
nipkow@15524: (*  Title:      HOL/Orderings.thy
nipkow@15524:     ID:         $Id$
nipkow@15524:     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
nipkow@15524: *)
nipkow@15524: 
haftmann@21329: header {* Syntactic and abstract orders *}
nipkow@15524: 
nipkow@15524: theory Orderings
haftmann@21329: imports HOL
nipkow@15524: begin
nipkow@15524: 
haftmann@21329: subsection {* Order syntax *}
nipkow@15524: 
haftmann@21194: class ord =
wenzelm@21656:   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
wenzelm@21656:     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
wenzelm@21204: begin
wenzelm@21204: 
wenzelm@21204: notation
wenzelm@21656:   less_eq  ("op \<^loc><=") and
haftmann@21620:   less_eq  ("(_/ \<^loc><= _)" [51, 51] 50) and
wenzelm@21656:   less  ("op \<^loc><") and
wenzelm@21656:   less  ("(_/ \<^loc>< _)"  [51, 51] 50)
haftmann@21620:   
wenzelm@21204: notation (xsymbols)
wenzelm@21404:   less_eq  ("op \<^loc>\<le>") and
wenzelm@21259:   less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
nipkow@15524: 
wenzelm@21204: notation (HTML output)
wenzelm@21404:   less_eq  ("op \<^loc>\<le>") and
wenzelm@21259:   less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
wenzelm@21204: 
wenzelm@21204: abbreviation (input)
wenzelm@21656:   greater  (infix "\<^loc>>" 50) where
haftmann@21620:   "x \<^loc>> y \<equiv> y \<^loc>< x"
haftmann@21620: 
wenzelm@21656: abbreviation (input)
wenzelm@21656:   greater_eq  (infix "\<^loc>>=" 50) where
wenzelm@21656:   "x \<^loc>>= y \<equiv> y \<^loc><= x"
wenzelm@21204: 
wenzelm@21656: notation (input)
wenzelm@21656:   greater_eq  (infix "\<^loc>\<ge>" 50)
wenzelm@21204: 
wenzelm@21204: end
wenzelm@21204: 
wenzelm@21204: notation
wenzelm@21656:   less_eq  ("op <=") and
haftmann@21620:   less_eq  ("(_/ <= _)" [51, 51] 50) and
wenzelm@21656:   less  ("op <") and
wenzelm@21656:   less  ("(_/ < _)"  [51, 51] 50)
wenzelm@21204:   
wenzelm@21204: notation (xsymbols)
wenzelm@21404:   less_eq  ("op \<le>") and
wenzelm@21259:   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
nipkow@15524: 
wenzelm@21204: notation (HTML output)
wenzelm@21404:   less_eq  ("op \<le>") and
wenzelm@21259:   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
haftmann@20714: 
wenzelm@19536: abbreviation (input)
wenzelm@21656:   greater  (infix ">" 50) where
haftmann@21620:   "x > y \<equiv> y < x"
haftmann@21620: 
wenzelm@21656: abbreviation (input)
wenzelm@21656:   greater_eq  (infix ">=" 50) where
wenzelm@21656:   "x >= y \<equiv> y <= x"
haftmann@21620: 
wenzelm@21656: notation (input)
wenzelm@21656:   greater_eq  (infix "\<ge>" 50)
nipkow@15524: 
nipkow@15524: 
haftmann@21329: subsection {* Quasiorders (preorders) *}
nipkow@15524: 
haftmann@22316: class preorder = ord +
haftmann@22316:   assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
haftmann@22316:   and refl [iff]: "x \<sqsubseteq> x"
haftmann@21216:   and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
haftmann@21248: begin
haftmann@21248: 
nipkow@15524: text {* Reflexivity. *}
nipkow@15524: 
haftmann@21248: lemma eq_refl: "x = y \<Longrightarrow> x \<sqsubseteq> y"
nipkow@15524:     -- {* This form is useful with the classical reasoner. *}
haftmann@21248:   by (erule ssubst) (rule refl)
nipkow@15524: 
haftmann@21248: lemma less_irrefl [iff]: "\<not> x \<sqsubset> x"
haftmann@21248:   by (simp add: less_le)
nipkow@15524: 
haftmann@21248: lemma le_less: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubset> y \<or> x = y"
nipkow@15524:     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
haftmann@21248:   by (simp add: less_le) blast
nipkow@15524: 
haftmann@21248: lemma le_imp_less_or_eq: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubset> y \<or> x = y"
haftmann@21248:   unfolding less_le by blast
nipkow@15524: 
haftmann@21248: lemma less_imp_le: "x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y"
haftmann@21248:   unfolding less_le by blast
haftmann@21248: 
haftmann@21329: lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
haftmann@21329:   by (erule contrapos_pn, erule subst, rule less_irrefl)
haftmann@21329: 
haftmann@21329: 
haftmann@21329: text {* Useful for simplification, but too risky to include by default. *}
haftmann@21329: 
haftmann@21329: lemma less_imp_not_eq: "x \<sqsubset> y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
haftmann@21329:   by auto
haftmann@21329: 
haftmann@21329: lemma less_imp_not_eq2: "x \<sqsubset> y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
haftmann@21329:   by auto
haftmann@21329: 
haftmann@21329: 
haftmann@21329: text {* Transitivity rules for calculational reasoning *}
haftmann@21329: 
haftmann@21329: lemma neq_le_trans: "\<lbrakk> a \<noteq> b; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
haftmann@21329:   by (simp add: less_le)
haftmann@21329: 
