author | haftmann |
Mon, 27 Nov 2006 13:42:33 +0100 | |
changeset 21546 | 268b6bed0cc8 |
parent 21524 | 7843e2fd14a9 |
child 21620 | 3d4bfc7f6363 |
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 HOL |
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begin |
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section {* Abstract orders *} |
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subsection {* Order syntax *} |
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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><=") and |
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less_eq ("(_/ \<^loc><= _)" [51, 51] 50) and |
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less ("op \<^loc><") and |
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less ("(_/ \<^loc>< _)" [51, 51] 50) |
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notation (xsymbols) |
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less_eq ("op \<^loc>\<le>") and |
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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>") and |
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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) where |
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"x \<^loc>> y \<equiv> y \<^loc>< x" |
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abbreviation (input) |
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greater_eq (infix "\<^loc>>=" 50) where |
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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 <=") and |
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less_eq ("(_/ <= _)" [51, 51] 50) and |
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less ("op <") and |
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less ("(_/ < _)" [51, 51] 50) |
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notation (xsymbols) |
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less_eq ("op \<le>") and |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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notation (HTML output) |
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less_eq ("op \<le>") and |
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less_eq ("(_/ \<le> _)" [51, 51] 50) |
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abbreviation (input) |
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greater (infix ">" 50) where |
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"x > y \<equiv> y < x" |
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abbreviation (input) |
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greater_eq (infix ">=" 50) where |
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"x >= y \<equiv> y <= x" |
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notation (xsymbols) |
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greater_eq (infix "\<ge>" 50) |
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subsection {* Quasiorders (preorders) *} |
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locale preorder = |
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fixes less_eq :: "'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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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 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) where |
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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) where |
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"x \<sqsupset> y \<equiv> y \<sqsubset> x" |
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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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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 {* Useful for simplification, but too risky to include by default. *} |
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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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end |
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subsection {* Partial orderings *} |
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locale partial_order = preorder + |
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assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y" |
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context partial_order |
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begin |
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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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text {* Transitivity rules for calculational reasoning *} |
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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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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 unfold_locales |
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apply (rule order_refl) |
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apply (erule (1) order_trans) |
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apply (rule order_less_le) |
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apply (erule (1) order_antisym) |
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done |
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subsection {* Linear (total) orders *} |
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locale linorder = partial_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 \<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 less_linear) |
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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 less_linear 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 less_linear, 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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(* min/max *) |
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definition |
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min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"min a b = (if a \<sqsubseteq> b then a else b)" |
