src/HOL/Orderings.thy
author nipkow
Sun Jun 03 16:57:51 2007 +0200 (2007-06-03)
changeset 23212 82881b1ae9c6
parent 23182 01fa88b79ddc
child 23247 b99dce43d252
permissions -rw-r--r--
tuned list comprehension, added lemma
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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 Code_Generator
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begin
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subsection {* Order syntax *}
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class ord = type +
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  fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
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    and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
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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 (input)
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  greater_eq  (infix "\<^loc>\<ge>" 50)
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text {*
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  syntactic min/max -- these definitions reach
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  their usual semantics in class linorder ahead.
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*}
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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 \<^loc>\<le> 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 \<^loc>\<le> b then b else a)"
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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 (input)
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  greater_eq  (infix "\<ge>" 50)
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lemmas min_def [code func, code unfold, code inline del] = min_def [folded ord_class.min]
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lemmas max_def [code func, code unfold, code inline del] = max_def [folded ord_class.max]
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subsection {* Partial orders *}
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class order = ord +
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  assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
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  and order_refl [iff]: "x \<sqsubseteq> x"
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  and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
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  assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
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begin
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text {* Reflexivity. *}
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lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
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    -- {* This form is useful with the classical reasoner. *}
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by (erule ssubst) (rule order_refl)
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lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
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by (simp add: less_le)
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lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"
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    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
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by (simp add: less_le) blast
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lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
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unfolding less_le by blast
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lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
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unfolding less_le by blast
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
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by (erule contrapos_pn, erule subst, rule less_irrefl)
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
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by auto
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lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
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by auto
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text {* Transitivity rules for calculational reasoning *}
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lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
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by (simp add: less_le)
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lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
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by (simp add: less_le)
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text {* Asymmetry. *}
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lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
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by (simp add: less_le antisym)
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lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
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by (drule less_not_sym, erule contrapos_np) simp
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lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
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by (blast intro: antisym)
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lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym)
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lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
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by (erule contrapos_pn, erule subst, rule less_irrefl)
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text {* Transitivity. *}
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lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
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by (simp add: less_le) (blast intro: order_trans antisym)
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text {* Useful for simplification, but too risky to include by default. *}
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lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
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by (blast elim: less_asym)
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text {* Transitivity rules for calculational reasoning *}
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lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
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by (rule less_asym)
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text {* Reverse order *}
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lemma order_reverse:
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  "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
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by unfold_locales
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   (simp add: less_le, auto intro: antisym order_trans)
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end
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subsection {* Linear (total) orders *}
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class linorder = order +
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  assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
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begin
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lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
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unfolding less_le using less_le linear by blast
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lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
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by (simp add: le_less less_linear)
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lemma le_cases [case_names le ge]:
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  "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
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using linear by blast
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lemma linorder_cases [case_names less equal greater]:
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  "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
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using less_linear by blast
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lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma not_less_iff_gr_or_eq:
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 "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"
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apply(simp add:not_less le_less)
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apply blast
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done
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lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
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apply (simp add: less_le)
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using linear apply (blast intro: antisym)
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done
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lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
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by (cut_tac x = x and y = y in less_linear, auto)
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lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
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by (simp add: neq_iff) blast
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lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
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by (blast intro: antisym dest: not_less [THEN iffD1])
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text{*Replacing the old Nat.leI*}
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lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
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unfolding not_less .
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lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
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unfolding not_less .
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(*FIXME inappropriate name (or delete altogether)*)
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lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
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unfolding not_le .