haftmann@21329: lemma le_neq_trans: "\<lbrakk> a \<sqsubseteq> b; a \<noteq> b \<rbrakk> \<Longrightarrow> a \<sqsubset> b"
haftmann@21329:   by (simp add: less_le)
haftmann@21329: 
haftmann@21329: end
haftmann@21329: 
haftmann@21329: 
haftmann@21329: subsection {* Partial orderings *}
haftmann@21329: 
haftmann@22316: class order = preorder + 
haftmann@21329:   assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
haftmann@21329: 
haftmann@22068: context order
haftmann@21329: begin
nipkow@15524: 
nipkow@15524: text {* Asymmetry. *}
nipkow@15524: 
haftmann@21248: lemma less_not_sym: "x \<sqsubset> y \<Longrightarrow> \<not> (y \<sqsubset> x)"
haftmann@21248:   by (simp add: less_le antisym)
nipkow@15524: 
haftmann@21248: lemma less_asym: "x \<sqsubset> y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<sqsubset> x) \<Longrightarrow> P"
haftmann@21248:   by (drule less_not_sym, erule contrapos_np) simp
nipkow@15524: 
haftmann@21248: lemma eq_iff: "x = y \<longleftrightarrow> x \<sqsubseteq> y \<and> y \<sqsubseteq> x"
haftmann@21248:   by (blast intro: antisym)
nipkow@15524: 
haftmann@21248: lemma antisym_conv: "y \<sqsubseteq> x \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
haftmann@21248:   by (blast intro: antisym)
nipkow@15524: 
haftmann@21248: lemma less_imp_neq: "x \<sqsubset> y \<Longrightarrow> x \<noteq> y"
haftmann@21248:   by (erule contrapos_pn, erule subst, rule less_irrefl)
haftmann@21248: 
haftmann@21083: 
nipkow@15524: text {* Transitivity. *}
nipkow@15524: 
haftmann@21248: lemma less_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
haftmann@21248:   by (simp add: less_le) (blast intro: trans antisym)
nipkow@15524: 
haftmann@21248: lemma le_less_trans: "\<lbrakk> x \<sqsubseteq> y; y \<sqsubset> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
haftmann@21248:   by (simp add: less_le) (blast intro: trans antisym)
nipkow@15524: 
haftmann@21248: lemma less_le_trans: "\<lbrakk> x \<sqsubset> y; y \<sqsubseteq> z \<rbrakk> \<Longrightarrow> x \<sqsubset> z"
haftmann@21248:   by (simp add: less_le) (blast intro: trans antisym)
nipkow@15524: 
nipkow@15524: 
nipkow@15524: text {* Useful for simplification, but too risky to include by default. *}
nipkow@15524: 
haftmann@21248: lemma less_imp_not_less: "x \<sqsubset> y \<Longrightarrow> (\<not> y \<sqsubset> x) \<longleftrightarrow> True"
haftmann@21248:   by (blast elim: less_asym)
nipkow@15524: 
haftmann@21248: lemma less_imp_triv: "x \<sqsubset> y \<Longrightarrow> (y \<sqsubset> x \<longrightarrow> P) \<longleftrightarrow> True"
haftmann@21248:   by (blast elim: less_asym)
nipkow@15524: 
haftmann@21248: 
haftmann@21083: text {* Transitivity rules for calculational reasoning *}
nipkow@15524: 
haftmann@21248: lemma less_asym': "\<lbrakk> a \<sqsubset> b; b \<sqsubset> a \<rbrakk> \<Longrightarrow> P"
haftmann@21248:   by (rule less_asym)
haftmann@21248: 
haftmann@21248: end
nipkow@15524: 
haftmann@21329: 
haftmann@21329: subsection {* Linear (total) orders *}
haftmann@21329: 
haftmann@22316: class linorder = order +
haftmann@21216:   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
haftmann@21248: begin
haftmann@21248: 
haftmann@21412: lemma less_linear: "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
haftmann@21248:   unfolding less_le using less_le linear by blast 
haftmann@21248: 
haftmann@21248: lemma le_less_linear: "x \<sqsubseteq> y \<or> y \<sqsubset> x"
haftmann@21412:   by (simp add: le_less less_linear)
haftmann@21248: 
haftmann@21248: lemma le_cases [case_names le ge]:
haftmann@21248:   "\<lbrakk> x \<sqsubseteq> y \<Longrightarrow> P; y \<sqsubseteq> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
haftmann@21248:   using linear by blast
haftmann@21248: 
haftmann@21248: lemma cases [case_names less equal greater]:
haftmann@21248:     "\<lbrakk> x \<sqsubset> y \<Longrightarrow> P; x = y \<Longrightarrow> P; y \<sqsubset> x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
haftmann@21412:   using less_linear by blast
haftmann@21248: 
haftmann@21248: lemma not_less: "\<not> x \<sqsubset> y \<longleftrightarrow> y \<sqsubseteq> x"
haftmann@21248:   apply (simp add: less_le)
haftmann@21248:   using linear apply (blast intro: antisym)
nipkow@15524:   done
nipkow@15524: 
haftmann@21248: lemma not_le: "\<not> x \<sqsubseteq> y \<longleftrightarrow> y \<sqsubset> x"
haftmann@21248:   apply (simp add: less_le)
haftmann@21248:   using linear apply (blast intro: antisym)
nipkow@15524:   done
nipkow@15524: 
haftmann@21248: lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<sqsubset> y \<or> y \<sqsubset> x"
haftmann@21412:   by (cut_tac x = x and y = y in less_linear, auto)
nipkow@15524: 
haftmann@21248: lemma neqE: "\<lbrakk> x \<noteq> y; x \<sqsubset> y \<Longrightarrow> R; y \<sqsubset> x \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
haftmann@21248:   by (simp add: neq_iff) blast
nipkow@15524: 
