wneuper/isa: src/HOL/Orderings.thy@2a45e400fdad

     1 (*  Title:      HOL/Orderings.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson

     4 *)

     6 header {* Syntactic and abstract orders *}

     8 theory Orderings

     9 imports Set Fun

    10 uses

    11   "~~/src/Provers/order.ML"

    12 begin

    14 subsection {* Partial orders *}

    16 class order = ord +

    17   assumes less_le: "x \<^loc>< y \<longleftrightarrow> x \<^loc>\<le> y \<and> x \<noteq> y"

    18   and order_refl [iff]: "x \<^loc>\<le> x"

    19   and order_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> z"

    20   assumes antisym: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y"

    22 begin

    24 notation (input)

    25   less_eq (infix "\<sqsubseteq>" 50)

    26 and

    27   less    (infix "\<sqsubset>" 50)

    29 text {* Reflexivity. *}

    31 lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"

    32     -- {* This form is useful with the classical reasoner. *}

    33 by (erule ssubst) (rule order_refl)

    35 lemma less_irrefl [iff]: "\<not> x \<^loc>< x"

    36 by (simp add: less_le)

    38 lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"

    39     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}

    40 by (simp add: less_le) blast

    42 lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"

    43 unfolding less_le by blast

    45 lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"

    46 unfolding less_le by blast

    48 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"

    49 by (erule contrapos_pn, erule subst, rule less_irrefl)

    52 text {* Useful for simplification, but too risky to include by default. *}

    54 lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"

    55 by auto

    57 lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"

    58 by auto

    61 text {* Transitivity rules for calculational reasoning *}

    63 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"

    64 by (simp add: less_le)

    66 lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"

    67 by (simp add: less_le)

    70 text {* Asymmetry. *}

    72 lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"

    73 by (simp add: less_le antisym)

    75 lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"

    76 by (drule less_not_sym, erule contrapos_np) simp

    78 lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"

    79 by (blast intro: antisym)

    81 lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"

    82 by (blast intro: antisym)

    84 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"

    85 by (erule contrapos_pn, erule subst, rule less_irrefl)

    88 text {* Transitivity. *}

    90 lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"

    91 by (simp add: less_le) (blast intro: order_trans antisym)

    93 lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"

    94 by (simp add: less_le) (blast intro: order_trans antisym)

    96 lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"

    97 by (simp add: less_le) (blast intro: order_trans antisym)

   100 text {* Useful for simplification, but too risky to include by default. *}

   102 lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"

   103 by (blast elim: less_asym)

   105 lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"

   106 by (blast elim: less_asym)

   109 text {* Transitivity rules for calculational reasoning *}

   111 lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"

   112 by (rule less_asym)

   115 text {* Reverse order *}

   117 lemma order_reverse:

   118   "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"

   119 by unfold_locales

   120    (simp add: less_le, auto intro: antisym order_trans)

   122 end

   125 subsection {* Linear (total) orders *}

   127 class linorder = order +

   128   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"

   129 begin

   131 lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"

   132 unfolding less_le using less_le linear by blast

   134 lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"

   135 by (simp add: le_less less_linear)

   137 lemma le_cases [case_names le ge]:

   138   "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"

   139 using linear by blast

   141 lemma linorder_cases [case_names less equal greater]:

   142   "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"

   143 using less_linear by blast

   145 lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"

   146 apply (simp add: less_le)

   147 using linear apply (blast intro: antisym)

   148 done

   150 lemma not_less_iff_gr_or_eq:

   151  "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"

   152 apply(simp add:not_less le_less)

   153 apply blast

   154 done

   156 lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"

   157 apply (simp add: less_le)

   158 using linear apply (blast intro: antisym)

   159 done

   161 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"

   162 by (cut_tac x = x and y = y in less_linear, auto)

   164 lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"

   165 by (simp add: neq_iff) blast

   167 lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"

   168 by (blast intro: antisym dest: not_less [THEN iffD1])

   170 lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"

   171 by (blast intro: antisym dest: not_less [THEN iffD1])

   173 lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"

   174 by (blast intro: antisym dest: not_less [THEN iffD1])

   176 text{*Replacing the old Nat.leI*}

   177 lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"

   178 unfolding not_less .