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definition |
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max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"max a b = (if a \<sqsubseteq> b then b else a)" |
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lemma min_le_iff_disj: |
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"min x y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<or> y \<sqsubseteq> z" |
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unfolding min_def using linear by (auto intro: trans) |
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lemma le_max_iff_disj: |
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"z \<sqsubseteq> max x y \<longleftrightarrow> z \<sqsubseteq> x \<or> z \<sqsubseteq> y" |
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unfolding max_def using linear by (auto intro: trans) |
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279 |
lemma min_less_iff_disj: |
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280 |
"min x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<or> y \<sqsubset> z" |
21412 | 281 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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|
282 |
|
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|
283 |
lemma less_max_iff_disj: |
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|
284 |
"z \<sqsubset> max x y \<longleftrightarrow> z \<sqsubset> x \<or> z \<sqsubset> y" |
21412 | 285 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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|
286 |
|
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|
287 |
lemma min_less_iff_conj [simp]: |
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288 |
"z \<sqsubset> min x y \<longleftrightarrow> z \<sqsubset> x \<and> z \<sqsubset> y" |
21412 | 289 |
unfolding min_def le_less using less_linear by (auto intro: less_trans) |
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|
290 |
|
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|
291 |
lemma max_less_iff_conj [simp]: |
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292 |
"max x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<and> y \<sqsubset> z" |
21412 | 293 |
unfolding max_def le_less using less_linear by (auto intro: less_trans) |
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|
294 |
|
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295 |
lemma split_min: |
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296 |
"P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)" |
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297 |
by (simp add: min_def) |
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298 |
|
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299 |
lemma split_max: |
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300 |
"P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)" |
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by (simp add: max_def) |
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302 |
|
21248 | 303 |
end |
304 |
||
21329 | 305 |
axclass linorder < order |
306 |
linorder_linear: "x <= y | y <= x" |
|
307 |
||
308 |
interpretation linorder: |
|
309 |
linorder ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op < \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool"] |
|
310 |
by unfold_locales (rule linorder_linear) |
|
311 |
||
21248 | 312 |
|
313 |
subsection {* Name duplicates *} |
|
314 |
||
315 |
lemmas order_eq_refl [where 'b = "?'a::order"] = order.eq_refl |
|
316 |
lemmas order_less_irrefl [where 'b = "?'a::order"] = order.less_irrefl |
|
317 |
lemmas order_le_less [where 'b = "?'a::order"] = order.le_less |
|
318 |
lemmas order_le_imp_less_or_eq [where 'b = "?'a::order"] = order.le_imp_less_or_eq |
|
319 |
lemmas order_less_imp_le [where 'b = "?'a::order"] = order.less_imp_le |
|
320 |
lemmas order_less_not_sym [where 'b = "?'a::order"] = order.less_not_sym |
|
321 |
lemmas order_less_asym [where 'b = "?'a::order"] = order.less_asym |
|
322 |
lemmas order_eq_iff [where 'b = "?'a::order"] = order.eq_iff |
|
323 |
lemmas order_antisym_conv [where 'b = "?'a::order"] = order.antisym_conv |
|
324 |
lemmas less_imp_neq [where 'b = "?'a::order"] = order.less_imp_neq |
|
325 |
lemmas order_less_trans [where 'b = "?'a::order"] = order.less_trans |
|
326 |
lemmas order_le_less_trans [where 'b = "?'a::order"] = order.le_less_trans |
|
327 |
lemmas order_less_le_trans [where 'b = "?'a::order"] = order.less_le_trans |
|
328 |
lemmas order_less_imp_not_less [where 'b = "?'a::order"] = order.less_imp_not_less |
|
329 |
lemmas order_less_imp_triv [where 'b = "?'a::order"] = order.less_imp_triv |
|
330 |
lemmas order_less_imp_not_eq [where 'b = "?'a::order"] = order.less_imp_not_eq |
|
331 |
lemmas order_less_imp_not_eq2 [where 'b = "?'a::order"] = order.less_imp_not_eq2 |
|
332 |
lemmas order_neq_le_trans [where 'b = "?'a::order"] = order.neq_le_trans |
|
333 |
lemmas order_le_neq_trans [where 'b = "?'a::order"] = order.le_neq_trans |
|
334 |
lemmas order_less_asym' [where 'b = "?'a::order"] = order.less_asym' |
|
21412 | 335 |
lemmas linorder_less_linear [where 'b = "?'a::linorder"] = linorder.less_linear |
21248 | 336 |
lemmas linorder_le_less_linear [where 'b = "?'a::linorder"] = linorder.le_less_linear |
337 |
lemmas linorder_le_cases [where 'b = "?'a::linorder"] = linorder.le_cases |
|
338 |
lemmas linorder_cases [where 'b = "?'a::linorder"] = linorder.cases |
|
339 |
lemmas linorder_not_less [where 'b = "?'a::linorder"] = linorder.not_less |
|
340 |
lemmas linorder_not_le [where 'b = "?'a::linorder"] = linorder.not_le |
|
341 |
lemmas linorder_neq_iff [where 'b = "?'a::linorder"] = linorder.neq_iff |
|
342 |
lemmas linorder_neqE [where 'b = "?'a::linorder"] = linorder.neqE |
|
343 |
lemmas linorder_antisym_conv1 [where 'b = "?'a::linorder"] = linorder.antisym_conv1 |
|
344 |
lemmas linorder_antisym_conv2 [where 'b = "?'a::linorder"] = linorder.antisym_conv2 |
|
345 |