haftmann@21248: lemma antisym_conv1: "\<not> x \<sqsubset> y \<Longrightarrow> x \<sqsubseteq> y \<longleftrightarrow> x = y"
haftmann@21248:   by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524: 
haftmann@21248: lemma antisym_conv2: "x \<sqsubseteq> y \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
haftmann@21248:   by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524: 
haftmann@21248: lemma antisym_conv3: "\<not> y \<sqsubset> x \<Longrightarrow> \<not> x \<sqsubset> y \<longleftrightarrow> x = y"
haftmann@21248:   by (blast intro: antisym dest: not_less [THEN iffD1])
nipkow@15524: 
paulson@16796: text{*Replacing the old Nat.leI*}
haftmann@21248: lemma leI: "\<not> x \<sqsubset> y \<Longrightarrow> y \<sqsubseteq> x"
haftmann@21248:   unfolding not_less .
paulson@16796: 
haftmann@21248: lemma leD: "y \<sqsubseteq> x \<Longrightarrow> \<not> x \<sqsubset> y"
haftmann@21248:   unfolding not_less .
paulson@16796: 
paulson@16796: (*FIXME inappropriate name (or delete altogether)*)
haftmann@21248: lemma not_leE: "\<not> y \<sqsubseteq> x \<Longrightarrow> x \<sqsubset> y"
haftmann@21248:   unfolding not_le .
haftmann@21248: 
haftmann@21383: (* min/max *)
haftmann@21383: 
haftmann@21383: definition
wenzelm@21404:   min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@21383:   "min a b = (if a \<sqsubseteq> b then a else b)"
wenzelm@21404: 
wenzelm@21404: definition
wenzelm@21404:   max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@21383:   "max a b = (if a \<sqsubseteq> b then b else a)"
haftmann@21383: 
haftmann@21383: lemma min_le_iff_disj:
haftmann@21383:   "min x y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<or> y \<sqsubseteq> z"
haftmann@21383:   unfolding min_def using linear by (auto intro: trans)
haftmann@21383: 
haftmann@21383: lemma le_max_iff_disj:
haftmann@21383:   "z \<sqsubseteq> max x y \<longleftrightarrow> z \<sqsubseteq> x \<or> z \<sqsubseteq> y"
haftmann@21383:   unfolding max_def using linear by (auto intro: trans)
haftmann@21383: 
haftmann@21383: lemma min_less_iff_disj:
haftmann@21383:   "min x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<or> y \<sqsubset> z"
haftmann@21412:   unfolding min_def le_less using less_linear by (auto intro: less_trans)
haftmann@21383: 
haftmann@21383: lemma less_max_iff_disj:
haftmann@21383:   "z \<sqsubset> max x y \<longleftrightarrow> z \<sqsubset> x \<or> z \<sqsubset> y"
haftmann@21412:   unfolding max_def le_less using less_linear by (auto intro: less_trans)
haftmann@21383: 
haftmann@21383: lemma min_less_iff_conj [simp]:
haftmann@21383:   "z \<sqsubset> min x y \<longleftrightarrow> z \<sqsubset> x \<and> z \<sqsubset> y"
haftmann@21412:   unfolding min_def le_less using less_linear by (auto intro: less_trans)
haftmann@21383: 
haftmann@21383: lemma max_less_iff_conj [simp]:
haftmann@21383:   "max x y \<sqsubset> z \<longleftrightarrow> x \<sqsubset> z \<and> y \<sqsubset> z"
haftmann@21412:   unfolding max_def le_less using less_linear by (auto intro: less_trans)
haftmann@21383: 
haftmann@21383: lemma split_min:
haftmann@21383:   "P (min i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P i) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P j)"
haftmann@21383:   by (simp add: min_def)
haftmann@21383: 
haftmann@21383: lemma split_max:
haftmann@21383:   "P (max i j) \<longleftrightarrow> (i \<sqsubseteq> j \<longrightarrow> P j) \<and> (\<not> i \<sqsubseteq> j \<longrightarrow> P i)"
haftmann@21383:   by (simp add: max_def)
haftmann@21383: 
haftmann@21248: end
haftmann@21248: 
haftmann@21248: 
haftmann@21248: subsection {* Name duplicates *}
haftmann@21248: 
haftmann@22316: lemmas order_refl [iff] = preorder_class.refl
haftmann@22316: lemmas order_trans = preorder_class.trans
haftmann@22316: lemmas order_less_le = preorder_class.less_le
haftmann@22316: lemmas order_eq_refl = preorder_class.eq_refl
haftmann@22316: lemmas order_less_irrefl = preorder_class.less_irrefl
haftmann@22316: lemmas order_le_less = preorder_class.le_less
haftmann@22316: lemmas order_le_imp_less_or_eq = preorder_class.le_imp_less_or_eq
haftmann@22316: lemmas order_less_imp_le = preorder_class.less_imp_le
haftmann@22316: lemmas order_less_imp_not_eq = preorder_class.less_imp_not_eq
haftmann@22316: lemmas order_less_imp_not_eq2 = preorder_class.less_imp_not_eq2
haftmann@22316: lemmas order_neq_le_trans = preorder_class.neq_le_trans
haftmann@22316: lemmas order_le_neq_trans = preorder_class.le_neq_trans
haftmann@22316: 
haftmann@22316: lemmas order_antisym = order_class.antisym
haftmann@22316: lemmas order_less_not_sym = order_class.less_not_sym
haftmann@22316: lemmas order_less_asym = order_class.less_asym
haftmann@22316: lemmas order_eq_iff = order_class.eq_iff
haftmann@22316: lemmas order_antisym_conv = order_class.antisym_conv
haftmann@22316: lemmas less_imp_neq = order_class.less_imp_neq