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text {* Reverse order *}
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lemma linorder_reverse:
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  "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
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by unfold_locales
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  (simp add: less_le, auto intro: antisym order_trans simp add: linear)
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text {* min/max properties *}
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lemma min_le_iff_disj:
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  "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
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unfolding min_def using linear by (auto intro: order_trans)
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lemma le_max_iff_disj:
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  "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
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unfolding max_def using linear by (auto intro: order_trans)
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lemma min_less_iff_disj:
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  "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
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unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma less_max_iff_disj:
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  "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
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unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma min_less_iff_conj [simp]:
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  "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
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unfolding min_def le_less using less_linear by (auto intro: less_trans)
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lemma max_less_iff_conj [simp]:
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  "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
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unfolding max_def le_less using less_linear by (auto intro: less_trans)
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lemma split_min:
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  "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
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by (simp add: min_def)
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lemma split_max:
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  "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
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by (simp add: max_def)
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end
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subsection {* Name duplicates -- including min/max interpretation *}
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lemmas order_less_le = less_le
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lemmas order_eq_refl = order_class.eq_refl
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lemmas order_less_irrefl = order_class.less_irrefl
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lemmas order_le_less = order_class.le_less
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lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
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lemmas order_less_imp_le = order_class.less_imp_le
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lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
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lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
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lemmas order_neq_le_trans = order_class.neq_le_trans
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lemmas order_le_neq_trans = order_class.le_neq_trans
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lemmas order_antisym = antisym
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lemmas order_less_not_sym = order_class.less_not_sym
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lemmas order_less_asym = order_class.less_asym
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lemmas order_eq_iff = order_class.eq_iff
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lemmas order_antisym_conv = order_class.antisym_conv
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lemmas order_less_trans = order_class.less_trans
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lemmas order_le_less_trans = order_class.le_less_trans
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lemmas order_less_le_trans = order_class.less_le_trans
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lemmas order_less_imp_not_less = order_class.less_imp_not_less
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lemmas order_less_imp_triv = order_class.less_imp_triv
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lemmas order_less_asym' = order_class.less_asym'
haftmann@22316
   325
haftmann@22384
   326
lemmas linorder_linear = linear
haftmann@22316
   327
lemmas linorder_less_linear = linorder_class.less_linear
haftmann@22316
   328
lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@22316
   329
lemmas linorder_le_cases = linorder_class.le_cases
haftmann@22316
   330
lemmas linorder_not_less = linorder_class.not_less
haftmann@22316
   331
lemmas linorder_not_le = linorder_class.not_le
haftmann@22316
   332
lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@22316
   333
lemmas linorder_neqE = linorder_class.neqE
haftmann@22316
   334
lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@22316
   335
lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@22316
   336
lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
paulson@16796
   337
haftmann@23087
   338
lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [folded ord_class.min]
haftmann@23087
   339
lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [folded ord_class.max]
haftmann@23087
   340
lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [folded ord_class.min]
haftmann@23087
   341
lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [folded ord_class.max]
haftmann@23087
   342
lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [folded ord_class.min]
haftmann@23087
   343
lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [folded ord_class.max]
