src/HOL/Orderings.thy
author haftmann
Thu May 10 10:21:44 2007 +0200 (2007-05-10)
changeset 22916 8caf6da610e2
parent 22886 cdff6ef76009
child 22997 d4f3b015b50b
permissions -rw-r--r--
tuned
     1 (*  Title:      HOL/Orderings.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
     4 *)
     5 
     6 header {* Syntactic and abstract orders *}
     7 
     8 theory Orderings
     9 imports Code_Generator
    10 begin
    11 
    12 subsection {* Order syntax *}
    13 
    14 class ord = type +
    15   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubseteq>" 50)
    16     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<sqsubset>" 50)
    17 begin
    18 
    19 notation
    20   less_eq  ("op \<^loc><=") and
    21   less_eq  ("(_/ \<^loc><= _)" [51, 51] 50) and
    22   less  ("op \<^loc><") and
    23   less  ("(_/ \<^loc>< _)"  [51, 51] 50)
    24   
    25 notation (xsymbols)
    26   less_eq  ("op \<^loc>\<le>") and
    27   less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
    28 
    29 notation (HTML output)
    30   less_eq  ("op \<^loc>\<le>") and
    31   less_eq  ("(_/ \<^loc>\<le> _)"  [51, 51] 50)
    32 
    33 abbreviation (input)
    34   greater  (infix "\<^loc>>" 50) where
    35   "x \<^loc>> y \<equiv> y \<^loc>< x"
    36 
    37 abbreviation (input)
    38   greater_eq  (infix "\<^loc>>=" 50) where
    39   "x \<^loc>>= y \<equiv> y \<^loc><= x"
    40 
    41 notation (input)
    42   greater_eq  (infix "\<^loc>\<ge>" 50)
    43 
    44 text {*
    45   syntactic min/max -- these definitions reach
    46   their usual semantics in class linorder ahead.
    47 *}
    48 
    49 definition
    50   min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
    51   "min a b = (if a \<^loc>\<le> b then a else b)"
    52 
    53 definition
    54   max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
    55   "max a b = (if a \<^loc>\<le> b then b else a)"
    56 
    57 end
    58 
    59 notation
    60   less_eq  ("op <=") and
    61   less_eq  ("(_/ <= _)" [51, 51] 50) and
    62   less  ("op <") and
    63   less  ("(_/ < _)"  [51, 51] 50)
    64   
    65 notation (xsymbols)
    66   less_eq  ("op \<le>") and
    67   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
    68 
    69 notation (HTML output)
    70   less_eq  ("op \<le>") and
    71   less_eq  ("(_/ \<le> _)"  [51, 51] 50)
    72 
    73 abbreviation (input)
    74   greater  (infix ">" 50) where
    75   "x > y \<equiv> y < x"
    76 
    77 abbreviation (input)
    78   greater_eq  (infix ">=" 50) where
    79   "x >= y \<equiv> y <= x"
    80 
    81 notation (input)
    82   greater_eq  (infix "\<ge>" 50)
    83 
    84 hide const min max
    85 
    86 definition
    87   min :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where
    88   "min a b = (if a \<le> b then a else b)"
    89 
    90 definition
    91   max :: "'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> 'a" where
    92   "max a b = (if a \<le> b then b else a)"
    93 
    94 lemma linorder_class_min:
    95   "ord.min (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = min"
    96   by rule+ (simp add: min_def ord_class.min_def)
    97 
    98 lemma linorder_class_max:
    99   "ord.max (op \<le> \<Colon> 'a\<Colon>ord \<Rightarrow> 'a \<Rightarrow> bool) = max"
   100   by rule+ (simp add: max_def ord_class.max_def)
   101 
   102 
   103 subsection {* Partial orders *}
   104 
   105 class order = ord +
   106   assumes less_le: "x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
   107   and order_refl [iff]: "x \<sqsubseteq> x"
   108   and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
   109   assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
   110 
   111 begin
   112 
   113 text {* Reflexivity. *}
   114 
   115 lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
   116     -- {* This form is useful with the classical reasoner. *}
   117   by (erule ssubst) (rule order_refl)
   118 
   119 lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
   120   by (simp add: less_le)
   121 
   122 lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"
   123     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
   124   by (simp add: less_le) blast
   125 
   126 lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
   127   unfolding less_le by blast
   128 
   129 lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
   130   unfolding less_le by blast
   131 
   132 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