lemmas linorder_antisym_conv3 [where 'b = "?'a::linorder"] = linorder.antisym_conv3 |
|
346 |
lemmas leI [where 'b = "?'a::linorder"] = linorder.leI |
|
347 |
lemmas leD [where 'b = "?'a::linorder"] = linorder.leD |
|
348 |
lemmas not_leE [where 'b = "?'a::linorder"] = linorder.not_leE |
|
16796 | 349 |
|
21083 | 350 |
|
351 |
subsection {* Reasoning tools setup *} |
|
352 |
||
21091 | 353 |
ML {* |
354 |
local |
|
355 |
||
356 |
fun decomp_gen sort thy (Trueprop $ t) = |
|
21248 | 357 |
let |
358 |
fun of_sort t = |
|
359 |
let |
|
360 |
val T = type_of t |
|
361 |
in |
|
21091 | 362 |
(* exclude numeric types: linear arithmetic subsumes transitivity *) |
21248 | 363 |
T <> HOLogic.natT andalso T <> HOLogic.intT |
364 |
andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort) |
|
365 |
end; |
|
366 |
fun dec (Const ("Not", _) $ t) = (case dec t |
|
367 |
of NONE => NONE |
|
368 |
| SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) |
|
369 |
| dec (Const ("op =", _) $ t1 $ t2) = |
|
370 |
if of_sort t1 |
|
371 |
then SOME (t1, "=", t2) |
|
372 |
else NONE |
|
373 |
| dec (Const ("Orderings.less_eq", _) $ t1 $ t2) = |
|
374 |
if of_sort t1 |
|
375 |
then SOME (t1, "<=", t2) |
|
376 |
else NONE |
|
377 |
| dec (Const ("Orderings.less", _) $ t1 $ t2) = |
|
378 |
if of_sort t1 |
|
379 |
then SOME (t1, "<", t2) |
|
380 |
else NONE |
|
381 |
| dec _ = NONE; |
|
21091 | 382 |
in dec t end; |
383 |
||
384 |
in |
|
385 |
||
386 |
(* The setting up of Quasi_Tac serves as a demo. Since there is no |
|
387 |
class for quasi orders, the tactics Quasi_Tac.trans_tac and |
|
388 |
Quasi_Tac.quasi_tac are not of much use. *) |
|
389 |
||
21248 | 390 |
structure Quasi_Tac = Quasi_Tac_Fun ( |
391 |
struct |
|
392 |
val le_trans = thm "order_trans"; |
|
393 |
val le_refl = thm "order_refl"; |
|
394 |
val eqD1 = thm "order_eq_refl"; |
|
395 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
396 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
397 |
val less_imp_le = thm "order_less_imp_le"; |
|
398 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
399 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
400 |
val less_imp_neq = thm "less_imp_neq"; |
|
401 |
val decomp_trans = decomp_gen ["Orderings.order"]; |
|
402 |
val decomp_quasi = decomp_gen ["Orderings.order"]; |
|
403 |
end); |
|
21091 | 404 |
|
405 |
structure Order_Tac = Order_Tac_Fun ( |
|
21248 | 406 |
struct |
407 |
val less_reflE = thm "order_less_irrefl" RS thm "notE"; |
|
408 |
val le_refl = thm "order_refl"; |
|
409 |
val less_imp_le = thm "order_less_imp_le"; |
|
410 |
val not_lessI = thm "linorder_not_less" RS thm "iffD2"; |
|
411 |
val not_leI = thm "linorder_not_le" RS thm "iffD2"; |
|
412 |
val not_lessD = thm "linorder_not_less" RS thm "iffD1"; |
|
413 |
val not_leD = thm "linorder_not_le" RS thm "iffD1"; |
|
414 |
val eqI = thm "order_antisym"; |
|
415 |
val eqD1 = thm "order_eq_refl"; |
|
416 |
val eqD2 = thm "sym" RS thm "order_eq_refl"; |
|
417 |
val less_trans = thm "order_less_trans"; |
|
418 |
val less_le_trans = thm "order_less_le_trans"; |
|
419 |
val le_less_trans = thm "order_le_less_trans"; |
|
420 |
val le_trans = thm "order_trans"; |
|
421 |
val le_neq_trans = thm "order_le_neq_trans"; |
|
422 |
val neq_le_trans = thm "order_neq_le_trans"; |
|
423 |
val less_imp_neq = thm "less_imp_neq"; |
|
424 |
val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq"; |
|
425 |
val not_sym = thm "not_sym"; |
|
426 |
val decomp_part = decomp_gen ["Orderings.order"]; |
|
427 |
val decomp_lin = decomp_gen ["Orderings.linorder"]; |
|
428 |
end); |
|
21091 | 429 |
|
430 |
end; |
|
431 |
*} |
|
432 |
||
21083 | 433 |
setup {* |
434 |
let |
|
435 |
||
436 |
val order_antisym_conv = thm "order_antisym_conv" |
|
437 |
val linorder_antisym_conv1 = thm "linorder_antisym_conv1" |
|
438 |
val linorder_antisym_conv2 = thm "linorder_antisym_conv2" |
|
439 |
val linorder_antisym_conv3 = thm "linorder_antisym_conv3" |
|
440 |
||
441 |
fun prp t thm = (#prop (rep_thm thm) = t); |
|
15524 | 442 |
|
21083 | 443 |
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = |
444 |
let val prems = prems_of_ss ss; |
|
445 |
val less = Const("Orderings.less",T); |
|
446 |
val t = HOLogic.mk_Trueprop(le $ s $ r); |
|
447 |
in case find_first (prp t) prems of |
|
448 |
NONE => |
|
449 |
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) |
|
450 |
in case find_first (prp t) prems of |
|
451 |
NONE => NONE |
|
452 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1)) |
|
453 |
end |
|
454 |
| SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv)) |
|
455 |
end |
|
456 |
handle THM _ => NONE; |
|
15524 | 457 |
|
21083 | 458 |
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = |
459 |
let val prems = prems_of_ss ss; |
|
460 |
val le = Const("Orderings.less_eq",T); |
|
461 |
val t = HOLogic.mk_Trueprop(le $ r $ s); |
|
462 |
in case find_first (prp t) prems of |
|
463 |
NONE => |
|
464 |
let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) |
|
465 |
in case find_first (prp t) prems of |
|
466 |
NONE => NONE |
|
467 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3)) |
|
468 |
end |
|
469 |
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2)) |
|
470 |
end |
|
471 |
handle THM _ => NONE; |
|
15524 | 472 |
|
21248 | 473 |
fun add_simprocs procs thy = |
474 |
(Simplifier.change_simpset_of thy (fn ss => ss |
|
475 |
addsimprocs (map (fn (name, raw_ts, proc) => |
|
476 |
Simplifier.simproc thy name raw_ts proc)) procs); thy); |
|
477 |
fun add_solver name tac thy = |
|
478 |
(Simplifier.change_simpset_of thy (fn ss => ss addSolver |
|
479 |
(mk_solver name (K tac))); thy); |
|
21083 | 480 |
|
481 |
in |
|
21248 | 482 |
add_simprocs [ |
483 |
("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), |
|
484 |
("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) |
|
485 |
] |
|
486 |
#> add_solver "Trans_linear" Order_Tac.linear_tac |
|
487 |
#> add_solver "Trans_partial" Order_Tac.partial_tac |
|
488 |
(* Adding the transitivity reasoners also as safe solvers showed a slight |
|
489 |
speed up, but the reasoning strength appears to be not higher (at least |
|
490 |