haftmann@23087
   344
lemmas split_min = linorder_class.split_min [folded ord_class.min]
haftmann@23087
   345
lemmas split_max = linorder_class.split_max [folded ord_class.max]
haftmann@22916
   346
haftmann@21083
   347
haftmann@21083
   348
subsection {* Reasoning tools setup *}
haftmann@21083
   349
haftmann@21091
   350
ML {*
haftmann@21091
   351
local
haftmann@21091
   352
haftmann@21091
   353
fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248
   354
  let
haftmann@21248
   355
    fun of_sort t =
haftmann@21248
   356
      let
haftmann@21248
   357
        val T = type_of t
haftmann@21248
   358
      in
haftmann@21091
   359
        (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248
   360
        T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248
   361
          andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248
   362
      end;
haftmann@22916
   363
    fun dec (Const (@{const_name Not}, _) $ t) = (case dec t
haftmann@21248
   364
          of NONE => NONE
haftmann@21248
   365
           | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@22916
   366
      | dec (Const (@{const_name "op ="},  _) $ t1 $ t2) =
haftmann@21248
   367
          if of_sort t1
haftmann@21248
   368
          then SOME (t1, "=", t2)
haftmann@21248
   369
          else NONE
haftmann@22997
   370
      | dec (Const (@{const_name Orderings.less_eq},  _) $ t1 $ t2) =
haftmann@21248
   371
          if of_sort t1
haftmann@21248
   372
          then SOME (t1, "<=", t2)
haftmann@21248
   373
          else NONE
haftmann@22997
   374
      | dec (Const (@{const_name Orderings.less},  _) $ t1 $ t2) =
haftmann@21248
   375
          if of_sort t1
haftmann@21248
   376
          then SOME (t1, "<", t2)
haftmann@21248
   377
          else NONE
haftmann@21248
   378
      | dec _ = NONE;
haftmann@21091
   379
  in dec t end;
haftmann@21091
   380
haftmann@21091
   381
in
haftmann@21091
   382
haftmann@22841
   383
(* sorry - there is no preorder class
haftmann@21248
   384
structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248
   385
struct
haftmann@21248
   386
  val le_trans = thm "order_trans";
haftmann@21248
   387
  val le_refl = thm "order_refl";
haftmann@21248
   388
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   389
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   390
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   391
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   392
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   393
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   394
  val less_imp_neq = thm "less_imp_neq";
haftmann@22738
   395
  val decomp_trans = decomp_gen ["Orderings.preorder"];
haftmann@22738
   396
  val decomp_quasi = decomp_gen ["Orderings.preorder"];
haftmann@22841
   397
end);*)
haftmann@21091
   398
haftmann@21091
   399
structure Order_Tac = Order_Tac_Fun (
haftmann@21248
   400
struct
haftmann@21248
   401
  val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248
   402
  val le_refl = thm "order_refl";
haftmann@21248
   403
  val less_imp_le = thm "order_less_imp_le";
haftmann@21248
   404
  val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248
   405
  val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248
   406
  val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248
   407
  val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248
   408
  val eqI = thm "order_antisym";
haftmann@21248
   409
  val eqD1 = thm "order_eq_refl";
haftmann@21248
   410
  val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248
   411
  val less_trans = thm "order_less_trans";
haftmann@21248
   412
  val less_le_trans = thm "order_less_le_trans";
haftmann@21248
   413
  val le_less_trans = thm "order_le_less_trans";
haftmann@21248
   414
  val le_trans = thm "order_trans";
haftmann@21248
   415
  val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248
   416
  val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248
   417
  val less_imp_neq = thm "less_imp_neq";
haftmann@21248
   418
  val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248
   419
  val not_sym = thm "not_sym";
haftmann@21248
   420
  val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248
   421
  val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248
   422
end);
haftmann@21091
   423
haftmann@21091
   424
end;
haftmann@21091
   425
*}
haftmann@21091
   426
haftmann@21083
   427
setup {*
haftmann@21083
   428
let
haftmann@21083
   429
haftmann@21083
   430
fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524
   431
haftmann@21083
   432
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083
   433
  let val prems = prems_of_ss ss;
haftmann@22916
   434
      val less = Const (@{const_name less}, T);
haftmann@21083
   435
      val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083
   436
  in case find_first (prp t) prems of
haftmann@21083
   437
       NONE =>
haftmann@21083
   438
         let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083
   439
         in case find_first (prp t) prems of
haftmann@21083
   440
              NONE => NONE
haftmann@22738
   441
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
haftmann@21083
   442
         end
haftmann@22738
   443
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
haftmann@21083
   444
  end
haftmann@21083
   445
  handle THM _ => NONE;
nipkow@15524
   446
haftmann@21083
   447
fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083
   448
  let val prems = prems_of_ss ss;
haftmann@22916
   449
      val le = Const (@{const_name less_eq}, T);
haftmann@21083
   450
      val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083
   451
  in case find_first (prp t) prems of
haftmann@21083
   452
       NONE =>
haftmann@21083
   453
         let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083
   454
         in case find_first (prp t) prems of
haftmann@21083
   455
              NONE => NONE
haftmann@22738
   456
            | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
haftmann@21083
   457
         end
haftmann@22738
   458
     | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2}))
haftmann@21083
   459
  end
haftmann@21083
   460
  handle THM _ => NONE;
nipkow@15524
   461
haftmann@21248
   462
fun add_simprocs procs thy =
haftmann@21248
   463
  (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248
   464
    addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248