   180 lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"

   181 unfolding not_less .

   183 (*FIXME inappropriate name (or delete altogether)*)

   184 lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"

   185 unfolding not_le .

   188 text {* Reverse order *}

   190 lemma linorder_reverse:

   191   "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"

   192 by unfold_locales

   193   (simp add: less_le, auto intro: antisym order_trans simp add: linear)

   196 text {* min/max *}

   198 text {* for historic reasons, definitions are done in context ord *}

   200 definition (in ord)

   201   min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where

   202   [code unfold, code inline del]: "min a b = (if a \<^loc>\<le> b then a else b)"

   204 definition (in ord)

   205   max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where

   206   [code unfold, code inline del]: "max a b = (if a \<^loc>\<le> b then b else a)"

   208 lemma min_le_iff_disj:

   209   "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"

   210 unfolding min_def using linear by (auto intro: order_trans)

   212 lemma le_max_iff_disj:

   213   "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"

   214 unfolding max_def using linear by (auto intro: order_trans)

   216 lemma min_less_iff_disj:

   217   "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"

   218 unfolding min_def le_less using less_linear by (auto intro: less_trans)

   220 lemma less_max_iff_disj:

   221   "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"

   222 unfolding max_def le_less using less_linear by (auto intro: less_trans)

   224 lemma min_less_iff_conj [simp]:

   225   "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"

   226 unfolding min_def le_less using less_linear by (auto intro: less_trans)

   228 lemma max_less_iff_conj [simp]:

   229   "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"

   230 unfolding max_def le_less using less_linear by (auto intro: less_trans)

   232 lemma split_min [noatp]:

   233   "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"

   234 by (simp add: min_def)

   236 lemma split_max [noatp]:

   237   "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"

   238 by (simp add: max_def)

   240 end

   243 subsection {* Reasoning tools setup *}

   245 ML {*

   247 signature ORDERS =

   248 sig

   249   val print_structures: Proof.context -> unit

   250   val setup: theory -> theory

   251   val order_tac: thm list -> Proof.context -> int -> tactic

   252 end;

   254 structure Orders: ORDERS =

   255 struct

   257 (** Theory and context data **)

   259 fun struct_eq ((s1: string, ts1), (s2, ts2)) =

   260   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);

   262 structure Data = GenericDataFun

   263 (

   264   type T = ((string * term list) * Order_Tac.less_arith) list;

   265     (* Order structures:

   266        identifier of the structure, list of operations and record of theorems

   267        needed to set up the transitivity reasoner,

   268        identifier and operations identify the structure uniquely. *)

   269   val empty = [];

   270   val extend = I;

   271   fun merge _ = AList.join struct_eq (K fst);

   272 );

   274 fun print_structures ctxt =

   275   let

   276     val structs = Data.get (Context.Proof ctxt);

   277     fun pretty_term t = Pretty.block

   278       [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,

   279         Pretty.str "::", Pretty.brk 1,

   280         Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];

   281     fun pretty_struct ((s, ts), _) = Pretty.block

   282       [Pretty.str s, Pretty.str ":", Pretty.brk 1,

   283        Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];

   284   in

   285     Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))

   286   end;

   289 (** Method **)

   291 fun struct_tac ((s, [eq, le, less]), thms) prems =

   292   let

   293     fun decomp thy (Trueprop $ t) =

   294       let

   295         fun excluded t =

   296           (* exclude numeric types: linear arithmetic subsumes transitivity *)

   297           let val T = type_of t

   298           in

   299 	    T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT

   300           end;

   301 	fun rel (bin_op $ t1 $ t2) =

   302               if excluded t1 then NONE

   303               else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)

   304               else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)

   305               else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)

   306               else NONE

   307 	  | rel _ = NONE;

   308 	fun dec (Const (@{const_name Not}, _) $ t) = (case rel t

   309 	      of NONE => NONE

   310 	       | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))

   311           | dec x = rel x;

   312       in dec t end;