no breaking of additional proofs in the entire HOL distribution, as |
|
491 |
of 5 March 2004, was observed). *) |
|
21083 | 492 |
end |
493 |
*} |
|
15524 | 494 |
|
495 |
||
21083 | 496 |
subsection {* Bounded quantifiers *} |
497 |
||
498 |
syntax |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
499 |
"_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
|
500 |
"_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
|
501 |
"_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
|
502 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
21083 | 503 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
504 |
"_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
|
505 |
"_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
|
506 |
"_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
|
507 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) |
21083 | 508 |
|
509 |
syntax (xsymbols) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
510 |
"_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
|
511 |
"_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
|
512 |
"_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
|
513 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 514 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
515 |
"_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
|
516 |
"_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
|
517 |
"_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
|
518 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 519 |
|
520 |
syntax (HOL) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
521 |
"_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
|
522 |
"_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
|
523 |
"_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
|
524 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
21083 | 525 |
|
526 |
syntax (HTML output) |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
527 |
"_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
|
528 |
"_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
|
529 |
"_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
|
530 |
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10) |
21083 | 531 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
532 |
"_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
|
533 |
"_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
|
534 |
"_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
|
535 |
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10) |
21083 | 536 |
|
537 |
translations |
|
538 |
"ALL x<y. P" => "ALL x. x < y \<longrightarrow> P" |
|
539 |
"EX x<y. P" => "EX x. x < y \<and> P" |
|
540 |
"ALL x<=y. P" => "ALL x. x <= y \<longrightarrow> P" |
|
541 |
"EX x<=y. P" => "EX x. x <= y \<and> P" |
|
542 |
"ALL x>y. P" => "ALL x. x > y \<longrightarrow> P" |
|
543 |
"EX x>y. P" => "EX x. x > y \<and> P" |
|
544 |
"ALL x>=y. P" => "ALL x. x >= y \<longrightarrow> P" |
|
545 |
"EX x>=y. P" => "EX x. x >= y \<and> P" |
|
546 |
||
547 |
print_translation {* |
|
548 |
let |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
549 |
val syntax_name = Sign.const_syntax_name (the_context ()); |
21524 | 550 |
val binder_name = Syntax.binder_name o syntax_name; |
551 |
val All_binder = binder_name "All"; |
|
552 |
val Ex_binder = binder_name "Ex"; |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
553 |
val impl = syntax_name "op -->"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
554 |
val conj = syntax_name "op &"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
555 |
val less = syntax_name "Orderings.less"; |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
556 |
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
|
557 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
558 |
val trans = |
21524 | 559 |
[((All_binder, impl, less), ("_All_less", "_All_greater")), |
560 |
((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), |
|
561 |
((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), |
|
562 |
((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
|
563 |
|
21083 | 564 |
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
|
565 |
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) |
21083 | 566 |
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
|
567 |
|
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
568 |
fun tr' q = (q, |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
569 |
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
|
570 |
(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
|
571 |
NONE => raise Match |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
572 |
| SOME (l, g) => |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
573 |
(case (t, u) of |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
574 |
(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
|
575 |
| (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
|
576 |
| _ => raise Match)) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
577 |
| _ => raise Match); |
21524 | 578 |
in [tr' All_binder, tr' Ex_binder] end |
21083 | 579 |
*} |
580 |
||
581 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
582 |
subsection {* Transitivity reasoning *} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
583 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
584 |
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
585 |
by (rule subst) |
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 ord_eq_le_trans: "a = b ==> b <= c ==> a <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
588 |
by (rule ssubst) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
589 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
590 |
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
591 |
by (rule subst) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
592 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
593 |
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
594 |
by (rule ssubst) |
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 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
|