   465
      Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248
   466
fun add_solver name tac thy =
haftmann@21248
   467
  (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248
   468
    (mk_solver name (K tac))); thy);
haftmann@21083
   469
haftmann@21083
   470
in
haftmann@21248
   471
  add_simprocs [
haftmann@21248
   472
       ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248
   473
       ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248
   474
     ]
haftmann@21248
   475
  #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248
   476
  #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248
   477
  (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248
   478
     speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248
   479
     no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248
   480
     of 5 March 2004, was observed). *)
haftmann@21083
   481
end
haftmann@21083
   482
*}
nipkow@15524
   483
nipkow@15524
   484
haftmann@21083
   485
subsection {* Bounded quantifiers *}
haftmann@21083
   486
haftmann@21083
   487
syntax
wenzelm@21180
   488
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   489
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   490
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   491
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   492
wenzelm@21180
   493
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   494
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   495
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   496
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083
   497
haftmann@21083
   498
syntax (xsymbols)
wenzelm@21180
   499
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   500
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   501
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   502
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   503
wenzelm@21180
   504
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   505
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   506
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   507
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   508
haftmann@21083
   509
syntax (HOL)
wenzelm@21180
   510
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   511
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   512
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180
   513
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083
   514
haftmann@21083
   515
syntax (HTML output)
wenzelm@21180
   516
  "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   517
  "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   518
  "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   519
  "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083
   520
wenzelm@21180
   521
  "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   522
  "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180
   523
  "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180
   524
  "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083
   525
haftmann@21083
   526
translations
haftmann@21083
   527
  "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083
   528
  "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083
   529
  "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083
   530
  "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083
   531
  "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083
   532
  "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083
   533
  "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083
   534
  "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083
   535
haftmann@21083
   536
print_translation {*
haftmann@21083
   537
let
haftmann@22916
   538
  val All_binder = Syntax.binder_name @{const_syntax All};
haftmann@22916
   539
  val Ex_binder = Syntax.binder_name @{const_syntax Ex};
wenzelm@22377
   540
  val impl = @{const_syntax "op -->"};
wenzelm@22377
   541
  val conj = @{const_syntax "op &"};
haftmann@22916
   542
  val less = @{const_syntax less};
haftmann@22916
   543
  val less_eq = @{const_syntax less_eq};
wenzelm@21180
   544
wenzelm@21180
   545
  val trans =
wenzelm@21524
   546
   [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524
   547
    ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524
   548
    ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524
   549
    ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180
   550
krauss@22344
   551
  fun matches_bound v t = 
krauss@22344
   552
     case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344
   553
              | _ => false
krauss@22344
   554
  fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344
   555
  fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180
   556
wenzelm@21180
   557
  fun tr' q = (q,
wenzelm@21180
   558
    fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180
   559
      (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180
   560
        NONE => raise Match
wenzelm@21180
   561
      | SOME (l, g) =>
krauss@22344
   562
          if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344
   563
          else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344
   564
          else raise Match)
wenzelm@21180
   565
     | _ => raise Match);
wenzelm@21524
   566
in [tr' All_binder, tr' Ex_binder] end
haftmann@21083
   567
*}
haftmann@21083
   568
haftmann@21083
   569
haftmann@21383
   570
subsection {* Transitivity reasoning *}
haftmann@21383
   571
haftmann@21383
   572
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
nipkow@23212
   573
by (rule subst)
haftmann@21383
   574
haftmann@21383
   575
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
nipkow@23212
   576
by (rule ssubst)
haftmann@21383
   577
haftmann@21383
   578
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
nipkow@23212
   579
by (rule subst)
haftmann@21383
   580
haftmann@21383
   581
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
nipkow@23212
   582
by (rule ssubst)
haftmann@21383
   583
haftmann@21383
   584
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383
   585
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   586
proof -
haftmann@21383
   587
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   588
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   589
  also assume "f b < c"
haftmann@21383
   590
  finally (order_less_trans) show ?thesis .