   313   in

   314     case s of

   315       "order" => Order_Tac.partial_tac decomp thms prems

   316     | "linorder" => Order_Tac.linear_tac decomp thms prems

   317     | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")

   318   end

   320 fun order_tac prems ctxt =

   321   FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));

   324 (** Attribute **)

   326 fun add_struct_thm s tag =

   327   Thm.declaration_attribute

   328     (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));

   329 fun del_struct s =

   330   Thm.declaration_attribute

   331     (fn _ => Data.map (AList.delete struct_eq s));

   333 val attribute = Attrib.syntax

   334      (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) ||

   335           Args.del >> K NONE) --| Args.colon (* FIXME ||

   336         Scan.succeed true *) ) -- Scan.lift Args.name --

   337       Scan.repeat Args.term

   338       >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag

   339            | ((NONE, n), ts) => del_struct (n, ts)));

   342 (** Diagnostic command **)

   344 val print = Toplevel.unknown_context o

   345   Toplevel.keep (Toplevel.node_case

   346     (Context.cases (print_structures o ProofContext.init) print_structures)

   347     (print_structures o Proof.context_of));

   349 val _ =

   350   OuterSyntax.improper_command "print_orders"

   351     "print order structures available to transitivity reasoner" OuterKeyword.diag

   352     (Scan.succeed (Toplevel.no_timing o print));

   355 (** Setup **)

   357 val setup =

   358   Method.add_methods

   359     [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #>

   360   Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")];

   362 end;

   364 *}

   366 setup Orders.setup

   369 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}

   371 (* The type constraint on @{term op =} below is necessary since the operation

   372    is not a parameter of the locale. *)

   373 lemmas (in order)

   374   [order add less_reflE: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   375   less_irrefl [THEN notE]

   376 lemmas (in order)

   377   [order add le_refl: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   378   order_refl

   379 lemmas (in order)

   380   [order add less_imp_le: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   381   less_imp_le

   382 lemmas (in order)

   383   [order add eqI: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   384   antisym

   385 lemmas (in order)

   386   [order add eqD1: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   387   eq_refl

   388 lemmas (in order)

   389   [order add eqD2: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   390   sym [THEN eq_refl]

   391 lemmas (in order)

   392   [order add less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   393   less_trans

   394 lemmas (in order)

   395   [order add less_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   396   less_le_trans

   397 lemmas (in order)

   398   [order add le_less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   399   le_less_trans

   400 lemmas (in order)

   401   [order add le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   402   order_trans

   403 lemmas (in order)

   404   [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   405   le_neq_trans

   406 lemmas (in order)

   407   [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   408   neq_le_trans

   409 lemmas (in order)

   410   [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   411   less_imp_neq

   412 lemmas (in order)

   413   [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   414    eq_neq_eq_imp_neq

   415 lemmas (in order)

   416   [order add not_sym: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   417   not_sym

   419 lemmas (in linorder) [order del: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = _

   421 lemmas (in linorder)

   422   [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   423   less_irrefl [THEN notE]

   424 lemmas (in linorder)

   425   [order add le_refl: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   426   order_refl

   427 lemmas (in linorder)

   428   [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   429   less_imp_le

   430 lemmas (in linorder)

   431   [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   432   not_less [THEN iffD2]

   433 lemmas (in linorder)

   434   [order add not_leI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   435   not_le [THEN iffD2]

   436 lemmas (in linorder)

   437   [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   438   not_less [THEN iffD1]

   439 lemmas (in linorder)

   440   [order add not_leD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   441   not_le [THEN iffD1]

   442 lemmas (in linorder)

   443   [order add eqI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   444   antisym

   445 lemmas (in linorder)

   446   [order add eqD1: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   447   eq_refl

   448 lemmas (in linorder)

   449   [order add eqD2: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   450   sym [THEN eq_refl]

   451 lemmas (in linorder)

   452   [order add less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   453   less_trans

   454 lemmas (in linorder)

   455   [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   456   less_le_trans

   457 lemmas (in linorder)

   458   [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   459   le_less_trans

   460 lemmas (in linorder)