   133   by (erule contrapos_pn, erule subst, rule less_irrefl)
   134 
   135 
   136 text {* Useful for simplification, but too risky to include by default. *}
   137 
   138 lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
   139   by auto
   140 
   141 lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
   142   by auto
   143 
   144 
   145 text {* Transitivity rules for calculational reasoning *}
   146 
   147 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
   148   by (simp add: less_le)
   149 
   150 lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
   151   by (simp add: less_le)
   152 
   153 
   154 text {* Asymmetry. *}
   155 
   156 lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
   157   by (simp add: less_le antisym)
   158 
   159 lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
   160   by (drule less_not_sym, erule contrapos_np) simp
   161 
   162 lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
   163   by (blast intro: antisym)
   164 
   165 lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
   166   by (blast intro: antisym)
   167 
   168 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
   169   by (erule contrapos_pn, erule subst, rule less_irrefl)
   170 
   171 
   172 text {* Transitivity. *}
   173 
   174 lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
   175   by (simp add: less_le) (blast intro: order_trans antisym)
   176 
   177 lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
   178   by (simp add: less_le) (blast intro: order_trans antisym)
   179 
   180 lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
   181   by (simp add: less_le) (blast intro: order_trans antisym)
   182 
   183 
   184 text {* Useful for simplification, but too risky to include by default. *}
   185 
   186 lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
   187   by (blast elim: less_asym)
   188 
   189 lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
   190   by (blast elim: less_asym)
   191 
   192 
   193 text {* Transitivity rules for calculational reasoning *}
   194 
   195 lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
   196   by (rule less_asym)
   197 
   198 
   199 text {* Reverse order *}
   200 
   201 lemma order_reverse:
   202   "order_pred (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
   203   by unfold_locales
   204     (simp add: less_le, auto intro: antisym order_trans)
   205 
   206 end
   207 
   208 
   209 subsection {* Linear (total) orders *}
   210 
   211 class linorder = order +
   212   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
   213 begin
   214 
   215 lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
   216   unfolding less_le using less_le linear by blast 
   217 
   218 lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
   219   by (simp add: le_less less_linear)
   220 
   221 lemma le_cases [case_names le ge]:
   222   "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
   223   using linear by blast
   224 
   225 lemma linorder_cases [case_names less equal greater]:
   226     "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
   227   using less_linear by blast
   228 
   229 lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
   230   apply (simp add: less_le)
   231   using linear apply (blast intro: antisym)
   232   done
   233 
   234 lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
   235   apply (simp add: less_le)
   236   using linear apply (blast intro: antisym)
   237   done
   238 
   239 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
   240   by (cut_tac x = x and y = y in less_linear, auto)
   241 
   242 lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
   243   by (simp add: neq_iff) blast
   244 
   245 lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
   246   by (blast intro: antisym dest: not_less [THEN iffD1])
   247 
   248 lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
   249   by (blast intro: antisym dest: not_less [THEN iffD1])
   250 
   251 lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
   252   by (blast intro: antisym dest: not_less [THEN iffD1])
   253 
   254 text{*Replacing the old Nat.leI*}
   255 lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
   256   unfolding not_less .
   257 
   258 lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
   259   unfolding not_less .
   260 
   261 (*FIXME inappropriate name (or delete altogether)*)
   262 lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
   263   unfolding not_le .