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 (order_less_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 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
|
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 (order_less_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 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
|
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 (order_le_less_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 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
|
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 (order_le_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 |
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
|
633 |
(!!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
|
634 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
635 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
636 |
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
|
637 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
638 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
639 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
640 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
641 |
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
|
642 |
(!!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
|
643 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
644 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
645 |
assume "a < f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
646 |
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
|
647 |
finally (order_less_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
648 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
649 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
650 |
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
|
651 |
(!!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
|
652 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
653 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
654 |
assume "a <= f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
655 |
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
|
656 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
657 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
658 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
659 |
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
|
660 |
(!!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
|
661 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
662 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
663 |
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
|
664 |
also assume "f b <= c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
665 |
finally (order_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
666 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
667 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
668 |
lemma ord_le_eq_subst: "a <= b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
669 |
(!!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
|
670 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
671 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
672 |
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
|
673 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
674 |
finally (ord_le_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
675 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
676 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
677 |
lemma ord_eq_le_subst: "a = f b ==> b <= c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
678 |
(!!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
|
679 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
680 |
assume r: "!!x y. x <= y ==> f x <= f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
681 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
682 |
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
|
683 |
finally (ord_eq_le_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
684 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
685 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
686 |
lemma ord_less_eq_subst: "a < b ==> f b = c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
687 |
(!!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
|
688 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
689 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
690 |
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
|
691 |
also assume "f b = c" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
692 |
finally (ord_less_eq_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
693 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
694 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
695 |
lemma ord_eq_less_subst: "a = f b ==> b < c ==> |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
696 |
(!!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
|
697 |
proof - |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
698 |
assume r: "!!x y. x < y ==> f x < f y" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
699 |
assume "a = f b" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
700 |
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
|
701 |
finally (ord_eq_less_trans) show ?thesis . |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
702 |
qed |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
703 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
704 |
text {* |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
705 |
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
|
706 |
*} |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
707 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
708 |
lemmas order_trans_rules [trans] = |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
709 |
order_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
710 |
order_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
711 |
order_le_less_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
712 |
order_le_less_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
713 |
order_less_le_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
714 |