   461   [order add le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   462   order_trans

   463 lemmas (in linorder)

   464   [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   465   le_neq_trans

   466 lemmas (in linorder)

   467   [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   468   neq_le_trans

   469 lemmas (in linorder)

   470   [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   471   less_imp_neq

   472 lemmas (in linorder)

   473   [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   474   eq_neq_eq_imp_neq

   475 lemmas (in linorder)

   476   [order add not_sym: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =

   477   not_sym

   480 setup {*

   481 let

   483 fun prp t thm = (#prop (rep_thm thm) = t);

   485 fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =

   486   let val prems = prems_of_ss ss;

   487       val less = Const (@{const_name less}, T);

   488       val t = HOLogic.mk_Trueprop(le $ s $ r);

   489   in case find_first (prp t) prems of

   490        NONE =>

   491          let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s))

   492          in case find_first (prp t) prems of

   493               NONE => NONE

   494             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))

   495          end

   496      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))

   497   end

   498   handle THM _ => NONE;

   500 fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =

   501   let val prems = prems_of_ss ss;

   502       val le = Const (@{const_name less_eq}, T);

   503       val t = HOLogic.mk_Trueprop(le $ r $ s);

   504   in case find_first (prp t) prems of

   505        NONE =>

   506          let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r))

   507          in case find_first (prp t) prems of

   508               NONE => NONE

   509             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))

   510          end

   511      | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))

   512   end

   513   handle THM _ => NONE;

   515 fun add_simprocs procs thy =

   516   (Simplifier.change_simpset_of thy (fn ss => ss

   517     addsimprocs (map (fn (name, raw_ts, proc) =>

   518       Simplifier.simproc thy name raw_ts proc)) procs); thy);

   519 fun add_solver name tac thy =

   520   (Simplifier.change_simpset_of thy (fn ss => ss addSolver

   521     (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy);

   523 in

   524   add_simprocs [

   525        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),

   526        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)

   527      ]

   528   #> add_solver "Transitivity" Orders.order_tac

   529   (* Adding the transitivity reasoners also as safe solvers showed a slight

   530      speed up, but the reasoning strength appears to be not higher (at least

   531      no breaking of additional proofs in the entire HOL distribution, as

   532      of 5 March 2004, was observed). *)

   533 end

   534 *}

   537 subsection {* Dense orders *}

   539 class dense_linear_order = linorder +

   540   assumes gt_ex: "\<exists>y. x \<sqsubset> y"

   541   and lt_ex: "\<exists>y. y \<sqsubset> x"

   542   and dense: "x \<sqsubset> y \<Longrightarrow> (\<exists>z. x \<sqsubset> z \<and> z \<sqsubset> y)"

   543   (*see further theory Dense_Linear_Order*)

   546 lemma interval_empty_iff:

   547   fixes x y z :: "'a\<Colon>dense_linear_order"

   548   shows "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"

   549   by (auto dest: dense)

   551 subsection {* Name duplicates *}

   553 lemmas order_less_le = less_le

   554 lemmas order_eq_refl = order_class.eq_refl

   555 lemmas order_less_irrefl = order_class.less_irrefl

   556 lemmas order_le_less = order_class.le_less

   557 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq

   558 lemmas order_less_imp_le = order_class.less_imp_le

   559 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq

   560 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2

   561 lemmas order_neq_le_trans = order_class.neq_le_trans

   562 lemmas order_le_neq_trans = order_class.le_neq_trans

   564 lemmas order_antisym = antisym

   565 lemmas order_less_not_sym = order_class.less_not_sym

   566 lemmas order_less_asym = order_class.less_asym

   567 lemmas order_eq_iff = order_class.eq_iff

   568 lemmas order_antisym_conv = order_class.antisym_conv

   569 lemmas order_less_trans = order_class.less_trans

   570 lemmas order_le_less_trans = order_class.le_less_trans

   571 lemmas order_less_le_trans = order_class.less_le_trans

   572 lemmas order_less_imp_not_less = order_class.less_imp_not_less

   573 lemmas order_less_imp_triv = order_class.less_imp_triv

   574 lemmas order_less_asym' = order_class.less_asym'