   264 
   265 
   266 text {* Reverse order *}
   267 
   268 lemma linorder_reverse:
   269   "linorder_pred (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
   270   by unfold_locales
   271     (simp add: less_le, auto intro: antisym order_trans simp add: linear)
   272 
   273 
   274 text {* min/max properties *}
   275 
   276 lemma min_le_iff_disj:
   277   "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
   278   unfolding min_def using linear by (auto intro: order_trans)
   279 
   280 lemma le_max_iff_disj:
   281   "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
   282   unfolding max_def using linear by (auto intro: order_trans)
   283 
   284 lemma min_less_iff_disj:
   285   "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
   286   unfolding min_def le_less using less_linear by (auto intro: less_trans)
   287 
   288 lemma less_max_iff_disj:
   289   "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
   290   unfolding max_def le_less using less_linear by (auto intro: less_trans)
   291 
   292 lemma min_less_iff_conj [simp]:
   293   "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
   294   unfolding min_def le_less using less_linear by (auto intro: less_trans)
   295 
   296 lemma max_less_iff_conj [simp]:
   297   "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
   298   unfolding max_def le_less using less_linear by (auto intro: less_trans)
   299 
   300 lemma split_min:
   301   "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
   302   by (simp add: min_def)
   303 
   304 lemma split_max:
   305   "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
   306   by (simp add: max_def)
   307 
   308 end
   309 
   310 subsection {* Name duplicates -- including min/max interpretation *}
   311 
   312 lemmas order_less_le = less_le
   313 lemmas order_eq_refl = order_class.eq_refl
   314 lemmas order_less_irrefl = order_class.less_irrefl
   315 lemmas order_le_less = order_class.le_less
   316 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
   317 lemmas order_less_imp_le = order_class.less_imp_le
   318 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
   319 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
   320 lemmas order_neq_le_trans = order_class.neq_le_trans
   321 lemmas order_le_neq_trans = order_class.le_neq_trans
   322 
   323 lemmas order_antisym = antisym
   324 lemmas order_less_not_sym = order_class.less_not_sym
   325 lemmas order_less_asym = order_class.less_asym
   326 lemmas order_eq_iff = order_class.eq_iff
   327 lemmas order_antisym_conv = order_class.antisym_conv
   328 lemmas less_imp_neq = order_class.less_imp_neq
   329 lemmas order_less_trans = order_class.less_trans
   330 lemmas order_le_less_trans = order_class.le_less_trans
   331 lemmas order_less_le_trans = order_class.less_le_trans
   332 lemmas order_less_imp_not_less = order_class.less_imp_not_less
   333 lemmas order_less_imp_triv = order_class.less_imp_triv
   334 lemmas order_less_asym' = order_class.less_asym'
   335 
   336 lemmas linorder_linear = linear
   337 lemmas linorder_less_linear = linorder_class.less_linear
   338 lemmas linorder_le_less_linear = linorder_class.le_less_linear
   339 lemmas linorder_le_cases = linorder_class.le_cases
   340 lemmas linorder_not_less = linorder_class.not_less
   341 lemmas linorder_not_le = linorder_class.not_le
   342 lemmas linorder_neq_iff = linorder_class.neq_iff
   343 lemmas linorder_neqE = linorder_class.neqE
   344 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
   345 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
   346 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
   347 lemmas leI = linorder_class.leI
   348 lemmas leD = linorder_class.leD
   349 lemmas not_leE = linorder_class.not_leE
   350 
   351 lemmas min_le_iff_disj = linorder_class.min_le_iff_disj [unfolded linorder_class_min]
   352 lemmas le_max_iff_disj = linorder_class.le_max_iff_disj [unfolded linorder_class_max]
   353 lemmas min_less_iff_disj = linorder_class.min_less_iff_disj [unfolded linorder_class_min]
   354 lemmas less_max_iff_disj = linorder_class.less_max_iff_disj [unfolded linorder_class_max]
   355 lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj [unfolded linorder_class_min]
   356 lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj [unfolded linorder_class_max]
   357 lemmas split_min = linorder_class.split_min [unfolded linorder_class_min]
   358 lemmas split_max = linorder_class.split_max [unfolded linorder_class_max]
   359 
   360 
   361 subsection {* Reasoning tools setup *}
   362 
   363 ML {*
   364 local
   365 
   366 fun decomp_gen sort thy (Trueprop $ t) =
   367   let
   368     fun of_sort t =
   369       let
   370         val T = type_of t
   371       in
   372         (* exclude numeric types: linear arithmetic subsumes transitivity *)
   373         T <> HOLogic.natT andalso T <> HOLogic.intT
   374           andalso T <> HOLogic.realT andalso Sign.of_sort thy (T, sort)
   375       end;
   376     fun dec (Const (@{const_name Not}, _) $ t) = (case dec t
   377           of NONE => NONE
   378            | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
   379       | dec (Const (@{const_name "op ="},  _) $ t1 $ t2) =
   380           if of_sort t1
   381           then SOME (t1, "=", t2)
   382           else NONE