haftmann@22316: lemmas order_less_trans = order_class.less_trans
haftmann@22316: lemmas order_le_less_trans = order_class.le_less_trans
haftmann@22316: lemmas order_less_le_trans = order_class.less_le_trans
haftmann@22316: lemmas order_less_imp_not_less = order_class.less_imp_not_less
haftmann@22316: lemmas order_less_imp_triv = order_class.less_imp_triv
haftmann@22316: lemmas order_less_asym' = order_class.less_asym'
haftmann@22316: 
haftmann@22316: lemmas linorder_linear = linorder_class.linear
haftmann@22316: lemmas linorder_less_linear = linorder_class.less_linear
haftmann@22316: lemmas linorder_le_less_linear = linorder_class.le_less_linear
haftmann@22316: lemmas linorder_le_cases = linorder_class.le_cases
haftmann@22316: lemmas linorder_cases = linorder_class.cases
haftmann@22316: lemmas linorder_not_less = linorder_class.not_less
haftmann@22316: lemmas linorder_not_le = linorder_class.not_le
haftmann@22316: lemmas linorder_neq_iff = linorder_class.neq_iff
haftmann@22316: lemmas linorder_neqE = linorder_class.neqE
haftmann@22316: lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
haftmann@22316: lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
haftmann@22316: lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
haftmann@22316: lemmas leI = linorder_class.leI
haftmann@22316: lemmas leD = linorder_class.leD
haftmann@22316: lemmas not_leE = linorder_class.not_leE
paulson@16796: 
haftmann@21083: 
haftmann@21083: subsection {* Reasoning tools setup *}
haftmann@21083: 
haftmann@21091: ML {*
haftmann@21091: local
haftmann@21091: 
haftmann@21091: fun decomp_gen sort thy (Trueprop $ t) =
haftmann@21248:   let
haftmann@21248:     fun of_sort t =
haftmann@21248:       let
haftmann@21248:         val T = type_of t
haftmann@21248:       in
haftmann@21091:         (* exclude numeric types: linear arithmetic subsumes transitivity *)
haftmann@21248:         T <> HOLogic.natT andalso T <> HOLogic.intT
haftmann@21248:           andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
haftmann@21248:       end;
haftmann@21248:     fun dec (Const ("Not", _) $ t) = (case dec t
haftmann@21248:           of NONE => NONE
haftmann@21248:            | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
haftmann@21248:       | dec (Const ("op =",  _) $ t1 $ t2) =
haftmann@21248:           if of_sort t1
haftmann@21248:           then SOME (t1, "=", t2)
haftmann@21248:           else NONE
haftmann@21248:       | dec (Const ("Orderings.less_eq",  _) $ t1 $ t2) =
haftmann@21248:           if of_sort t1
haftmann@21248:           then SOME (t1, "<=", t2)
haftmann@21248:           else NONE
haftmann@21248:       | dec (Const ("Orderings.less",  _) $ t1 $ t2) =
haftmann@21248:           if of_sort t1
haftmann@21248:           then SOME (t1, "<", t2)
haftmann@21248:           else NONE
haftmann@21248:       | dec _ = NONE;
haftmann@21091:   in dec t end;
haftmann@21091: 
haftmann@21091: in
haftmann@21091: 
haftmann@21091: (* The setting up of Quasi_Tac serves as a demo.  Since there is no
haftmann@21091:    class for quasi orders, the tactics Quasi_Tac.trans_tac and
haftmann@21091:    Quasi_Tac.quasi_tac are not of much use. *)
haftmann@21091: 
haftmann@21248: structure Quasi_Tac = Quasi_Tac_Fun (
haftmann@21248: struct
haftmann@21248:   val le_trans = thm "order_trans";
haftmann@21248:   val le_refl = thm "order_refl";
haftmann@21248:   val eqD1 = thm "order_eq_refl";
haftmann@21248:   val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248:   val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248:   val less_imp_le = thm "order_less_imp_le";
haftmann@21248:   val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248:   val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248:   val less_imp_neq = thm "less_imp_neq";
haftmann@21248:   val decomp_trans = decomp_gen ["Orderings.order"];
haftmann@21248:   val decomp_quasi = decomp_gen ["Orderings.order"];
haftmann@21248: end);
haftmann@21091: 
haftmann@21091: structure Order_Tac = Order_Tac_Fun (
haftmann@21248: struct
haftmann@21248:   val less_reflE = thm "order_less_irrefl" RS thm "notE";
haftmann@21248:   val le_refl = thm "order_refl";
haftmann@21248:   val less_imp_le = thm "order_less_imp_le";
haftmann@21248:   val not_lessI = thm "linorder_not_less" RS thm "iffD2";
haftmann@21248:   val not_leI = thm "linorder_not_le" RS thm "iffD2";
haftmann@21248:   val not_lessD = thm "linorder_not_less" RS thm "iffD1";
haftmann@21248:   val not_leD = thm "linorder_not_le" RS thm "iffD1";
haftmann@21248:   val eqI = thm "order_antisym";
haftmann@21248:   val eqD1 = thm "order_eq_refl";
haftmann@21248:   val eqD2 = thm "sym" RS thm "order_eq_refl";
haftmann@21248:   val less_trans = thm "order_less_trans";
haftmann@21248:   val less_le_trans = thm "order_less_le_trans";
haftmann@21248:   val le_less_trans = thm "order_le_less_trans";