order_less_le_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
715 |
order_subst2 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
716 |
order_subst1 |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
717 |
ord_le_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
718 |
ord_eq_le_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
719 |
ord_less_eq_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
720 |
ord_eq_less_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
721 |
forw_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
722 |
back_subst |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
723 |
rev_mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
724 |
mp |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
725 |
order_neq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
726 |
order_le_neq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
727 |
order_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
728 |
order_less_asym' |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
729 |
order_le_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
730 |
order_less_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
731 |
order_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
732 |
order_antisym |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
733 |
ord_le_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
734 |
ord_eq_le_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
735 |
ord_less_eq_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
736 |
ord_eq_less_trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
737 |
trans |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
738 |
|
21083 | 739 |
|
21180
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
740 |
(* FIXME cleanup *) |
f27f12bcafb8
fixed print_translation for ALL/EX and <, <=, etc.; tuned syntax names;
wenzelm
parents:
21091
diff
changeset
|
741 |
|
21083 | 742 |
text {* These support proving chains of decreasing inequalities |
743 |
a >= b >= c ... in Isar proofs. *} |
|
744 |
||
745 |
lemma xt1: |
|
746 |
"a = b ==> b > c ==> a > c" |
|
747 |
"a > b ==> b = c ==> a > c" |
|
748 |
"a = b ==> b >= c ==> a >= c" |
|
749 |
"a >= b ==> b = c ==> a >= c" |
|
750 |
"(x::'a::order) >= y ==> y >= x ==> x = y" |
|
751 |
"(x::'a::order) >= y ==> y >= z ==> x >= z" |
|
752 |
"(x::'a::order) > y ==> y >= z ==> x > z" |
|
753 |
"(x::'a::order) >= y ==> y > z ==> x > z" |
|
754 |
"(a::'a::order) > b ==> b > a ==> ?P" |
|
755 |
"(x::'a::order) > y ==> y > z ==> x > z" |
|
756 |
"(a::'a::order) >= b ==> a ~= b ==> a > b" |
|
757 |
"(a::'a::order) ~= b ==> a >= b ==> a > b" |
|
758 |
"a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" |
|
759 |
"a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" |
|
760 |
"a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
761 |
"a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
762 |
by auto |
|
763 |
||
764 |
lemma xt2: |
|
765 |
"(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" |
|
766 |
by (subgoal_tac "f b >= f c", force, force) |
|
767 |
||
768 |
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> |
|
769 |
(!!x y. x >= y ==> f x >= f y) ==> f a >= c" |
|
770 |
by (subgoal_tac "f a >= f b", force, force) |
|
771 |
||
772 |
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> |
|
773 |
(!!x y. x >= y ==> f x >= f y) ==> a > f c" |
|
774 |
by (subgoal_tac "f b >= f c", force, force) |
|
775 |
||
776 |
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> |
|
777 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
778 |
by (subgoal_tac "f a > f b", force, force) |
|
779 |
||
780 |
lemma xt6: "(a::'a::order) >= f b ==> b > c ==> |
|
781 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
782 |
by (subgoal_tac "f b > f c", force, force) |
|
783 |
||
784 |
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> |
|
785 |
(!!x y. x >= y ==> f x >= f y) ==> f a > c" |
|
786 |
by (subgoal_tac "f a >= f b", force, force) |
|
787 |
||
788 |
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> |
|
789 |
(!!x y. x > y ==> f x > f y) ==> a > f c" |
|
790 |
by (subgoal_tac "f b > f c", force, force) |
|
791 |
||
792 |
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> |
|
793 |
(!!x y. x > y ==> f x > f y) ==> f a > c" |
|
794 |
by (subgoal_tac "f a > f b", force, force) |
|
795 |
||
796 |
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 |
|
797 |
||
798 |
(* |
|
799 |
Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands |
|
800 |
for the wrong thing in an Isar proof. |
|
801 |
||
802 |
The extra transitivity rules can be used as follows: |
|
803 |
||
804 |
lemma "(a::'a::order) > z" |
|
805 |
proof - |
|
806 |
have "a >= b" (is "_ >= ?rhs") |
|
807 |
sorry |
|
808 |
also have "?rhs >= c" (is "_ >= ?rhs") |
|
809 |
sorry |
|
810 |
also (xtrans) have "?rhs = d" (is "_ = ?rhs") |
|
811 |
sorry |
|
812 |
also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") |
|
813 |
sorry |
|
814 |
also (xtrans) have "?rhs > f" (is "_ > ?rhs") |
|
815 |
sorry |
|
816 |
also (xtrans) have "?rhs > z" |
|
817 |
sorry |
|
818 |
finally (xtrans) show ?thesis . |
|
819 |
qed |
|
820 |
||
821 |
Alternatively, one can use "declare xtrans [trans]" and then |
|
822 |
leave out the "(xtrans)" above. |
|
823 |
*) |
|
824 |
||
21546 | 825 |
subsection {* Order on bool *} |
826 |
||
827 |
instance bool :: linorder |
|
828 |
le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q" |
|
829 |
less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q" |
|
830 |
by default (auto simp add: le_bool_def less_bool_def) |
|
831 |
||
832 |
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q" |
|
833 |
by (simp add: le_bool_def) |
|
834 |
||
835 |
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q" |
|
836 |
by (simp add: le_bool_def) |
|
837 |
||
838 |
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" |
|
839 |
by (simp add: le_bool_def) |
|
840 |
||
841 |
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q" |
|
842 |
by (simp add: le_bool_def) |
|
843 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
844 |