   383       | dec (Const (@{const_name less_eq},  _) $ t1 $ t2) =
   384           if of_sort t1
   385           then SOME (t1, "<=", t2)
   386           else NONE
   387       | dec (Const (@{const_name less},  _) $ t1 $ t2) =
   388           if of_sort t1
   389           then SOME (t1, "<", t2)
   390           else NONE
   391       | dec _ = NONE;
   392   in dec t end;
   393 
   394 in
   395 
   396 (* sorry - there is no preorder class
   397 structure Quasi_Tac = Quasi_Tac_Fun (
   398 struct
   399   val le_trans = thm "order_trans";
   400   val le_refl = thm "order_refl";
   401   val eqD1 = thm "order_eq_refl";
   402   val eqD2 = thm "sym" RS thm "order_eq_refl";
   403   val less_reflE = thm "order_less_irrefl" RS thm "notE";
   404   val less_imp_le = thm "order_less_imp_le";
   405   val le_neq_trans = thm "order_le_neq_trans";
   406   val neq_le_trans = thm "order_neq_le_trans";
   407   val less_imp_neq = thm "less_imp_neq";
   408   val decomp_trans = decomp_gen ["Orderings.preorder"];
   409   val decomp_quasi = decomp_gen ["Orderings.preorder"];
   410 end);*)
   411 
   412 structure Order_Tac = Order_Tac_Fun (
   413 struct
   414   val less_reflE = thm "order_less_irrefl" RS thm "notE";
   415   val le_refl = thm "order_refl";
   416   val less_imp_le = thm "order_less_imp_le";
   417   val not_lessI = thm "linorder_not_less" RS thm "iffD2";
   418   val not_leI = thm "linorder_not_le" RS thm "iffD2";
   419   val not_lessD = thm "linorder_not_less" RS thm "iffD1";
   420   val not_leD = thm "linorder_not_le" RS thm "iffD1";
   421   val eqI = thm "order_antisym";
   422   val eqD1 = thm "order_eq_refl";
   423   val eqD2 = thm "sym" RS thm "order_eq_refl";
   424   val less_trans = thm "order_less_trans";
   425   val less_le_trans = thm "order_less_le_trans";
   426   val le_less_trans = thm "order_le_less_trans";
   427   val le_trans = thm "order_trans";
   428   val le_neq_trans = thm "order_le_neq_trans";
   429   val neq_le_trans = thm "order_neq_le_trans";
   430   val less_imp_neq = thm "less_imp_neq";
   431   val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
   432   val not_sym = thm "not_sym";
   433   val decomp_part = decomp_gen ["Orderings.order"];
   434   val decomp_lin = decomp_gen ["Orderings.linorder"];
   435 end);
   436 
   437 end;
   438 *}
   439 
   440 setup {*
   441 let
   442 
   443 fun prp t thm = (#prop (rep_thm thm) = t);
   444 
   445 fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
   446   let val prems = prems_of_ss ss;
   447       val less = Const (@{const_name less}, T);
   448       val t = HOLogic.mk_Trueprop(le $ s $ r);
   449   in case find_first (prp t) prems of
   450        NONE =>
   451          let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))
   452          in case find_first (prp t) prems of
   453               NONE => NONE
   454             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv1}))
   455          end
   456      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_antisym_conv}))
   457   end
   458   handle THM _ => NONE;
   459 
   460 fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =
   461   let val prems = prems_of_ss ss;
   462       val le = Const (@{const_name less_eq}, T);
   463       val t = HOLogic.mk_Trueprop(le $ r $ s);
   464   in case find_first (prp t) prems of
   465        NONE =>
   466          let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))
   467          in case find_first (prp t) prems of
   468               NONE => NONE
   469             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv3}))
   470          end
   471      | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_antisym_conv2}))
   472   end
   473   handle THM _ => NONE;
   474 
   475 fun add_simprocs procs thy =
   476   (Simplifier.change_simpset_of thy (fn ss => ss
   477     addsimprocs (map (fn (name, raw_ts, proc) =>
   478       Simplifier.simproc thy name raw_ts proc)) procs); thy);
   479 fun add_solver name tac thy =
   480   (Simplifier.change_simpset_of thy (fn ss => ss addSolver
   481     (mk_solver name (K tac))); thy);
   482 
   483 in
   484   add_simprocs [
   485        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
   486        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
   487      ]
   488   #> add_solver "Trans_linear" Order_Tac.linear_tac
   489   #> add_solver "Trans_partial" Order_Tac.partial_tac
   490   (* Adding the transitivity reasoners also as safe solvers showed a slight
   491      speed up, but the reasoning strength appears to be not higher (at least
   492      no breaking of additional proofs in the entire HOL distribution, as
   493      of 5 March 2004, was observed). *)
   494 end
   495 *}
   496 
   497 
   498 subsection {* Bounded quantifiers *}
   499 
   500 syntax
   501   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   502   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
   503   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   504   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
   505 
   506   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
   507   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
   508   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
   509   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
   510 
   511 syntax (xsymbols)
