author | wenzelm |
Thu, 09 Nov 2006 11:58:13 +0100 | |
changeset 21259 | 63ab016c99ca |
parent 21248 | 3fd22b0939ff |
child 21329 | 7338206d75f1 |
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 {* Abstract orderings *} |
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theory Orderings |
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imports Code_Generator |
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begin |
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section {* Abstract orderings *} |
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subsection {* Order signatures *} |
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class ord = |
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fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" |
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begin |
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notation |
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less_eq ("op \<^loc><=") |
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less_eq ("(_/ \<^loc><= _)" [51, 51] 50) |
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less ("op \<^loc><") |
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less ("(_/ \<^loc>< _)" [51, 51] 50) |
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notation (xsymbols) |
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less_eq ("op \<^loc>\<le>") |
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) |
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notation (HTML output) |
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less_eq ("op \<^loc>\<le>") |
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less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50) |
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abbreviation (input) |
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greater (infix "\<^loc>>" 50) |
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"x \<^loc>> y \<equiv> y \<^loc>< x" |
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greater_eq (infix "\<^loc>>=" 50) |
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"x \<^loc>>= y \<equiv> y \<^loc><= x" |
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notation (xsymbols) |
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greater_eq (infix "\<^loc>\<ge>" 50) |
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end |
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notation |
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less_eq ("op <=") |
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less_eq ("(_/ <= _)" [51, 51] 50) |
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less ("op <") |
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less ("(_/ < _)" [51, 51] 50) |
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notation (xsymbols) |
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parents:
19637
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less_eq ("op \<le>") |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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notation (HTML output) |
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less_eq ("op \<le>") |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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abbreviation (input) |
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greater (infix ">" 50) |
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"x > y \<equiv> y < x" |
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greater_eq (infix ">=" 50) |
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"x >= y \<equiv> y <= x" |
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notation (xsymbols) |
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greater_eq (infix "\<ge>" 50) |
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subsection {* Partial orderings *} |
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locale partial_order = |
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fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50) |
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fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubset>" 50) |
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parents:
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assumes refl [iff]: "x \<sqsubseteq> x" |
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and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z" |
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and antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" |
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and less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y" |
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begin |
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abbreviation (input) |
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greater_eq (infixl "\<sqsupseteq>" 50) |
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"x \<sqsupseteq> y \<equiv> y \<sqsubseteq> x" |
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abbreviation (input) |
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greater (infixl "\<sqsupset>" 50) |
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"x \<sqsupset> y \<equiv> y \<sqsubset> x" |
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end |
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parents:
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axclass order < ord |
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order_refl [iff]: "x <= x" |
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order_trans: "x <= y ==> y <= z ==> x <= z" |
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order_antisym: "x <= y ==> y <= x ==> x = y" |
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order_less_le: "(x < y) = (x <= y & x ~= y)" |