haftmann@21383
   591
qed
haftmann@21383
   592
haftmann@21383
   593
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383
   594
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   595
proof -
haftmann@21383
   596
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   597
  assume "a < f b"
haftmann@21383
   598
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   599
  finally (order_less_trans) show ?thesis .
haftmann@21383
   600
qed
haftmann@21383
   601
haftmann@21383
   602
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383
   603
  (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383
   604
proof -
haftmann@21383
   605
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   606
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   607
  also assume "f b < c"
haftmann@21383
   608
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   609
qed
haftmann@21383
   610
haftmann@21383
   611
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383
   612
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   613
proof -
haftmann@21383
   614
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   615
  assume "a <= f b"
haftmann@21383
   616
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   617
  finally (order_le_less_trans) show ?thesis .
haftmann@21383
   618
qed
haftmann@21383
   619
haftmann@21383
   620
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383
   621
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   622
proof -
haftmann@21383
   623
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   624
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   625
  also assume "f b <= c"
haftmann@21383
   626
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   627
qed
haftmann@21383
   628
haftmann@21383
   629
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383
   630
  (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383
   631
proof -
haftmann@21383
   632
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   633
  assume "a < f b"
haftmann@21383
   634
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   635
  finally (order_less_le_trans) show ?thesis .
haftmann@21383
   636
qed
haftmann@21383
   637
haftmann@21383
   638
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383
   639
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   640
proof -
haftmann@21383
   641
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   642
  assume "a <= f b"
haftmann@21383
   643
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   644
  finally (order_trans) show ?thesis .
haftmann@21383
   645
qed
haftmann@21383
   646
haftmann@21383
   647
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383
   648
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   649
proof -
haftmann@21383
   650
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   651
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   652
  also assume "f b <= c"
haftmann@21383
   653
  finally (order_trans) show ?thesis .
haftmann@21383
   654
qed
haftmann@21383
   655
haftmann@21383
   656
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383
   657
  (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383
   658
proof -
haftmann@21383
   659
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   660
  assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383
   661
  also assume "f b = c"
haftmann@21383
   662
  finally (ord_le_eq_trans) show ?thesis .
haftmann@21383
   663
qed
haftmann@21383
   664
haftmann@21383
   665
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383
   666
  (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383
   667
proof -
haftmann@21383
   668
  assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383
   669
  assume "a = f b"
haftmann@21383
   670
  also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383
   671
  finally (ord_eq_le_trans) show ?thesis .
haftmann@21383
   672
qed
haftmann@21383
   673
haftmann@21383
   674
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383
   675
  (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383
   676
proof -
haftmann@21383
   677
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   678
  assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383
   679
  also assume "f b = c"
haftmann@21383
   680
  finally (ord_less_eq_trans) show ?thesis .
haftmann@21383
   681
qed
haftmann@21383
   682
haftmann@21383
   683
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383
   684
  (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383
   685
proof -
haftmann@21383
   686
  assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383
   687
  assume "a = f b"
haftmann@21383
   688
  also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383
   689
  finally (ord_eq_less_trans) show ?thesis .
haftmann@21383
   690
qed
haftmann@21383
   691
haftmann@21383
   692
text {*
haftmann@21383
   693
  Note that this list of rules is in reverse order of priorities.