   512   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
   513   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
   514   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
   515   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
   516 
   517   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
   518   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
   519   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
   520   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
   521 
   522 syntax (HOL)
   523   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
   524   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
   525   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
   526   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
   527 
   528 syntax (HTML output)
   529   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
   530   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
   531   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
   532   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
   533 
   534   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
   535   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
   536   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
   537   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
   538 
   539 translations
   540   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
   541   "EX x<y. P"    =>  "EX x. x < y \<and> P"
   542   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
   543   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
   544   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
   545   "EX x>y. P"    =>  "EX x. x > y \<and> P"
   546   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
   547   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
   548 
   549 print_translation {*
   550 let
   551   val All_binder = Syntax.binder_name @{const_syntax All};
   552   val Ex_binder = Syntax.binder_name @{const_syntax Ex};
   553   val impl = @{const_syntax "op -->"};
   554   val conj = @{const_syntax "op &"};
   555   val less = @{const_syntax less};
   556   val less_eq = @{const_syntax less_eq};
   557 
   558   val trans =
   559    [((All_binder, impl, less), ("_All_less", "_All_greater")),
   560     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
   561     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
   562     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
   563 
   564   fun matches_bound v t = 
   565      case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')
   566               | _ => false
   567   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
   568   fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P
   569 
   570   fun tr' q = (q,
   571     fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
   572       (case AList.lookup (op =) trans (q, c, d) of
   573         NONE => raise Match
   574       | SOME (l, g) =>
   575           if matches_bound v t andalso not (contains_var v u) then mk v l u P
   576           else if matches_bound v u andalso not (contains_var v t) then mk v g t P
   577           else raise Match)
   578      | _ => raise Match);
   579 in [tr' All_binder, tr' Ex_binder] end
   580 *}
   581 
   582 
   583 subsection {* Transitivity reasoning *}
   584 
   585 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
   586   by (rule subst)
   587 
   588 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
   589   by (rule ssubst)
   590 
   591 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
   592   by (rule subst)
   593 
   594 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
   595   by (rule ssubst)
   596 
   597 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
   598   (!!x y. x < y ==> f x < f y) ==> f a < c"
   599 proof -
   600   assume r: "!!x y. x < y ==> f x < f y"
   601   assume "a < b" hence "f a < f b" by (rule r)
   602   also assume "f b < c"
   603   finally (order_less_trans) show ?thesis .
   604 qed
   605 
   606 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
   607   (!!x y. x < y ==> f x < f y) ==> a < f c"
   608 proof -
   609   assume r: "!!x y. x < y ==> f x < f y"
   610   assume "a < f b"
   611   also assume "b < c" hence "f b < f c" by (rule r)
   612   finally (order_less_trans) show ?thesis .
   613 qed
   614 
   615 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
   616   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
   617 proof -
   618   assume r: "!!x y. x <= y ==> f x <= f y"
   619   assume "a <= b" hence "f a <= f b" by (rule r)
   620   also assume "f b < c"
   621   finally (order_le_less_trans) show ?thesis .
   622 qed
   623 
   624 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
   625   (!!x y. x < y ==> f x < f y) ==> a < f c"
   626 proof -
   627   assume r: "!!x y. x < y ==> f x < f y"
   628   assume "a <= f b"
   629   also assume "b < c" hence "f b < f c" by (rule r)
   630   finally (order_le_less_trans) show ?thesis .
   631 qed
   632 
   633 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
   634   (!!x y. x < y ==> f x < f y) ==> f a < c"
   635 proof -
   636   assume r: "!!x y. x < y ==> f x < f y"
   637   assume "a < b" hence "f a < f b" by (rule r)
   638   also assume "f b <= c"
   639   finally (order_less_le_trans) show ?thesis .