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interpretation order: |
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partial_order ["op \<le> \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>order \<Rightarrow> 'a \<Rightarrow> bool"] |
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apply(rule partial_order.intro) |
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apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym, rule order_less_le) |
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done |
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context partial_order |
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begin |
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text {* Reflexivity. *} |
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lemma eq_refl: "x = y \<Longrightarrow> x \<sqsubseteq> y" |
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-- {* This form is useful with the classical reasoner. *} |
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by (erule ssubst) (rule refl) |
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lemma less_irrefl [iff]: "\<not> x \<sqsubset> x" |
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by (simp add: less_le) |
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lemma le_less: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubset> 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 \<sqsubseteq> y \<Longrightarrow> x \<sqsubset> y \<or> x = y" |
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unfolding less_le by blast |
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lemma less_imp_le: "x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y" |
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unfolding less_le by blast |
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text {* Asymmetry. *} |
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lemma less_not_sym: "x \<sqsubset> y \<Longrightarrow> \<not> (y \<sqsubset> x)" |
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by (simp add: less_le antisym) |
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lemma less_asym: "x \<sqsubset> y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<sqsubset> 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 \<sqsubseteq> y \<and> y \<sqsubseteq> x" |
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by (blast intro: antisym) |
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lemma antisym_conv: "y \<sqsubseteq> x \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y" |
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by (blast intro: antisym) |
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lemma less_imp_neq: "x \<sqsubset> 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: "\<lbrakk> x \<sqsubset> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" |
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by (simp add: less_le) (blast intro: trans antisym) |
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lemma le_less_trans: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" |
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by (simp add: less_le) (blast intro: trans antisym) |
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lemma less_le_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z" |
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by (simp add: less_le) (blast intro: 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 \<sqsubset> y \<Longrightarrow> (\<not> y \<sqsubset> x) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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lemma less_imp_triv: "x \<sqsubset> y \<Longrightarrow> (y \<sqsubset> x \<longrightarrow> P) \<longleftrightarrow> True" |
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by (blast elim: less_asym) |
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lemma less_imp_not_eq: "x \<sqsubset> y \<Longrightarrow> (x = y) \<longleftrightarrow> False" |
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by auto |
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lemma less_imp_not_eq2: "x \<sqsubset> 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: "\<lbrakk> a \<noteq> b; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b" |
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by (simp add: less_le) |
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lemma le_neq_trans: "\<lbrakk> a \<sqsubseteq> b; a \<noteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b" |
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by (simp add: less_le) |
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lemma less_asym': "\<lbrakk> a \<sqsubset> b; b \<sqsubset> a \<rbrakk> \<Longrightarrow> P" |
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by (rule less_asym) |
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end |
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subsection {* Linear (total) orderings *} |
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locale linear_order = partial_order + |
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assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x" |
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axclass linorder < order |
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linorder_linear: "x <= y | y <= x" |
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interpretation linorder: |
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linear_order ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"] |
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by unfold_locales (rule linorder_linear) |
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context linear_order |
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begin |
