|
415 |
#> add_solver "Trans_linear" Order_Tac.linear_tac |
|
416 |
#> add_solver "Trans_partial" Order_Tac.partial_tac |
|
417 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
|
418 |
speed up, but the reasoning strength appears to be not higher (at least |
|
419 |
no breaking of additional proofs in the entire HOL distribution, as |
|
420 |
of 5 March 2004, was observed). *) |
|
21083 | 421 |
end |
422 |
*} |
|
15524 | 423 |
|
424 |
||
21083 | 425 |
subsection {* Bounded quantifiers *} |
426 |
||
427 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
428 |
"_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
|
429 |
"_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
|
430 |
"_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
|
431 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 432 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
433 |
"_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
|
434 |
"_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
|
435 |
"_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
|
436 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 437 |
|
438 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
439 |
"_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
|
440 |
"_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
|
441 |
"_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
|
442 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 443 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
444 |
"_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
|
445 |
"_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
|
446 |
"_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
|
447 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 448 |
|
449 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
450 |
"_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
|
451 |
"_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
|
452 |
"_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
|
453 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 454 |
|
455 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
456 |
"_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
|
457 |
"_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
|
458 |
"_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
|
459 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 460 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
461 |
"_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
|
462 |
"_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
|
463 |
"_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
|
464 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 465 |
|
466 |
translations |
|
467 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
468 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
469 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
470 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
471 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
472 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
473 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
474 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
475 |
||
476 |
print_translation {* |
|
477 |
let |
|
22916 | 478 |
val All_binder = Syntax.binder_name @{const_syntax All}; |
479 |
val Ex_binder = Syntax.binder_name @{const_syntax Ex}; |
|
22377 | 480 |
val impl = @{const_syntax "op -->"}; |
481 |
val conj = @{const_syntax "op &"}; |
|
22916 | 482 |
val less = @{const_syntax less}; |
483 |
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
|
484 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
485 |
val trans = |
21524 | 486 |
[((All_binder, impl, less), ("_All_less", "_All_greater")), |
487 |
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
|
488 |
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), |
|
489 |
((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
|
490 |
|
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
491 |
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
|
492 |
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
|
493 |
| _ => false |
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
494 |
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
|
495 |
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
|
496 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
497 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
498 |
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
|
499 |
(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
|
500 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
501 |
| SOME (l, g) => |
22344
eddeabf16b5d
Fixed print translations for quantifiers a la "ALL x>=t. P x". These used
krauss
parents:
22316
diff
changeset
|
502 |
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
|
503 |
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
|
504 |
else raise Match) |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
505 |
| _ => raise Match); |
21524 | 506 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 507 |
*} |
508 |
||
509 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
510 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
511 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
512 |
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" |
23212 | 513 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
514 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
515 |
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" |
23212 | 516 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
517 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
518 |
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" |
23212 | 519 |
by (rule subst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
520 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
521 |
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" |
23212 | 522 |
by (rule ssubst) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
523 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
524 |
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
|
525 |
(!!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
|
526 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
527 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
528 |
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
|
529 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
530 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
531 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
532 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
533 |
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
|
534 |
(!!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
|
535 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
536 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
537 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
538 |
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
|
539 |
finally (order_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
540 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
541 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
542 |
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
|
543 |
(!!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
|
544 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
545 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
546 |
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
|
547 |
also assume "f b < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
548 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
549 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
550 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
551 |
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
|
552 |
(!!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
|
553 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
554 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
555 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
556 |
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
|
557 |
finally (order_le_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
558 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
559 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
560 |
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
|
561 |
(!!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
|