   576 lemmas linorder_linear = linear

   577 lemmas linorder_less_linear = linorder_class.less_linear

   578 lemmas linorder_le_less_linear = linorder_class.le_less_linear

   579 lemmas linorder_le_cases = linorder_class.le_cases

   580 lemmas linorder_not_less = linorder_class.not_less

   581 lemmas linorder_not_le = linorder_class.not_le

   582 lemmas linorder_neq_iff = linorder_class.neq_iff

   583 lemmas linorder_neqE = linorder_class.neqE

   584 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1

   585 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2

   586 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3

   588 lemmas min_le_iff_disj = linorder_class.min_le_iff_disj

   589 lemmas le_max_iff_disj = linorder_class.le_max_iff_disj

   590 lemmas min_less_iff_disj = linorder_class.min_less_iff_disj

   591 lemmas less_max_iff_disj = linorder_class.less_max_iff_disj

   592 lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj

   593 lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj

   594 lemmas split_min = linorder_class.split_min

   595 lemmas split_max = linorder_class.split_max

   598 subsection {* Bounded quantifiers *}

   600 syntax

   601   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)

   602   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)

   603   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)

   604   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)

   606   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)

   607   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)

   608   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)

   609   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)

   611 syntax (xsymbols)

   612   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)

   613   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)

   614   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)

   615   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)

   617   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)

   618   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)

   619   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)

   620   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)

   622 syntax (HOL)

   623   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)

   624   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)

   625   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)

   626   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)

   628 syntax (HTML output)

   629   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)

   630   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)

   631   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)

   632   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)

   634   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)

   635   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)

   636   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)

   637   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)

   639 translations

   640   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"

   641   "EX x<y. P"    =>  "EX x. x < y \<and> P"

   642   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"

   643   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"

   644   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"

   645   "EX x>y. P"    =>  "EX x. x > y \<and> P"

   646   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"

   647   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"

   649 print_translation {*

   650 let

   651   val All_binder = Syntax.binder_name @{const_syntax All};

   652   val Ex_binder = Syntax.binder_name @{const_syntax Ex};

   653   val impl = @{const_syntax "op -->"};

   654   val conj = @{const_syntax "op &"};

   655   val less = @{const_syntax less};

   656   val less_eq = @{const_syntax less_eq};

   658   val trans =

   659    [((All_binder, impl, less), ("_All_less", "_All_greater")),

   660     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),

   661     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),

   662     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];

   664   fun matches_bound v t =

   665      case t of (Const ("_bound", _) $ Free (v', _)) => (v = v')

   666               | _ => false

   667   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)

   668   fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P

   670   fun tr' q = (q,

   671     fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>

   672       (case AList.lookup (op =) trans (q, c, d) of

   673         NONE => raise Match

   674       | SOME (l, g) =>

   675           if matches_bound v t andalso not (contains_var v u) then mk v l u P

   676           else if matches_bound v u andalso not (contains_var v t) then mk v g t P

   677           else raise Match)

   678      | _ => raise Match);

   679 in [tr' All_binder, tr' Ex_binder] end

   680 *}

   683 subsection {* Transitivity reasoning *}

   685 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"

   686 by (rule subst)

   688 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"

   689 by (rule ssubst)

   691 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"

   692 by (rule subst)

   694 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"

   695 by (rule ssubst)

   697 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>

   698   (!!x y. x < y ==> f x < f y) ==> f a < c"

   699 proof -

   700   assume r: "!!x y. x < y ==> f x < f y"

   701   assume "a < b" hence "f a < f b" by (rule r)

   702   also assume "f b < c"

   703   finally (order_less_trans) show ?thesis .

   704 qed

   706 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>

   707   (!!x y. x < y ==> f x < f y) ==> a < f c"

   708 proof -

   709   assume r: "!!x y. x < y ==> f x < f y"

   710   assume "a < f b"

   711   also assume "b < c" hence "f b < f c" by (rule r)

   712   finally (order_less_trans) show ?thesis .