haftmann@21383
   694
*}
haftmann@21383
   695
haftmann@21383
   696
lemmas order_trans_rules [trans] =
haftmann@21383
   697
  order_less_subst2
haftmann@21383
   698
  order_less_subst1
haftmann@21383
   699
  order_le_less_subst2
haftmann@21383
   700
  order_le_less_subst1
haftmann@21383
   701
  order_less_le_subst2
haftmann@21383
   702
  order_less_le_subst1
haftmann@21383
   703
  order_subst2
haftmann@21383
   704
  order_subst1
haftmann@21383
   705
  ord_le_eq_subst
haftmann@21383
   706
  ord_eq_le_subst
haftmann@21383
   707
  ord_less_eq_subst
haftmann@21383
   708
  ord_eq_less_subst
haftmann@21383
   709
  forw_subst
haftmann@21383
   710
  back_subst
haftmann@21383
   711
  rev_mp
haftmann@21383
   712
  mp
haftmann@21383
   713
  order_neq_le_trans
haftmann@21383
   714
  order_le_neq_trans
haftmann@21383
   715
  order_less_trans
haftmann@21383
   716
  order_less_asym'
haftmann@21383
   717
  order_le_less_trans
haftmann@21383
   718
  order_less_le_trans
haftmann@21383
   719
  order_trans
haftmann@21383
   720
  order_antisym
haftmann@21383
   721
  ord_le_eq_trans
haftmann@21383
   722
  ord_eq_le_trans
haftmann@21383
   723
  ord_less_eq_trans
haftmann@21383
   724
  ord_eq_less_trans
haftmann@21383
   725
  trans
haftmann@21383
   726
haftmann@21083
   727
wenzelm@21180
   728
(* FIXME cleanup *)
wenzelm@21180
   729
haftmann@21083
   730
text {* These support proving chains of decreasing inequalities
haftmann@21083
   731
    a >= b >= c ... in Isar proofs. *}
haftmann@21083
   732
haftmann@21083
   733
lemma xt1:
haftmann@21083
   734
  "a = b ==> b > c ==> a > c"
haftmann@21083
   735
  "a > b ==> b = c ==> a > c"
haftmann@21083
   736
  "a = b ==> b >= c ==> a >= c"
haftmann@21083
   737
  "a >= b ==> b = c ==> a >= c"
haftmann@21083
   738
  "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083
   739
  "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083
   740
  "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083
   741
  "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083
   742
  "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083
   743
  "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083
   744
  "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083
   745
  "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083
   746
  "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083
   747
  "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   748
  "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   749
  "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   750
by auto
haftmann@21083
   751
haftmann@21083
   752
lemma xt2:
haftmann@21083
   753
  "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083
   754
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   755
haftmann@21083
   756
lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083
   757
    (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083
   758
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   759
haftmann@21083
   760
lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083
   761
  (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083
   762
by (subgoal_tac "f b >= f c", force, force)
haftmann@21083
   763
haftmann@21083
   764
lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083
   765
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   766
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   767
haftmann@21083
   768
lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083
   769
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   770
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   771
haftmann@21083
   772
lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083
   773
    (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083
   774
by (subgoal_tac "f a >= f b", force, force)
haftmann@21083
   775
haftmann@21083
   776
lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083
   777
    (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083
   778
by (subgoal_tac "f b > f c", force, force)
haftmann@21083
   779
haftmann@21083
   780
lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083
   781
    (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083
   782
by (subgoal_tac "f a > f b", force, force)
haftmann@21083
   783
haftmann@21083
   784
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083
   785
haftmann@21083
   786
(* 
haftmann@21083
   787
  Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083
   788
  for the wrong thing in an Isar proof.
haftmann@21083
   789
haftmann@21083
   790
  The extra transitivity rules can be used as follows: 
haftmann@21083
   791
haftmann@21083
   792
lemma "(a::'a::order) > z"
haftmann@21083
   793
proof -
haftmann@21083
   794
  have "a >= b" (is "_ >= ?rhs")
haftmann@21083
   795
    sorry
haftmann@21083
   796
  also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083
   797
    sorry
haftmann@21083
   798
  also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083
   799
    sorry
haftmann@21083
   800
  also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083
   801
    sorry
haftmann@21083
   802
  also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083
   803
    sorry
haftmann@21083
   804
  also (xtrans) have "?rhs > z"
haftmann@21083
   805
    sorry
haftmann@21083
   806
  finally (xtrans) show ?thesis .
haftmann@21083
   807
qed
haftmann@21083
   808
haftmann@21083
   809
  Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083
   810
  leave out the "(xtrans)" above.