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lemma trichotomy: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x" |
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unfolding less_le using less_le linear by blast |
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lemma le_less_linear: "x \<sqsubseteq> y \<or> y \<sqsubset> x" |
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by (simp add: le_less trichotomy) |
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lemma le_cases [case_names le ge]: |
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"\<lbrakk> x \<sqsubseteq> y \<Longrightarrow> P; y \<sqsubseteq> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
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using linear by blast |
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lemma cases [case_names less equal greater]: |
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"\<lbrakk> x \<sqsubset> y \<Longrightarrow> P; x = y \<Longrightarrow> P; y \<sqsubset> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" |
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using trichotomy by blast |
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lemma not_less: "\<not> x \<sqsubset> y \<longleftrightarrow> y \<sqsubseteq> 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_le: "\<not> x \<sqsubseteq> y \<longleftrightarrow> y \<sqsubset> 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 \<sqsubset> y \<or> y \<sqsubset> x" |
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by (cut_tac x = x and y = y in trichotomy, auto) |
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lemma neqE: "\<lbrakk> x \<noteq> y; x \<sqsubset> y \<Longrightarrow> R; y \<sqsubset> x \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" |
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by (simp add: neq_iff) blast |
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lemma antisym_conv1: "\<not> x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> 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 \<sqsubseteq> y \<Longrightarrow> \<not> x \<sqsubset> 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 \<sqsubset> x \<Longrightarrow> \<not> x \<sqsubset> 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 \<sqsubset> y \<Longrightarrow> y \<sqsubseteq> x" |
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unfolding not_less . |
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lemma leD: "y \<sqsubseteq> x \<Longrightarrow> \<not> x \<sqsubset> 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 \<sqsubseteq> x \<Longrightarrow> x \<sqsubset> y" |
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unfolding not_le . |
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end |
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subsection {* Name duplicates *} |
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lemmas order_eq_refl [where 'b = "?'a::order"] = order.eq_refl |
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lemmas order_less_irrefl [where 'b = "?'a::order"] = order.less_irrefl |
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lemmas order_le_less [where 'b = "?'a::order"] = order.le_less |
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lemmas order_le_imp_less_or_eq [where 'b = "?'a::order"] = order.le_imp_less_or_eq |
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lemmas order_less_imp_le [where 'b = "?'a::order"] = order.less_imp_le |
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lemmas order_less_not_sym [where 'b = "?'a::order"] = order.less_not_sym |
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lemmas order_less_asym [where 'b = "?'a::order"] = order.less_asym |
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lemmas order_eq_iff [where 'b = "?'a::order"] = order.eq_iff |
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lemmas order_antisym_conv [where 'b = "?'a::order"] = order.antisym_conv |
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lemmas less_imp_neq [where 'b = "?'a::order"] = order.less_imp_neq |
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lemmas order_less_trans [where 'b = "?'a::order"] = order.less_trans |
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lemmas order_le_less_trans [where 'b = "?'a::order"] = order.le_less_trans |
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lemmas order_less_le_trans [where 'b = "?'a::order"] = order.less_le_trans |
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lemmas order_less_imp_not_less [where 'b = "?'a::order"] = order.less_imp_not_less |
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lemmas order_less_imp_triv [where 'b = "?'a::order"] = order.less_imp_triv |
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lemmas order_less_imp_not_eq [where 'b = "?'a::order"] = order.less_imp_not_eq |
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lemmas order_less_imp_not_eq2 [where 'b = "?'a::order"] = order.less_imp_not_eq2 |
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lemmas order_neq_le_trans [where 'b = "?'a::order"] = order.neq_le_trans |
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lemmas order_le_neq_trans [where 'b = "?'a::order"] = order.le_neq_trans |
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lemmas order_less_asym' [where 'b = "?'a::order"] = order.less_asym' |
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lemmas linorder_less_linear [where 'b = "?'a::linorder"] = linorder.trichotomy |
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lemmas linorder_le_less_linear [where 'b = "?'a::linorder"] = linorder.le_less_linear |
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lemmas linorder_le_cases [where 'b = "?'a::linorder"] = linorder.le_cases |
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lemmas linorder_cases [where 'b = "?'a::linorder"] = linorder.cases |