   713 qed

   715 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>

   716   (!!x y. x <= y ==> f x <= f y) ==> f a < c"

   717 proof -

   718   assume r: "!!x y. x <= y ==> f x <= f y"

   719   assume "a <= b" hence "f a <= f b" by (rule r)

   720   also assume "f b < c"

   721   finally (order_le_less_trans) show ?thesis .

   722 qed

   724 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>

   725   (!!x y. x < y ==> f x < f y) ==> a < f c"

   726 proof -

   727   assume r: "!!x y. x < y ==> f x < f y"

   728   assume "a <= f b"

   729   also assume "b < c" hence "f b < f c" by (rule r)

   730   finally (order_le_less_trans) show ?thesis .

   731 qed

   733 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>

   734   (!!x y. x < y ==> f x < f y) ==> f a < c"

   735 proof -

   736   assume r: "!!x y. x < y ==> f x < f y"

   737   assume "a < b" hence "f a < f b" by (rule r)

   738   also assume "f b <= c"

   739   finally (order_less_le_trans) show ?thesis .

   740 qed

   742 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>

   743   (!!x y. x <= y ==> f x <= f y) ==> a < f c"

   744 proof -

   745   assume r: "!!x y. x <= y ==> f x <= f y"

   746   assume "a < f b"

   747   also assume "b <= c" hence "f b <= f c" by (rule r)

   748   finally (order_less_le_trans) show ?thesis .

   749 qed

   751 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>

   752   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"

   753 proof -

   754   assume r: "!!x y. x <= y ==> f x <= f y"

   755   assume "a <= f b"

   756   also assume "b <= c" hence "f b <= f c" by (rule r)

   757   finally (order_trans) show ?thesis .

   758 qed

   760 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>

   761   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"

   762 proof -

   763   assume r: "!!x y. x <= y ==> f x <= f y"

   764   assume "a <= b" hence "f a <= f b" by (rule r)

   765   also assume "f b <= c"

   766   finally (order_trans) show ?thesis .

   767 qed

   769 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>

   770   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"

   771 proof -

   772   assume r: "!!x y. x <= y ==> f x <= f y"

   773   assume "a <= b" hence "f a <= f b" by (rule r)

   774   also assume "f b = c"

   775   finally (ord_le_eq_trans) show ?thesis .

   776 qed

   778 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>

   779   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"

   780 proof -

   781   assume r: "!!x y. x <= y ==> f x <= f y"

   782   assume "a = f b"

   783   also assume "b <= c" hence "f b <= f c" by (rule r)

   784   finally (ord_eq_le_trans) show ?thesis .

   785 qed

   787 lemma ord_less_eq_subst: "a < b ==> f b = c ==>

   788   (!!x y. x < y ==> f x < f y) ==> f a < c"

   789 proof -

   790   assume r: "!!x y. x < y ==> f x < f y"

   791   assume "a < b" hence "f a < f b" by (rule r)

   792   also assume "f b = c"

   793   finally (ord_less_eq_trans) show ?thesis .

   794 qed

   796 lemma ord_eq_less_subst: "a = f b ==> b < c ==>

   797   (!!x y. x < y ==> f x < f y) ==> a < f c"

   798 proof -

   799   assume r: "!!x y. x < y ==> f x < f y"

   800   assume "a = f b"

   801   also assume "b < c" hence "f b < f c" by (rule r)

   802   finally (ord_eq_less_trans) show ?thesis .

   803 qed

   805 text {*

   806   Note that this list of rules is in reverse order of priorities.

   807 *}

   809 lemmas order_trans_rules [trans] =

   810   order_less_subst2

   811   order_less_subst1

   812   order_le_less_subst2

   813   order_le_less_subst1

   814   order_less_le_subst2

   815   order_less_le_subst1

   816   order_subst2

   817   order_subst1

   818   ord_le_eq_subst

   819   ord_eq_le_subst

   820   ord_less_eq_subst

   821   ord_eq_less_subst

   822   forw_subst

   823   back_subst

   824   rev_mp

   825   mp

   826   order_neq_le_trans

   827   order_le_neq_trans

   828   order_less_trans

   829   order_less_asym'