haftmann@21083
   811
*)
haftmann@21083
   812
haftmann@21546
   813
subsection {* Order on bool *}
haftmann@21546
   814
haftmann@22886
   815
instance bool :: order 
haftmann@21546
   816
  le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546
   817
  less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@22916
   818
  by intro_classes (auto simp add: le_bool_def less_bool_def)
haftmann@21546
   819
haftmann@21546
   820
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
nipkow@23212
   821
by (simp add: le_bool_def)
haftmann@21546
   822
haftmann@21546
   823
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
nipkow@23212
   824
by (simp add: le_bool_def)
haftmann@21546
   825
haftmann@21546
   826
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23212
   827
by (simp add: le_bool_def)
haftmann@21546
   828
haftmann@21546
   829
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
nipkow@23212
   830
by (simp add: le_bool_def)
haftmann@21546
   831
haftmann@22348
   832
lemma [code func]:
haftmann@22348
   833
  "False \<le> b \<longleftrightarrow> True"
haftmann@22348
   834
  "True \<le> b \<longleftrightarrow> b"
haftmann@22348
   835
  "False < b \<longleftrightarrow> b"
haftmann@22348
   836
  "True < b \<longleftrightarrow> False"
haftmann@22348
   837
  unfolding le_bool_def less_bool_def by simp_all
haftmann@22348
   838
haftmann@22424
   839
haftmann@21383
   840
subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083
   841
haftmann@21216
   842
locale mono =
haftmann@21216
   843
  fixes f
haftmann@21216
   844
  assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216
   845
haftmann@21216
   846
lemmas monoI [intro?] = mono.intro
haftmann@21216
   847
  and monoD [dest?] = mono.mono
haftmann@21083
   848
haftmann@21083
   849
constdefs
haftmann@21083
   850
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083
   851
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083
   852
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083
   853
haftmann@21383
   854
lemma LeastI2_order:
haftmann@21383
   855
  "[| P (x::'a::order);
haftmann@21383
   856
      !!y. P y ==> x <= y;
haftmann@21383
   857
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383
   858
   ==> Q (Least P)"
nipkow@23212
   859
apply (unfold Least_def)
nipkow@23212
   860
apply (rule theI2)
nipkow@23212
   861
  apply (blast intro: order_antisym)+
nipkow@23212
   862
done
haftmann@21383
   863
haftmann@21383
   864
lemma Least_equality:
nipkow@23212
   865
  "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
nipkow@23212
   866
apply (simp add: Least_def)
nipkow@23212
   867
apply (rule the_equality)
nipkow@23212
   868
apply (auto intro!: order_antisym)
nipkow@23212
   869
done
haftmann@21383
   870
haftmann@21383
   871
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
nipkow@23212
   872
by (simp add: min_def)
haftmann@21383
   873
haftmann@21383
   874
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
nipkow@23212
   875
by (simp add: max_def)
haftmann@21383
   876
haftmann@21383
   877
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
nipkow@23212
   878
apply (simp add: min_def)
nipkow@23212
   879
apply (blast intro: order_antisym)
nipkow@23212
   880
done
haftmann@21383
   881
haftmann@21383
   882
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
nipkow@23212
   883
apply (simp add: max_def)
nipkow@23212
   884
apply (blast intro: order_antisym)
nipkow@23212
   885
done
haftmann@21383
   886
haftmann@21383
   887
lemma min_of_mono:
nipkow@23212
   888
  "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
nipkow@23212
   889
by (simp add: min_def)
haftmann@21383
   890
haftmann@21383
   891
lemma max_of_mono:
nipkow@23212
   892
  "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
nipkow@23212
   893
by (simp add: max_def)
haftmann@21383
   894
haftmann@22548
   895
haftmann@22548
   896
subsection {* legacy ML bindings *}
wenzelm@21673
   897
wenzelm@21673
   898
ML {*
haftmann@22548
   899
val monoI = @{thm monoI};
haftmann@22886
   900
*}
wenzelm@21673
   901
nipkow@15524
   902
end