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lemmas linorder_not_less [where 'b = "?'a::linorder"] = linorder.not_less |
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lemmas linorder_not_le [where 'b = "?'a::linorder"] = linorder.not_le |
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lemmas linorder_neq_iff [where 'b = "?'a::linorder"] = linorder.neq_iff |
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lemmas linorder_neqE [where 'b = "?'a::linorder"] = linorder.neqE |
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lemmas linorder_antisym_conv1 [where 'b = "?'a::linorder"] = linorder.antisym_conv1 |
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lemmas linorder_antisym_conv2 [where 'b = "?'a::linorder"] = linorder.antisym_conv2 |
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lemmas linorder_antisym_conv3 [where 'b = "?'a::linorder"] = linorder.antisym_conv3 |
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lemmas leI [where 'b = "?'a::linorder"] = linorder.leI |
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lemmas leD [where 'b = "?'a::linorder"] = linorder.leD |
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lemmas not_leE [where 'b = "?'a::linorder"] = linorder.not_leE |
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subsection {* Reasoning tools setup *} |
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||
21091 | 293 |
ML {* |
294 |
local |
|
295 |
||
296 |
fun decomp_gen sort thy (Trueprop $ t) = |
|
21248 | 297 |
let |
298 |
fun of_sort t = |
|
299 |
let |
|
300 |
val T = type_of t |
|
301 |
in |
|
21091 | 302 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
21248 | 303 |
T <> HOLogic.natT andalso T <> HOLogic.intT |
304 |
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) |
|
305 |
end; |
|
306 |
fun dec (Const ("Not", _) $ t) = (case dec t |
|
307 |
of NONE => NONE |
|
308 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
309 |
| dec (Const ("op =", _) $ t1 $ t2) = |
|
310 |
if of_sort t1 |
|
311 |
then SOME (t1, "=", t2) |
|
312 |
else NONE |
|
313 |
| dec (Const ("Orderings.less_eq", _) $ t1 $ t2) = |
|
314 |
if of_sort t1 |
|
315 |
then SOME (t1, "<=", t2) |
|
316 |
else NONE |
|
317 |
| dec (Const ("Orderings.less", _) $ t1 $ t2) = |
|
318 |
if of_sort t1 |
|
319 |
then SOME (t1, "<", t2) |
|
320 |
else NONE |
|
321 |
| dec _ = NONE; |
|
21091 | 322 |
in dec t end; |
323 |
||
324 |
in |
|
325 |
||
326 |
(* The setting up of Quasi_Tac serves as a demo. Since there is no |
|
327 |
class for quasi orders, the tactics Quasi_Tac.trans_tac and |
|
328 |
Quasi_Tac.quasi_tac are not of much use. *) |
|
329 |
||
21248 | 330 |
structure Quasi_Tac = Quasi_Tac_Fun ( |
331 |
struct |
|
332 |
val le_trans = thm "order_trans"; |
|
333 |
val le_refl = thm "order_refl"; |
|
334 |
val eqD1 = thm "order_eq_refl"; |
|
335 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
336 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
337 |
val less_imp_le = thm "order_less_imp_le"; |
|
338 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
339 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
340 |
val less_imp_neq = thm "less_imp_neq"; |
|
341 |
val decomp_trans = decomp_gen ["Orderings.order"]; |
|
342 |
val decomp_quasi = decomp_gen ["Orderings.order"]; |
|
343 |
end); |
|
21091 | 344 |
|
345 |
structure Order_Tac = Order_Tac_Fun ( |
|
21248 | 346 |
struct |
347 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
348 |
val le_refl = thm "order_refl"; |
|
349 |
val less_imp_le = thm "order_less_imp_le"; |
|
350 |
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; |
|
351 |
val not_leI = thm "linorder_not_le" RS thm "iffD2"; |
|
352 |
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; |
|
353 |
val not_leD = thm "linorder_not_le" RS thm "iffD1"; |
|
354 |
val eqI = thm "order_antisym"; |
|
355 |
val eqD1 = thm "order_eq_refl"; |
|
356 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
357 |
val less_trans = thm "order_less_trans"; |
|
358 |
val less_le_trans = thm "order_less_le_trans"; |
|
359 |
val le_less_trans = thm "order_le_less_trans"; |
|
360 |
val le_trans = thm "order_trans"; |
|
361 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
362 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
363 |
val less_imp_neq = thm "less_imp_neq"; |
|
364 |
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; |
|
365 |
val not_sym = thm "not_sym"; |
|
366 |
val decomp_part = decomp_gen ["Orderings.order"]; |
|
367 |
val decomp_lin = decomp_gen ["Orderings.linorder"]; |
|
368 |
end); |
|
21091 | 369 |
|
370 |
end; |
|
371 |
*} |
|
372 |
||
21083 | 373 |
setup {* |
374 |
let |
|
375 |
||
376 |
val order_antisym_conv = thm "order_antisym_conv" |
|
377 |
val linorder_antisym_conv1 = thm "linorder_antisym_conv1" |
|
378 |
val linorder_antisym_conv2 = thm "linorder_antisym_conv2" |
|
379 |
val linorder_antisym_conv3 = thm "linorder_antisym_conv3" |
|
380 |
||
381 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 382 |
|
21083 | 383 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
384 |
let val prems = prems_of_ss ss; |
|
385 |
val less = Const("Orderings.less",T); |
|
386 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
|
387 |
in case find_first (prp t) prems of |
|
388 |
NONE => |
|
389 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
390 |
in case find_first (prp t) prems of |
|
391 |
NONE => NONE |
|
392 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1)) |
|
393 |
end |
|
394 |
| SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv)) |
|
395 |
end |
|
396 |
handle THM _ => NONE; |
|
15524 | 397 |
|
21083 | 398 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
399 |
let val prems = prems_of_ss ss; |
|
400 |
val le = Const("Orderings.less_eq",T); |
|