subsection {* Monotonicity, syntactic least value operator and min/max *} |
21083 | 845 |
|
21216
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
846 |
locale mono = |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
847 |
fixes f |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
848 |
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
|
849 |
|
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
850 |
lemmas monoI [intro?] = mono.intro |
1c8580913738
made locale partial_order compatible with axclass order; changed import order; consecutive changes
haftmann
parents:
21204
diff
changeset
|
851 |
and monoD [dest?] = mono.mono |
21083 | 852 |
|
853 |
constdefs |
|
854 |
Least :: "('a::ord => bool) => 'a" (binder "LEAST " 10) |
|
855 |
"Least P == THE x. P x & (ALL y. P y --> x <= y)" |
|
856 |
-- {* We can no longer use LeastM because the latter requires Hilbert-AC. *} |
|
857 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
858 |
lemma LeastI2_order: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
859 |
"[| P (x::'a::order); |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
860 |
!!y. P y ==> x <= y; |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
861 |
!!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
|
862 |
==> Q (Least P)" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
863 |
apply (unfold Least_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
864 |
apply (rule theI2) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
865 |
apply (blast intro: order_antisym)+ |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
866 |
done |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
867 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
868 |
lemma Least_equality: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
869 |
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
870 |
apply (simp add: Least_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
871 |
apply (rule the_equality) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
872 |
apply (auto intro!: order_antisym) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
873 |
done |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
874 |
|
21083 | 875 |
constdefs |
876 |
min :: "['a::ord, 'a] => 'a" |
|
877 |
"min a b == (if a <= b then a else b)" |
|
878 |
max :: "['a::ord, 'a] => 'a" |
|
879 |
"max a b == (if a <= b then b else a)" |
|
880 |
||
21383
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
881 |
lemma min_linorder: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
882 |
"linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
883 |
by (rule+) (simp add: min_def linorder.min_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
884 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
885 |
lemma max_linorder: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
886 |
"linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
887 |
by (rule+) (simp add: max_def linorder.max_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
888 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
889 |
lemmas min_le_iff_disj = linorder.min_le_iff_disj [where 'b = "?'a::linorder", simplified min_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
890 |
lemmas le_max_iff_disj = linorder.le_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
891 |
lemmas min_less_iff_disj = linorder.min_less_iff_disj [where 'b = "?'a::linorder", simplified min_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
892 |
lemmas less_max_iff_disj = linorder.less_max_iff_disj [where 'b = "?'a::linorder", simplified max_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
893 |
lemmas min_less_iff_conj [simp] = linorder.min_less_iff_conj [where 'b = "?'a::linorder", simplified min_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
894 |
lemmas max_less_iff_conj [simp] = linorder.max_less_iff_conj [where 'b = "?'a::linorder", simplified max_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
895 |
lemmas split_min = linorder.split_min [where 'b = "?'a::linorder", simplified min_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
896 |
lemmas split_max = linorder.split_max [where 'b = "?'a::linorder", simplified max_linorder] |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
897 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
898 |
lemma min_leastL: "(!!x. least <= x) ==> min least x = least" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
899 |
by (simp add: min_def) |
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_leastL: "(!!x. least <= x) ==> max least x = x" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
902 |
by (simp add: max_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
903 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
904 |
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
905 |
apply (simp add: min_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
906 |
apply (blast intro: order_antisym) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
907 |
done |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
908 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
909 |
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
910 |
apply (simp add: max_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
911 |
apply (blast intro: order_antisym) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
912 |
done |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
913 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
914 |
lemma min_of_mono: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
915 |
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
916 |
by (simp add: min_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
917 |
|
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
918 |
lemma max_of_mono: |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
919 |
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)" |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
920 |
by (simp add: max_def) |
17e6275e13f5
added transitivity rules, reworking of min/max lemmas
haftmann
parents:
21329
diff
changeset
|
921 |
|
15524 | 922 |
end |