   830   order_le_less_trans

   831   order_less_le_trans

   832   order_trans

   833   order_antisym

   834   ord_le_eq_trans

   835   ord_eq_le_trans

   836   ord_less_eq_trans

   837   ord_eq_less_trans

   838   trans

   841 (* FIXME cleanup *)

   843 text {* These support proving chains of decreasing inequalities

   844     a >= b >= c ... in Isar proofs. *}

   846 lemma xt1:

   847   "a = b ==> b > c ==> a > c"

   848   "a > b ==> b = c ==> a > c"

   849   "a = b ==> b >= c ==> a >= c"

   850   "a >= b ==> b = c ==> a >= c"

   851   "(x::'a::order) >= y ==> y >= x ==> x = y"

   852   "(x::'a::order) >= y ==> y >= z ==> x >= z"

   853   "(x::'a::order) > y ==> y >= z ==> x > z"

   854   "(x::'a::order) >= y ==> y > z ==> x > z"

   855   "(a::'a::order) > b ==> b > a ==> P"

   856   "(x::'a::order) > y ==> y > z ==> x > z"

   857   "(a::'a::order) >= b ==> a ~= b ==> a > b"

   858   "(a::'a::order) ~= b ==> a >= b ==> a > b"

   859   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c"

   860   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"

   861   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"

   862   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"

   863 by auto

   865 lemma xt2:

   866   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"

   867 by (subgoal_tac "f b >= f c", force, force)

   869 lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>

   870     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"

   871 by (subgoal_tac "f a >= f b", force, force)

   873 lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>

   874   (!!x y. x >= y ==> f x >= f y) ==> a > f c"

   875 by (subgoal_tac "f b >= f c", force, force)

   877 lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>

   878     (!!x y. x > y ==> f x > f y) ==> f a > c"

   879 by (subgoal_tac "f a > f b", force, force)

   881 lemma xt6: "(a::'a::order) >= f b ==> b > c ==>

   882     (!!x y. x > y ==> f x > f y) ==> a > f c"

   883 by (subgoal_tac "f b > f c", force, force)

   885 lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>

   886     (!!x y. x >= y ==> f x >= f y) ==> f a > c"

   887 by (subgoal_tac "f a >= f b", force, force)

   889 lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>

   890     (!!x y. x > y ==> f x > f y) ==> a > f c"

   891 by (subgoal_tac "f b > f c", force, force)

   893 lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>

   894     (!!x y. x > y ==> f x > f y) ==> f a > c"

   895 by (subgoal_tac "f a > f b", force, force)

   897 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9

   899 (*

   900   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands

   901   for the wrong thing in an Isar proof.

   903   The extra transitivity rules can be used as follows:

   905 lemma "(a::'a::order) > z"

   906 proof -

   907   have "a >= b" (is "_ >= ?rhs")

   908     sorry

   909   also have "?rhs >= c" (is "_ >= ?rhs")

   910     sorry

   911   also (xtrans) have "?rhs = d" (is "_ = ?rhs")

   912     sorry

   913   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")

   914     sorry

   915   also (xtrans) have "?rhs > f" (is "_ > ?rhs")

   916     sorry

   917   also (xtrans) have "?rhs > z"

   918     sorry

   919   finally (xtrans) show ?thesis .

   920 qed

   922   Alternatively, one can use "declare xtrans [trans]" and then

   923   leave out the "(xtrans)" above.

   924 *)

   926 subsection {* Order on bool *}

   928 instance bool :: order

   929   le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"

   930   less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"

   931   by intro_classes (auto simp add: le_bool_def less_bool_def)

   932 lemmas [code func del] = le_bool_def less_bool_def

   934 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"

   935 by (simp add: le_bool_def)

   937 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"

   938 by (simp add: le_bool_def)

   940 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"

   941 by (simp add: le_bool_def)

   943 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"

   944 by (simp add: le_bool_def)

   946 lemma [code func]:

   947   "False \<le> b \<longleftrightarrow> True"

   948   "True \<le> b \<longleftrightarrow> b"

   949   "False < b \<longleftrightarrow> b"

   950   "True < b \<longleftrightarrow> False"

   951   unfolding le_bool_def less_bool_def by simp_all

   954 subsection {* Order on sets *}

   956 instance set :: (type) order

   957   by (intro_classes,

   958       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)

   960 lemmas basic_trans_rules [trans] =

   961   order_trans_rules set_rev_mp set_mp

   964 subsection {* Order on functions *}

   966 instance "fun" :: (type, ord) ord

   967   le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"

   968   less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..