haftmann@21248:   val le_trans = thm "order_trans";
haftmann@21248:   val le_neq_trans = thm "order_le_neq_trans";
haftmann@21248:   val neq_le_trans = thm "order_neq_le_trans";
haftmann@21248:   val less_imp_neq = thm "less_imp_neq";
haftmann@21248:   val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
haftmann@21248:   val not_sym = thm "not_sym";
haftmann@21248:   val decomp_part = decomp_gen ["Orderings.order"];
haftmann@21248:   val decomp_lin = decomp_gen ["Orderings.linorder"];
haftmann@21248: end);
haftmann@21091: 
haftmann@21091: end;
haftmann@21091: *}
haftmann@21091: 
haftmann@21083: setup {*
haftmann@21083: let
haftmann@21083: 
haftmann@21083: val order_antisym_conv = thm "order_antisym_conv"
haftmann@21083: val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
haftmann@21083: val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
haftmann@21083: val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
haftmann@21083: 
haftmann@21083: fun prp t thm = (#prop (rep_thm thm) = t);
nipkow@15524: 
haftmann@21083: fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
haftmann@21083:   let val prems = prems_of_ss ss;
haftmann@21083:       val less = Const("Orderings.less",T);
haftmann@21083:       val t = HOLogic.mk_Trueprop(le $ s $ r);
haftmann@21083:   in case find_first (prp t) prems of
haftmann@21083:        NONE =>
haftmann@21083:          let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
haftmann@21083:          in case find_first (prp t) prems of
haftmann@21083:               NONE => NONE
haftmann@21083:             | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv1))
haftmann@21083:          end
haftmann@21083:      | SOME thm => SOME(mk_meta_eq(thm RS order_antisym_conv))
haftmann@21083:   end
haftmann@21083:   handle THM _ => NONE;
nipkow@15524: 
haftmann@21083: fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
haftmann@21083:   let val prems = prems_of_ss ss;
haftmann@21083:       val le = Const("Orderings.less_eq",T);
haftmann@21083:       val t = HOLogic.mk_Trueprop(le $ r $ s);
haftmann@21083:   in case find_first (prp t) prems of
haftmann@21083:        NONE =>
haftmann@21083:          let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
haftmann@21083:          in case find_first (prp t) prems of
haftmann@21083:               NONE => NONE
haftmann@21083:             | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv3))
haftmann@21083:          end
haftmann@21083:      | SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
haftmann@21083:   end
haftmann@21083:   handle THM _ => NONE;
nipkow@15524: 
haftmann@21248: fun add_simprocs procs thy =
haftmann@21248:   (Simplifier.change_simpset_of thy (fn ss => ss
haftmann@21248:     addsimprocs (map (fn (name, raw_ts, proc) =>
haftmann@21248:       Simplifier.simproc thy name raw_ts proc)) procs); thy);
haftmann@21248: fun add_solver name tac thy =
haftmann@21248:   (Simplifier.change_simpset_of thy (fn ss => ss addSolver
haftmann@21248:     (mk_solver name (K tac))); thy);
haftmann@21083: 
haftmann@21083: in
haftmann@21248:   add_simprocs [
haftmann@21248:        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
haftmann@21248:        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
haftmann@21248:      ]
haftmann@21248:   #> add_solver "Trans_linear" Order_Tac.linear_tac
haftmann@21248:   #> add_solver "Trans_partial" Order_Tac.partial_tac
haftmann@21248:   (* Adding the transitivity reasoners also as safe solvers showed a slight
haftmann@21248:      speed up, but the reasoning strength appears to be not higher (at least
haftmann@21248:      no breaking of additional proofs in the entire HOL distribution, as
haftmann@21248:      of 5 March 2004, was observed). *)
haftmann@21083: end
haftmann@21083: *}
nipkow@15524: 
nipkow@15524: 
haftmann@21083: subsection {* Bounded quantifiers *}
haftmann@21083: 
haftmann@21083: syntax
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (xsymbols)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (HOL)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: syntax (HTML output)
wenzelm@21180:   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
wenzelm@21180:   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
wenzelm@21180:   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
wenzelm@21180:   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
haftmann@21083: 
haftmann@21083: translations
haftmann@21083:   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
haftmann@21083:   "EX x<y. P"    =>  "EX x. x < y \<and> P"
haftmann@21083:   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
haftmann@21083:   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
haftmann@21083:   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
haftmann@21083:   "EX x>y. P"    =>  "EX x. x > y \<and> P"
haftmann@21083:   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