   640 qed
   641 
   642 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
   643   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
   644 proof -
   645   assume r: "!!x y. x <= y ==> f x <= f y"
   646   assume "a < f b"
   647   also assume "b <= c" hence "f b <= f c" by (rule r)
   648   finally (order_less_le_trans) show ?thesis .
   649 qed
   650 
   651 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
   652   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
   653 proof -
   654   assume r: "!!x y. x <= y ==> f x <= f y"
   655   assume "a <= f b"
   656   also assume "b <= c" hence "f b <= f c" by (rule r)
   657   finally (order_trans) show ?thesis .
   658 qed
   659 
   660 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
   661   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
   662 proof -
   663   assume r: "!!x y. x <= y ==> f x <= f y"
   664   assume "a <= b" hence "f a <= f b" by (rule r)
   665   also assume "f b <= c"
   666   finally (order_trans) show ?thesis .
   667 qed
   668 
   669 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
   670   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
   671 proof -
   672   assume r: "!!x y. x <= y ==> f x <= f y"
   673   assume "a <= b" hence "f a <= f b" by (rule r)
   674   also assume "f b = c"
   675   finally (ord_le_eq_trans) show ?thesis .
   676 qed
   677 
   678 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
   679   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
   680 proof -
   681   assume r: "!!x y. x <= y ==> f x <= f y"
   682   assume "a = f b"
   683   also assume "b <= c" hence "f b <= f c" by (rule r)
   684   finally (ord_eq_le_trans) show ?thesis .
   685 qed
   686 
   687 lemma ord_less_eq_subst: "a < b ==> f b = c ==>
   688   (!!x y. x < y ==> f x < f y) ==> f a < c"
   689 proof -
   690   assume r: "!!x y. x < y ==> f x < f y"
   691   assume "a < b" hence "f a < f b" by (rule r)
   692   also assume "f b = c"
   693   finally (ord_less_eq_trans) show ?thesis .
   694 qed
   695 
   696 lemma ord_eq_less_subst: "a = f b ==> b < c ==>
   697   (!!x y. x < y ==> f x < f y) ==> a < f c"
   698 proof -
   699   assume r: "!!x y. x < y ==> f x < f y"
   700   assume "a = f b"
   701   also assume "b < c" hence "f b < f c" by (rule r)
   702   finally (ord_eq_less_trans) show ?thesis .
   703 qed
   704 
   705 text {*
   706   Note that this list of rules is in reverse order of priorities.
   707 *}
   708 
   709 lemmas order_trans_rules [trans] =
   710   order_less_subst2
   711   order_less_subst1
   712   order_le_less_subst2
   713   order_le_less_subst1
   714   order_less_le_subst2
   715   order_less_le_subst1
   716   order_subst2
   717   order_subst1
   718   ord_le_eq_subst
   719   ord_eq_le_subst
   720   ord_less_eq_subst
   721   ord_eq_less_subst
   722   forw_subst
   723   back_subst
   724   rev_mp
   725   mp
   726   order_neq_le_trans
   727   order_le_neq_trans
   728   order_less_trans
   729   order_less_asym'
   730   order_le_less_trans
   731   order_less_le_trans
   732   order_trans
   733   order_antisym
   734   ord_le_eq_trans
   735   ord_eq_le_trans
   736   ord_less_eq_trans
   737   ord_eq_less_trans
   738   trans
   739 
   740 
   741 (* FIXME cleanup *)
   742 
   743 text {* These support proving chains of decreasing inequalities
   744     a >= b >= c ... in Isar proofs. *}
   745 
   746 lemma xt1:
   747   "a = b ==> b > c ==> a > c"
   748   "a > b ==> b = c ==> a > c"
   749   "a = b ==> b >= c ==> a >= c"
   750   "a >= b ==> b = c ==> a >= c"
   751   "(x::'a::order) >= y ==> y >= x ==> x = y"
   752   "(x::'a::order) >= y ==> y >= z ==> x >= z"
   753   "(x::'a::order) > y ==> y >= z ==> x > z"
   754   "(x::'a::order) >= y ==> y > z ==> x > z"
   755   "(a::'a::order) > b ==> b > a ==> ?P"
   756   "(x::'a::order) > y ==> y > z ==> x > z"
   757   "(a::'a::order) >= b ==> a ~= b ==> a > b"
   758   "(a::'a::order) ~= b ==> a >= b ==> a > b"
   759   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 
   760   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
   761   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
   762   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
   763 by auto
   764 
   765 lemma xt2:
   766   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
   767 by (subgoal_tac "f b >= f c", force, force)
   768 
   769 lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> 
   770     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
   771 by (subgoal_tac "f a >= f b", force, force)
   772 
   773 lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
   774   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
   775 by (subgoal_tac "f b >= f c", force, force)
   776 
   777 lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
   778     (!!x y. x > y ==> f x > f y) ==> f a > c"
   779 by (subgoal_tac "f a > f b", force, force)
   780 
   781 lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
   782     (!!x y. x > y ==> f x > f y) ==> a > f c"
   783 by (subgoal_tac "f b > f c", force, force)
   784 
   785 lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
   786     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
   787 by (subgoal_tac "f a >= f b", force, force)
   788 
   789 lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
   790     (!!x y. x > y ==> f x > f y) ==> a > f c"
   791 by (subgoal_tac "f b > f c", force, force)
   792 
   793 lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
   794     (!!x y. x > y ==> f x > f y) ==> f a > c"
   795 by (subgoal_tac "f a > f b", force, force)
   796 
   797 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
   798 
   799 (* 
   800   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
   801   for the wrong thing in an Isar proof.