   970 lemmas [code func del] = le_fun_def less_fun_def

   972 instance "fun" :: (type, order) order

   973   by default

   974     (auto simp add: le_fun_def less_fun_def expand_fun_eq

   975        intro: order_trans order_antisym)

   977 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"

   978   unfolding le_fun_def by simp

   980 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"

   981   unfolding le_fun_def by simp

   983 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"

   984   unfolding le_fun_def by simp

   986 text {*

   987   Handy introduction and elimination rules for @{text "\<le>"}

   988   on unary and binary predicates

   989 *}

   991 lemma predicate1I [Pure.intro!, intro!]:

   992   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"

   993   shows "P \<le> Q"

   994   apply (rule le_funI)

   995   apply (rule le_boolI)

   996   apply (rule PQ)

   997   apply assumption

   998   done

  1000 lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"

  1001   apply (erule le_funE)

  1002   apply (erule le_boolE)

  1003   apply assumption+

  1004   done

  1006 lemma predicate2I [Pure.intro!, intro!]:

  1007   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"

  1008   shows "P \<le> Q"

  1009   apply (rule le_funI)+

  1010   apply (rule le_boolI)

  1011   apply (rule PQ)

  1012   apply assumption

  1013   done

  1015 lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"

  1016   apply (erule le_funE)+

  1017   apply (erule le_boolE)

  1018   apply assumption+

  1019   done

  1021 lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"

  1022   by (rule predicate1D)

  1024 lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"

  1025   by (rule predicate2D)

  1028 subsection {* Monotonicity, least value operator and min/max *}

  1030 locale mono =

  1031   fixes f

  1032   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"

  1034 lemmas monoI [intro?] = mono.intro

  1035   and monoD [dest?] = mono.mono

  1037 lemma LeastI2_order:

  1038   "[| P (x::'a::order);

  1039       !!y. P y ==> x <= y;

  1040       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]

  1041    ==> Q (Least P)"

  1042 apply (unfold Least_def)

  1043 apply (rule theI2)

  1044   apply (blast intro: order_antisym)+

  1045 done

  1047 lemma Least_mono:

  1048   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y

  1049     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"

  1050     -- {* Courtesy of Stephan Merz *}

  1051   apply clarify

  1052   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)

  1053   apply (rule LeastI2_order)

  1054   apply (auto elim: monoD intro!: order_antisym)

  1055   done

  1057 lemma Least_equality:

  1058   "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"

  1059 apply (simp add: Least_def)

  1060 apply (rule the_equality)

  1061 apply (auto intro!: order_antisym)

  1062 done

  1064 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"

  1065 by (simp add: min_def)

  1067 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"

  1068 by (simp add: max_def)

  1070 lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"

  1071 apply (simp add: min_def)

  1072 apply (blast intro: order_antisym)

  1073 done

  1075 lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"

  1076 apply (simp add: max_def)

  1077 apply (blast intro: order_antisym)

  1078 done

  1080 lemma min_of_mono:

  1081   "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"

  1082 by (simp add: min_def)

  1084 lemma max_of_mono:

  1085   "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"

  1086 by (simp add: max_def)

  1089 subsection {* legacy ML bindings *}

  1091 ML {*

  1092 val monoI = @{thm monoI};

  1093 *}

  1095 end

author	wenzelm
	Tue, 09 Oct 2007 00:20:13 +0200
changeset 24920	2a45e400fdad
parent 24867	e5b55d7be9bb
child 25062	af5ef0d4d655
permissions	-rw-r--r--