haftmann@21083:   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
haftmann@21083: 
haftmann@21083: print_translation {*
haftmann@21083: let
wenzelm@21180:   val syntax_name = Sign.const_syntax_name (the_context ());
wenzelm@21524:   val binder_name = Syntax.binder_name o syntax_name;
wenzelm@21524:   val All_binder = binder_name "All";
wenzelm@21524:   val Ex_binder = binder_name "Ex";
wenzelm@21180:   val impl = syntax_name "op -->";
wenzelm@21180:   val conj = syntax_name "op &";
wenzelm@21180:   val less = syntax_name "Orderings.less";
wenzelm@21180:   val less_eq = syntax_name "Orderings.less_eq";
wenzelm@21180: 
wenzelm@21180:   val trans =
wenzelm@21524:    [((All_binder, impl, less), ("_All_less", "_All_greater")),
wenzelm@21524:     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
wenzelm@21524:     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
wenzelm@21524:     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
wenzelm@21180: 
krauss@22344:   fun matches_bound v t = 
krauss@22344:      case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
krauss@22344:               | _ => false
krauss@22344:   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
krauss@22344:   fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
wenzelm@21180: 
wenzelm@21180:   fun tr' q = (q,
wenzelm@21180:     fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
wenzelm@21180:       (case AList.lookup (op =) trans (q, c, d) of
wenzelm@21180:         NONE => raise Match
wenzelm@21180:       | SOME (l, g) =>
krauss@22344:           if matches_bound v t andalso not (contains_var v u) then mk v l u P
krauss@22344:           else if matches_bound v u andalso not (contains_var v t) then mk v g t P
krauss@22344:           else raise Match)
wenzelm@21180:      | _ => raise Match);
wenzelm@21524: in [tr' All_binder, tr' Ex_binder] end
haftmann@21083: *}
haftmann@21083: 
haftmann@21083: 
haftmann@21383: subsection {* Transitivity reasoning *}
haftmann@21383: 
haftmann@21383: lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
haftmann@21383:   by (rule subst)
haftmann@21383: 
haftmann@21383: lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
haftmann@21383:   by (rule ssubst)
haftmann@21383: 
haftmann@21383: lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
haftmann@21383:   by (rule subst)
haftmann@21383: 
haftmann@21383: lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
haftmann@21383:   by (rule ssubst)
haftmann@21383: 
haftmann@21383: lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383:   also assume "f b < c"
haftmann@21383:   finally (order_less_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a < f b"
haftmann@21383:   also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383:   finally (order_less_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383:   also assume "f b < c"
haftmann@21383:   finally (order_le_less_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a <= f b"
haftmann@21383:   also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383:   finally (order_le_less_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383:   also assume "f b <= c"
haftmann@21383:   finally (order_less_le_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a < f b"
haftmann@21383:   also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383:   finally (order_less_le_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a <= f b"
haftmann@21383:   also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383:   finally (order_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383:   also assume "f b <= c"
haftmann@21383:   finally (order_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a <= b" hence "f a <= f b" by (rule r)
haftmann@21383:   also assume "f b = c"
haftmann@21383:   finally (ord_le_eq_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
haftmann@21383:   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x <= y ==> f x <= f y"
haftmann@21383:   assume "a = f b"
haftmann@21383:   also assume "b <= c" hence "f b <= f c" by (rule r)
haftmann@21383:   finally (ord_eq_le_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma ord_less_eq_subst: "a < b ==> f b = c ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> f a < c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a < b" hence "f a < f b" by (rule r)
haftmann@21383:   also assume "f b = c"
haftmann@21383:   finally (ord_less_eq_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: lemma ord_eq_less_subst: "a = f b ==> b < c ==>
haftmann@21383:   (!!x y. x < y ==> f x < f y) ==> a < f c"
haftmann@21383: proof -
haftmann@21383:   assume r: "!!x y. x < y ==> f x < f y"
haftmann@21383:   assume "a = f b"
haftmann@21383:   also assume "b < c" hence "f b < f c" by (rule r)
haftmann@21383:   finally (ord_eq_less_trans) show ?thesis .