   802 
   803   The extra transitivity rules can be used as follows: 
   804 
   805 lemma "(a::'a::order) > z"
   806 proof -
   807   have "a >= b" (is "_ >= ?rhs")
   808     sorry
   809   also have "?rhs >= c" (is "_ >= ?rhs")
   810     sorry
   811   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
   812     sorry
   813   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
   814     sorry
   815   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
   816     sorry
   817   also (xtrans) have "?rhs > z"
   818     sorry
   819   finally (xtrans) show ?thesis .
   820 qed
   821 
   822   Alternatively, one can use "declare xtrans [trans]" and then
   823   leave out the "(xtrans)" above.
   824 *)
   825 
   826 subsection {* Order on bool *}
   827 
   828 instance bool :: order 
   829   le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
   830   less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
   831   by intro_classes (auto simp add: le_bool_def less_bool_def)
   832 
   833 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
   834   by (simp add: le_bool_def)
   835 
   836 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
   837   by (simp add: le_bool_def)
   838 
   839 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
   840   by (simp add: le_bool_def)
   841 
   842 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
   843   by (simp add: le_bool_def)
   844 
   845 lemma [code func]:
   846   "False \<le> b \<longleftrightarrow> True"
   847   "True \<le> b \<longleftrightarrow> b"
   848   "False < b \<longleftrightarrow> b"
   849   "True < b \<longleftrightarrow> False"
   850   unfolding le_bool_def less_bool_def by simp_all
   851 
   852 
   853 subsection {* Monotonicity, syntactic least value operator and min/max *}
   854 
   855 locale mono =
   856   fixes f
   857   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
   858 
   859 lemmas monoI [intro?] = mono.intro
   860   and monoD [dest?] = mono.mono
   861 
   862 constdefs
   863   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
   864   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
   865     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
   866 
   867 lemma LeastI2_order:
   868   "[| P (x::'a::order);
   869       !!y. P y ==> x <= y;
   870       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
   871    ==> Q (Least P)"
   872   apply (unfold Least_def)
   873   apply (rule theI2)
   874     apply (blast intro: order_antisym)+
   875   done
   876 
   877 lemma Least_equality:
   878     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
   879   apply (simp add: Least_def)
   880   apply (rule the_equality)
   881   apply (auto intro!: order_antisym)
   882   done
   883 
   884 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
   885   by (simp add: min_def)
   886 
   887 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
   888   by (simp add: max_def)
   889 
   890 lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
   891   apply (simp add: min_def)
   892   apply (blast intro: order_antisym)
   893   done
   894 
   895 lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
   896   apply (simp add: max_def)
   897   apply (blast intro: order_antisym)
   898   done
   899 
   900 lemma min_of_mono:
   901     "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
   902   by (simp add: min_def)
   903 
   904 lemma max_of_mono:
   905     "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
   906   by (simp add: max_def)
   907 
   908 
   909 subsection {* legacy ML bindings *}
   910 
   911 ML {*
   912 val monoI = @{thm monoI};
   913 *}
   914 
   915 end