401 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
|
402 |
in case find_first (prp t) prems of |
|
403 |
NONE => |
|
404 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
405 |
in case find_first (prp t) prems of |
|
406 |
NONE => NONE |
|
407 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3)) |
|
408 |
end |
|
409 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2)) |
|
410 |
end |
|
411 |
handle THM _ => NONE; |
|
15524 | 412 |
|
21248 | 413 |
fun add_simprocs procs thy = |
414 |
(Simplifier.change_simpset_of thy (fn ss => ss |
|
415 |
addsimprocs (map (fn (name, raw_ts, proc) => |
|
416 |
Simplifier.simproc thy name raw_ts proc)) procs); thy); |
|
417 |
fun add_solver name tac thy = |
|
418 |
(Simplifier.change_simpset_of thy (fn ss => ss addSolver |
|
419 |
(mk_solver name (K tac))); thy); |
|
21083 | 420 |
|
421 |
in |
|
21248 | 422 |
add_simprocs [ |
423 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
424 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
425 |
] |
|
426 |
#> add_solver "Trans_linear" Order_Tac.linear_tac |
|
427 |
#> add_solver "Trans_partial" Order_Tac.partial_tac |
|
428 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
|
429 |
speed up, but the reasoning strength appears to be not higher (at least |
|
430 |
no breaking of additional proofs in the entire HOL distribution, as |
|
431 |
of 5 March 2004, was observed). *) |
|
21083 | 432 |
end |
433 |
*} |
|
15524 | 434 |
|
435 |
||
21083 | 436 |
subsection {* Bounded quantifiers *} |
437 |
||
438 |
syntax |
|
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" ("(3ALL _<_./ _)" [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" ("(3EX _<_./ _)" [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" ("(3ALL _<=_./ _)" [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" ("(3EX _<=_./ _)" [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" ("(3ALL _>_./ _)" [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" ("(3EX _>_./ _)" [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" ("(3ALL _>=_./ _)" [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" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 448 |
|
449 |
syntax (xsymbols) |
|
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\<forall>_<_./ _)" [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\<exists>_<_./ _)" [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\<forall>_\<le>_./ _)" [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\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 454 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
455 |
"_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
|
456 |
"_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
|
457 |
"_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
|
458 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 459 |
|
460 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
461 |
"_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
|
462 |
"_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
|
463 |
"_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
|
464 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 465 |
|
466 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
467 |
"_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
|
468 |
"_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
|
469 |
"_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
|
470 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 471 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
472 |
"_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
|
473 |
"_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
|
474 |
"_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
|
475 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 476 |
|
477 |
translations |
|
478 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
479 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
480 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
481 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
482 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
483 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
484 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
485 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
486 |
||
487 |
print_translation {* |
|
488 |
let |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
489 |
val syntax_name = Sign.const_syntax_name (the_context ()); |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
490 |
val impl = syntax_name "op -->"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
491 |
val conj = syntax_name "op &"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
492 |
val less = syntax_name "Orderings.less"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
493 |
val less_eq = syntax_name "Orderings.less_eq"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
494 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
495 |
val trans = |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
496 |
[(("ALL ", impl, less), ("_All_less", "_All_greater")), |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
497 |
(("ALL ", impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
498 |
(("EX ", conj, less), ("_Ex_less", "_Ex_greater")), |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
499 |
(("EX ", conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
500 |
|
21083 | 501 |
fun mk v v' c n P = |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
502 |
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) |
21083 | 503 |
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; |
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
504 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
505 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
506 |
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
|
507 |
(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