haftmann@21383: qed
haftmann@21383: 
haftmann@21383: text {*
haftmann@21383:   Note that this list of rules is in reverse order of priorities.
haftmann@21383: *}
haftmann@21383: 
haftmann@21383: lemmas order_trans_rules [trans] =
haftmann@21383:   order_less_subst2
haftmann@21383:   order_less_subst1
haftmann@21383:   order_le_less_subst2
haftmann@21383:   order_le_less_subst1
haftmann@21383:   order_less_le_subst2
haftmann@21383:   order_less_le_subst1
haftmann@21383:   order_subst2
haftmann@21383:   order_subst1
haftmann@21383:   ord_le_eq_subst
haftmann@21383:   ord_eq_le_subst
haftmann@21383:   ord_less_eq_subst
haftmann@21383:   ord_eq_less_subst
haftmann@21383:   forw_subst
haftmann@21383:   back_subst
haftmann@21383:   rev_mp
haftmann@21383:   mp
haftmann@21383:   order_neq_le_trans
haftmann@21383:   order_le_neq_trans
haftmann@21383:   order_less_trans
haftmann@21383:   order_less_asym'
haftmann@21383:   order_le_less_trans
haftmann@21383:   order_less_le_trans
haftmann@21383:   order_trans
haftmann@21383:   order_antisym
haftmann@21383:   ord_le_eq_trans
haftmann@21383:   ord_eq_le_trans
haftmann@21383:   ord_less_eq_trans
haftmann@21383:   ord_eq_less_trans
haftmann@21383:   trans
haftmann@21383: 
haftmann@21083: 
wenzelm@21180: (* FIXME cleanup *)
wenzelm@21180: 
haftmann@21083: text {* These support proving chains of decreasing inequalities
haftmann@21083:     a >= b >= c ... in Isar proofs. *}
haftmann@21083: 
haftmann@21083: lemma xt1:
haftmann@21083:   "a = b ==> b > c ==> a > c"
haftmann@21083:   "a > b ==> b = c ==> a > c"
haftmann@21083:   "a = b ==> b >= c ==> a >= c"
haftmann@21083:   "a >= b ==> b = c ==> a >= c"
haftmann@21083:   "(x::'a::order) >= y ==> y >= x ==> x = y"
haftmann@21083:   "(x::'a::order) >= y ==> y >= z ==> x >= z"
haftmann@21083:   "(x::'a::order) > y ==> y >= z ==> x > z"
haftmann@21083:   "(x::'a::order) >= y ==> y > z ==> x > z"
haftmann@21083:   "(a::'a::order) > b ==> b > a ==> ?P"
haftmann@21083:   "(x::'a::order) > y ==> y > z ==> x > z"
haftmann@21083:   "(a::'a::order) >= b ==> a ~= b ==> a > b"
haftmann@21083:   "(a::'a::order) ~= b ==> a >= b ==> a > b"
haftmann@21083:   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
haftmann@21083:   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083:   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083:   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083: by auto
haftmann@21083: 
haftmann@21083: lemma xt2:
haftmann@21083:   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
haftmann@21083: by (subgoal_tac "f b >= f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
haftmann@21083:     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
haftmann@21083: by (subgoal_tac "f a >= f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
haftmann@21083:   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b >= f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a > f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b > f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
haftmann@21083:     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a >= f b", force, force)
haftmann@21083: 
haftmann@21083: lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> a > f c"
haftmann@21083: by (subgoal_tac "f b > f c", force, force)
haftmann@21083: 
haftmann@21083: lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
haftmann@21083:     (!!x y. x > y ==> f x > f y) ==> f a > c"
haftmann@21083: by (subgoal_tac "f a > f b", force, force)
haftmann@21083: 
haftmann@21083: lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
haftmann@21083: 
haftmann@21083: (* 
haftmann@21083:   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
haftmann@21083:   for the wrong thing in an Isar proof.
haftmann@21083: 
haftmann@21083:   The extra transitivity rules can be used as follows: 
haftmann@21083: 
haftmann@21083: lemma "(a::'a::order) > z"
haftmann@21083: proof -
haftmann@21083:   have "a >= b" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also have "?rhs >= c" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
haftmann@21083:     sorry
haftmann@21083:   also (xtrans) have "?rhs > z"
haftmann@21083:     sorry
haftmann@21083:   finally (xtrans) show ?thesis .
haftmann@21083: qed
haftmann@21083: 
haftmann@21083:   Alternatively, one can use "declare xtrans [trans]" and then
haftmann@21083:   leave out the "(xtrans)" above.