562 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
563 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
564 |
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
|
565 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
566 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
567 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
568 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
569 |
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
|
570 |
(!!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
|
571 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
572 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
573 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
574 |
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
|
575 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
576 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
577 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
578 |
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
|
579 |
(!!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
|
580 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
581 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
582 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
583 |
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
|
584 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
585 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
586 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
587 |
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
|
588 |
(!!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
|
589 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
590 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
591 |
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
|
592 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
593 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
594 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
595 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
596 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
597 |
(!!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
|
598 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
599 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
600 |
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
|
601 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
602 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
603 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
604 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
605 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
606 |
(!!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
|
607 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
608 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
609 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
610 |
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
|
611 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
612 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
613 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
614 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
615 |
(!!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
|
616 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
617 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
618 |
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
|
619 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
620 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
621 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
622 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
623 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
624 |
(!!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
|
625 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
626 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
627 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
628 |
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
|
629 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
630 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
631 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
632 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
633 |
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
|
634 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
635 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
636 |
lemmas order_trans_rules [trans] = |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
637 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
638 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
639 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
640 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
641 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
642 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
643 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
644 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
645 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
646 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
647 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
648 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
649 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
650 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
651 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
652 |
mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
653 |
order_neq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
654 |
order_le_neq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
655 |
order_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
656 |
order_less_asym' |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
657 |
order_le_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
658 |
order_less_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
659 |
order_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
660 |
order_antisym |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
661 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
662 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
663 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
664 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
665 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
666 |
|
21083 | 667 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
668 |
(* FIXME cleanup *) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
669 |
|
21083 | 670 |
text {* These support proving chains of decreasing inequalities |
671 |
a >= b >= c ... in Isar proofs. *} |
|
672 |
||
673 |
lemma xt1: |
|
674 |
"a = b ==> b > c ==> a > c" |
|
675 |
"a > b ==> b = c ==> a > c" |
|
676 |
"a = b ==> b >= c ==> a >= c" |
|
677 |
"a >= b ==> b = c ==> a >= c" |
|
678 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
679 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
680 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
681 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
23417 | 682 |
"(a::'a::order) > b ==> b > a ==> P" |
21083 | 683 |
"(x::'a::order) > y ==> y > z ==> x > z" |
684 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
685 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
686 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
687 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
688 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
689 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
690 |
by auto |
|
691 |
||
692 |
lemma xt2: |
|
693 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
694 |
by (subgoal_tac "f b >= f c", force, force) |
|
695 |
||
696 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
697 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
698 |
by (subgoal_tac "f a >= f b", force, force) |
|
699 |
||
700 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
701 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
702 |
by (subgoal_tac "f b >= f c", force, force) |
|
703 |
||
704 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
705 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
706 |
by (subgoal_tac "f a > f b", force, force) |
|
707 |
||
708 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
709 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
710 |
by (subgoal_tac "f b > f c", force, force) |
|
711 |
||
712 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
713 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