|
508 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
509 |
| SOME (l, g) => |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
510 |
(case (t, u) of |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
511 |
(Const ("_bound", _) $ Free (v', _), n) => mk v v' l n P |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
512 |
| (n, Const ("_bound", _) $ Free (v', _)) => mk v v' g n P |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
513 |
| _ => raise Match)) |
f27f12bcafb8
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changeset
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514 |
| _ => raise Match); |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
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parents:
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changeset
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515 |
in [tr' "ALL ", tr' "EX "] end |
21083 | 516 |
*} |
517 |
||
518 |
||
519 |
subsection {* Transitivity reasoning on decreasing inequalities *} |
|
520 |
||
21180
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521 |
(* FIXME cleanup *) |
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fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
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522 |
|
21083 | 523 |
text {* These support proving chains of decreasing inequalities |
524 |
a >= b >= c ... in Isar proofs. *} |
|
525 |
||
526 |
lemma xt1: |
|
527 |
"a = b ==> b > c ==> a > c" |
|
528 |
"a > b ==> b = c ==> a > c" |
|
529 |
"a = b ==> b >= c ==> a >= c" |
|
530 |
"a >= b ==> b = c ==> a >= c" |
|
531 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
532 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
533 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
534 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
535 |
"(a::'a::order) > b ==> b > a ==> ?P" |
|
536 |
"(x::'a::order) > y ==> y > z ==> x > z" |
|
537 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
538 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
539 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
540 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
541 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
542 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
543 |
by auto |
|
544 |
||
545 |
lemma xt2: |
|
546 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
547 |
by (subgoal_tac "f b >= f c", force, force) |
|
548 |
||
549 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
550 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
551 |
by (subgoal_tac "f a >= f b", force, force) |
|
552 |
||
553 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
554 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
555 |
by (subgoal_tac "f b >= f c", force, force) |
|
556 |
||
557 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
558 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
559 |
by (subgoal_tac "f a > f b", force, force) |
|
560 |
||
561 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
562 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
563 |
by (subgoal_tac "f b > f c", force, force) |
|
564 |
||
565 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
566 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
567 |
by (subgoal_tac "f a >= f b", force, force) |
|
568 |
||
569 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
570 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
571 |
by (subgoal_tac "f b > f c", force, force) |
|
572 |
||
573 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
574 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
575 |
by (subgoal_tac "f a > f b", force, force) |
|
576 |
||
577 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
578 |
||
579 |
(* |
|
580 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
581 |
for the wrong thing in an Isar proof. |
|
582 |
||
583 |
The extra transitivity rules can be used as follows: |
|
584 |
||
585 |
lemma "(a::'a::order) > z" |
|
586 |
proof - |
|
587 |
have "a >= b" (is "_ >= ?rhs") |
|
588 |
sorry |
|
589 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
590 |
sorry |
|
591 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
592 |
sorry |
|
593 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
594 |
sorry |
|
595 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
596 |
sorry |
|
597 |
also (xtrans) have "?rhs > z" |
|
598 |
sorry |
|
599 |
finally (xtrans) show ?thesis . |
|
600 |
qed |
|
601 |
||
602 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
603 |
leave out the "(xtrans)" above. |
|
604 |
*) |
|
605 |
||
21216
1c8580913738
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parents:
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diff
changeset
|
606 |
subsection {* Monotonicity, syntactic least value operator and syntactic min/max *} |
21083 | 607 |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
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diff
changeset
|
608 |
locale mono = |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
609 |
fixes f |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
610 |
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
|
611 |
|
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
612 |
lemmas monoI [intro?] = mono.intro |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
613 |
and monoD [dest?] = mono.mono |
21083 | 614 |
|
615 |
constdefs |
|
616 |
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) |
|
617 |
"Least P == THE x. P x & (ALL y. P y --> x <= y)" |
|
618 |
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *} |
|
619 |
||
620 |
constdefs |
|
621 |
min :: "['a::ord, 'a] => 'a" |
|
622 |
"min a b == (if a <= b then a else b)" |
|
623 |
max :: "['a::ord, 'a] => 'a" |
|
624 |
"max a b == (if a <= b then b else a)" |
|
625 |
||
15524 | 626 |
end |