haftmann@21083: *)
haftmann@21083: 
haftmann@21546: subsection {* Order on bool *}
haftmann@21546: 
haftmann@21546: instance bool :: linorder 
haftmann@21546:   le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
haftmann@21546:   less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
haftmann@21546:   by default (auto simp add: le_bool_def less_bool_def)
haftmann@21546: 
haftmann@21546: lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
haftmann@21546:   by (simp add: le_bool_def)
haftmann@21546: 
haftmann@21546: lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
haftmann@21546:   by (simp add: le_bool_def)
haftmann@21546: 
haftmann@21546: lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@21546:   by (simp add: le_bool_def)
haftmann@21546: 
haftmann@21546: lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
haftmann@21546:   by (simp add: le_bool_def)
haftmann@21546: 
haftmann@21383: subsection {* Monotonicity, syntactic least value operator and min/max *}
haftmann@21083: 
haftmann@21216: locale mono =
haftmann@21216:   fixes f
haftmann@21216:   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
haftmann@21216: 
haftmann@21216: lemmas monoI [intro?] = mono.intro
haftmann@21216:   and monoD [dest?] = mono.mono
haftmann@21083: 
haftmann@21083: constdefs
haftmann@21083:   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
haftmann@21083:   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
haftmann@21083:     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
haftmann@21083: 
haftmann@21383: lemma LeastI2_order:
haftmann@21383:   "[| P (x::'a::order);
haftmann@21383:       !!y. P y ==> x <= y;
haftmann@21383:       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
haftmann@21383:    ==> Q (Least P)"
haftmann@21383:   apply (unfold Least_def)
haftmann@21383:   apply (rule theI2)
haftmann@21383:     apply (blast intro: order_antisym)+
haftmann@21383:   done
haftmann@21383: 
haftmann@21383: lemma Least_equality:
haftmann@21383:     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
haftmann@21383:   apply (simp add: Least_def)
haftmann@21383:   apply (rule the_equality)
haftmann@21383:   apply (auto intro!: order_antisym)
haftmann@21383:   done
haftmann@21383: 
haftmann@21083: constdefs
haftmann@21083:   min :: "['a::ord, 'a] => 'a"
haftmann@21083:   "min a b == (if a <= b then a else b)"
haftmann@21083:   max :: "['a::ord, 'a] => 'a"
haftmann@21083:   "max a b == (if a <= b then b else a)"
haftmann@21083: 
haftmann@21383: lemma min_linorder:
haftmann@21383:   "linorder.min (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = min"
haftmann@22316:   by rule+ (simp add: min_def linorder_class.min_def)
haftmann@21383: 
haftmann@21383: lemma max_linorder:
haftmann@21383:   "linorder.max (op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool) = max"
haftmann@22316:   by rule+ (simp add: max_def linorder_class.max_def)
haftmann@21383: 
haftmann@22316: lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded min_linorder]
haftmann@22316: lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded max_linorder]
haftmann@22316: lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded min_linorder]
haftmann@22316: lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded max_linorder]
haftmann@22316: lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded min_linorder]
haftmann@22316: lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded max_linorder]
haftmann@22316: lemmas split_min = linorder_class.split_min [unfolded min_linorder]
haftmann@22316: lemmas split_max = linorder_class.split_max [unfolded max_linorder]
haftmann@21383: 
haftmann@21383: lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
haftmann@21383:   by (simp add: min_def)
haftmann@21383: 
haftmann@21383: lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
haftmann@21383:   by (simp add: max_def)
haftmann@21383: 
haftmann@21383: lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
haftmann@21383:   apply (simp add: min_def)
haftmann@21383:   apply (blast intro: order_antisym)
haftmann@21383:   done
haftmann@21383: 
haftmann@21383: lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
haftmann@21383:   apply (simp add: max_def)
haftmann@21383:   apply (blast intro: order_antisym)
haftmann@21383:   done
haftmann@21383: 
haftmann@21383: lemma min_of_mono:
haftmann@21383:     "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
haftmann@21383:   by (simp add: min_def)
haftmann@21383: 
haftmann@21383: lemma max_of_mono:
haftmann@21383:     "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
haftmann@21383:   by (simp add: max_def)
haftmann@21383: 
wenzelm@21673: 
wenzelm@21673: subsection {* Basic ML bindings *}
wenzelm@21673: 
wenzelm@21673: ML {*
wenzelm@21673: val leD = thm "leD";
wenzelm@21673: val leI = thm "leI";
wenzelm@21673: val linorder_neqE = thm "linorder_neqE";
wenzelm@21673: val linorder_neq_iff = thm "linorder_neq_iff";
wenzelm@21673: val linorder_not_le = thm "linorder_not_le";
wenzelm@21673: val linorder_not_less = thm "linorder_not_less";
wenzelm@21673: val monoD = thm "monoD";
wenzelm@21673: val monoI = thm "monoI";
wenzelm@21673: val order_antisym = thm "order_antisym";
wenzelm@21673: val order_less_irrefl = thm "order_less_irrefl";
wenzelm@21673: val order_refl = thm "order_refl";
wenzelm@21673: val order_trans = thm "order_trans";
wenzelm@21673: val split_max = thm "split_max";
wenzelm@21673: val split_min = thm "split_min";
wenzelm@21673: *}
wenzelm@21673: 
wenzelm@21673: ML {*
wenzelm@21673: structure HOL =
wenzelm@21673: struct
wenzelm@21673:   val thy = theory "HOL";
wenzelm@21673: end;
wenzelm@21673: *}  -- "belongs to theory HOL"
wenzelm@21673: 
nipkow@15524: end