714 |
by (subgoal_tac "f a >= f b", force, force) |
|
715 |
||
716 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
717 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
718 |
by (subgoal_tac "f b > f c", force, force) |
|
719 |
||
720 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
721 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
722 |
by (subgoal_tac "f a > f b", force, force) |
|
723 |
||
724 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
725 |
||
726 |
(* |
|
727 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
728 |
for the wrong thing in an Isar proof. |
|
729 |
||
730 |
The extra transitivity rules can be used as follows: |
|
731 |
||
732 |
lemma "(a::'a::order) > z" |
|
733 |
proof - |
|
734 |
have "a >= b" (is "_ >= ?rhs") |
|
735 |
sorry |
|
736 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
737 |
sorry |
|
738 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
739 |
sorry |
|
740 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
741 |
sorry |
|
742 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
743 |
sorry |
|
744 |
also (xtrans) have "?rhs > z" |
|
745 |
sorry |
|
746 |
finally (xtrans) show ?thesis . |
|
747 |
qed |
|
748 |
||
749 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
750 |
leave out the "(xtrans)" above. |
|
751 |
*) |
|
752 |
||
21546 | 753 |
subsection {* Order on bool *} |
754 |
||
22886 | 755 |
instance bool :: order |
21546 | 756 |
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" |
757 |
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" |
|
22916 | 758 |
by intro_classes (auto simp add: le_bool_def less_bool_def) |
21546 | 759 |
|
760 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
23212 | 761 |
by (simp add: le_bool_def) |
21546 | 762 |
|
763 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
23212 | 764 |
by (simp add: le_bool_def) |
21546 | 765 |
|
766 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
23212 | 767 |
by (simp add: le_bool_def) |
21546 | 768 |
|
769 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
23212 | 770 |
by (simp add: le_bool_def) |
21546 | 771 |
|
22348 | 772 |
lemma [code func]: |
773 |
"False \<le> b \<longleftrightarrow> True" |
|
774 |
"True \<le> b \<longleftrightarrow> b" |
|
775 |
"False < b \<longleftrightarrow> b" |
|
776 |
"True < b \<longleftrightarrow> False" |
|
777 |
unfolding le_bool_def less_bool_def by simp_all |
|
778 |
||
22424 | 779 |
|
23881 | 780 |
subsection {* Order on sets *} |
781 |
||
782 |
instance set :: (type) order |
|
783 |
by (intro_classes, |
|
784 |
(assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+) |
|
785 |
||
786 |
lemmas basic_trans_rules [trans] = |
|
787 |
order_trans_rules set_rev_mp set_mp |
|
788 |
||
789 |
||
790 |
subsection {* Order on functions *} |
|
791 |
||
792 |
instance "fun" :: (type, ord) ord |
|
793 |
le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x" |
|
794 |
less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" .. |
|
795 |
||
796 |
lemmas [code func del] = le_fun_def less_fun_def |
|
797 |
||
798 |
instance "fun" :: (type, order) order |
|
799 |
by default |
|
800 |
(auto simp add: le_fun_def less_fun_def expand_fun_eq |
|
801 |
intro: order_trans order_antisym) |
|
802 |
||
803 |
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g" |
|
804 |
unfolding le_fun_def by simp |
|
805 |
||
806 |
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P" |
|
807 |
unfolding le_fun_def by simp |
|
808 |
||
809 |
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x" |
|
810 |
unfolding le_fun_def by simp |
|
811 |
||
812 |
text {* |
|
813 |
Handy introduction and elimination rules for @{text "\<le>"} |
|
814 |
on unary and binary predicates |
|
815 |
*} |
|
816 |
||
817 |
lemma predicate1I [Pure.intro!, intro!]: |
|
818 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
|
819 |
shows "P \<le> Q" |
|
820 |
apply (rule le_funI) |
|
821 |
apply (rule le_boolI) |
|
822 |
apply (rule PQ) |
|
823 |
apply assumption |
|
824 |
done |
|
825 |
||
826 |
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
|
827 |
apply (erule le_funE) |
|
828 |
apply (erule le_boolE) |
|
829 |
apply assumption+ |
|
830 |
done |
|
831 |
||
832 |
lemma predicate2I [Pure.intro!, intro!]: |
|
833 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
|
834 |
shows "P \<le> Q" |
|
835 |
apply (rule le_funI)+ |
|
836 |
apply (rule le_boolI) |
|
837 |
apply (rule PQ) |
|
838 |
apply assumption |
|
839 |
done |
|
840 |
||
841 |
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
|
842 |
apply (erule le_funE)+ |
|
843 |
apply (erule le_boolE) |
|
844 |
apply assumption+ |
|
845 |
done |
|
846 |
||
847 |
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" |
|
848 |
by (rule predicate1D) |
|
849 |
||
850 |
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" |
|
851 |
by (rule predicate2D) |
|
852 |
||
853 |
||
854 |
subsection {* Monotonicity, least value operator and min/max *} |
|
21083 | 855 |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
856 |
locale mono = |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
857 |
fixes f |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
858 |
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
|
859 |
|
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
860 |
lemmas monoI [intro?] = mono.intro |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
861 |
and monoD [dest?] = mono.mono |
21083 | 862 |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
lemma LeastI2_order: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
"[| P (x::'a::order); |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
!!y. P y ==> x <= y; |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
866 |
!!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
|
867 |
==> Q (Least P)" |
23212 | 868 |
apply (unfold Least_def) |
869 |
apply (rule theI2) |
|
870 |
apply (blast intro: order_antisym)+ |
|
871 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
872 |
|
23881 | 873 |
lemma Least_mono: |
874 |
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y |
|
875 |
==> (LEAST y. y : f ` S) = f (LEAST x. x : S)" |
|
876 |
-- {* Courtesy of Stephan Merz *} |
|
877 |
apply clarify |
|
878 |
apply (erule_tac P = "%x. x : S" in LeastI2_order, fast) |
|
879 |
apply (rule LeastI2_order) |
|
880 |
apply (auto elim: monoD intro!: order_antisym) |
|
881 |
done |
|
882 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
883 |
lemma Least_equality: |
23212 | 884 |
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" |
885 |
apply (simp add: Least_def) |
|
886 |
apply (rule the_equality) |
|
887 |
apply (auto intro!: order_antisym) |
|
888 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
889 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
890 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
23212 | 891 |
by (simp add: min_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
892 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
893 |
lemma max_leastL: "(!!x. least <= x) ==> max least x = x" |
23212 | 894 |
by (simp add: max_def) |
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
895 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
896 |
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" |
23212 | 897 |
apply (simp add: min_def) |
898 |
apply (blast intro: order_antisym) |
|
899 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
900 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
901 |
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" |
23212 | 902 |
apply (simp add: max_def) |
903 |
apply (blast intro: order_antisym) |
|
904 |
done |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
905 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
906 |
lemma min_of_mono: |
23212 | 907 |
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)" |
908 |
by (simp add: min_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
909 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
910 |
lemma max_of_mono: |
23212 | 911 |
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)" |
912 |
by (simp add: max_def) |
|
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
913 |
|
22548 | 914 |
|
915 |
subsection {* legacy ML bindings *} |
|
21673 | 916 |
|
917 |
ML {* |
|
22548 | 918 |
val monoI = @{thm monoI}; |
22886 | 919 |
*} |
21673 | 920 |
|
15524 | 921 |
end |