src/HOL/Orderings.thy
 author haftmann Fri Oct 19 19:45:31 2007 +0200 (2007-10-19) changeset 25103 1ee419a5a30f parent 25076 a50b36401c61 child 25193 e2e1a4b00de3 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 Set Fun
```
```    10 uses
```
```    11   "~~/src/Provers/order.ML"
```
```    12 begin
```
```    13
```
```    14 subsection {* Partial orders *}
```
```    15
```
```    16 class order = ord +
```
```    17   assumes less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
```
```    18   and order_refl [iff]: "x \<le> x"
```
```    19   and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
```
```    20   assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
```
```    21 begin
```
```    22
```
```    23 notation (input)
```
```    24   less_eq (infix "\<sqsubseteq>" 50)
```
```    25 and
```
```    26   less    (infix "\<sqsubset>" 50)
```
```    27
```
```    28 text {* Reflexivity. *}
```
```    29
```
```    30 lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
```
```    31     -- {* This form is useful with the classical reasoner. *}
```
```    32 by (erule ssubst) (rule order_refl)
```
```    33
```
```    34 lemma less_irrefl [iff]: "\<not> x < x"
```
```    35 by (simp add: less_le)
```
```    36
```
```    37 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
```
```    38     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
```
```    39 by (simp add: less_le) blast
```
```    40
```
```    41 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
```
```    42 unfolding less_le by blast
```
```    43
```
```    44 lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
```
```    45 unfolding less_le by blast
```
```    46
```
```    47 lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
```
```    48 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    49
```
```    50
```
```    51 text {* Useful for simplification, but too risky to include by default. *}
```
```    52
```
```    53 lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
```
```    54 by auto
```
```    55
```
```    56 lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
```
```    57 by auto
```
```    58
```
```    59
```
```    60 text {* Transitivity rules for calculational reasoning *}
```
```    61
```
```    62 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
```
```    63 by (simp add: less_le)
```
```    64
```
```    65 lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
```
```    66 by (simp add: less_le)
```
```    67
```
```    68
```
```    69 text {* Asymmetry. *}
```
```    70
```
```    71 lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
```
```    72 by (simp add: less_le antisym)
```
```    73
```
```    74 lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
```
```    75 by (drule less_not_sym, erule contrapos_np) simp
```
```    76
```
```    77 lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
```
```    78 by (blast intro: antisym)
```
```    79
```
```    80 lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
```
```    81 by (blast intro: antisym)
```
```    82
```
```    83 lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
```
```    84 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    85
```
```    86
```
```    87 text {* Transitivity. *}
```
```    88
```
```    89 lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
```
```    90 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    91
```
```    92 lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
```
```    93 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    94
```
```    95 lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
```
```    96 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    97
```
```    98
```
```    99 text {* Useful for simplification, but too risky to include by default. *}
```
```   100
```
```   101 lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
```
```   102 by (blast elim: less_asym)
```
```   103
```
```   104 lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
```
```   105 by (blast elim: less_asym)
```
```   106
```
```   107
```
```   108 text {* Transitivity rules for calculational reasoning *}
```
```   109
```
```   110 lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
```
```   111 by (rule less_asym)
```
```   112
```
```   113
```
```   114 text {* Reverse order *}
```
```   115
```
```   116 lemma order_reverse:
```
```   117   "order (op \<ge>) (op >)"
```
```   118 by unfold_locales
```
```   119    (simp add: less_le, auto intro: antisym order_trans)
```
```   120
```
```   121 end
```
```   122
```
```   123
```
```   124 subsection {* Linear (total) orders *}
```
```   125
```
```   126 class linorder = order +
```
```   127   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
```
```   128 begin
```
```   129
```
```   130 lemma less_linear: "x < y \<or> x = y \<or> y < x"
```
```   131 unfolding less_le using less_le linear by blast
```
```   132
```
```   133 lemma le_less_linear: "x \<le> y \<or> y < x"
```
```   134 by (simp add: le_less less_linear)
```
```   135
```
```   136 lemma le_cases [case_names le ge]:
```
```   137   "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   138 using linear by blast
```
```   139
```
```   140 lemma linorder_cases [case_names less equal greater]:
```
```   141   "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   142 using less_linear by blast
```
```   143
```
```   144 lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
```
```   145 apply (simp add: less_le)
```
```   146 using linear apply (blast intro: antisym)
```
```   147 done
```
```   148
```
```   149 lemma not_less_iff_gr_or_eq:
```
```   150  "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
```
```   151 apply(simp add:not_less le_less)
```
```   152 apply blast
```
```   153 done
```
```   154
```
```   155 lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
```
```   156 apply (simp add: less_le)
```
```   157 using linear apply (blast intro: antisym)
```
```   158 done
```
```   159
```
```   160 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
```
```   161 by (cut_tac x = x and y = y in less_linear, auto)
```
```   162
```
```   163 lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
```
```   164 by (simp add: neq_iff) blast
```
```   165
```
```   166 lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
```
```   167 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   168
```
```   169 lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
```
```   170 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   171
```
```   172 lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
```
```   173 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   174
```
```   175 text{*Replacing the old Nat.leI*}
```
```   176 lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
```
```   177 unfolding not_less .
```
```   178
```
```   179 lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
```
```   180 unfolding not_less .
```
```   181
```
```   182 (*FIXME inappropriate name (or delete altogether)*)
```
```   183 lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
```
```   184 unfolding not_le .
```
```   185
```
```   186
```
```   187 text {* Reverse order *}
```
```   188
```
```   189 lemma linorder_reverse:
```
```   190   "linorder (op \<ge>) (op >)"
```
```   191 by unfold_locales
```
```   192   (simp add: less_le, auto intro: antisym order_trans simp add: linear)
```
```   193
```
```   194
```
```   195 text {* min/max *}
```
```   196
```
```   197 text {* for historic reasons, definitions are done in context ord *}
```
```   198
```
```   199 definition (in ord)
```
```   200   min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   201   [code unfold, code inline del]: "min a b = (if a \<le> b then a else b)"
```
```   202
```
```   203 definition (in ord)
```
```   204   max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   205   [code unfold, code inline del]: "max a b = (if a \<le> b then b else a)"
```
```   206
```
```   207 lemma min_le_iff_disj:
```
```   208   "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
```
```   209 unfolding min_def using linear by (auto intro: order_trans)
```
```   210
```
```   211 lemma le_max_iff_disj:
```
```   212   "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
```
```   213 unfolding max_def using linear by (auto intro: order_trans)
```
```   214
```
```   215 lemma min_less_iff_disj:
```
```   216   "min x y < z \<longleftrightarrow> x < z \<or> y < z"
```
```   217 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   218
```
```   219 lemma less_max_iff_disj:
```
```   220   "z < max x y \<longleftrightarrow> z < x \<or> z < y"
```
```   221 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   222
```
```   223 lemma min_less_iff_conj [simp]:
```
```   224   "z < min x y \<longleftrightarrow> z < x \<and> z < y"
```
```   225 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   226
```
```   227 lemma max_less_iff_conj [simp]:
```
```   228   "max x y < z \<longleftrightarrow> x < z \<and> y < z"
```
```   229 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   230
```
```   231 lemma split_min [noatp]:
```
```   232   "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
```
```   233 by (simp add: min_def)
```
```   234
```
```   235 lemma split_max [noatp]:
```
```   236   "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
```
```   237 by (simp add: max_def)
```
```   238
```
```   239 end
```
```   240
```
```   241
```
```   242 subsection {* Reasoning tools setup *}
```
```   243
```
```   244 ML {*
```
```   245
```
```   246 signature ORDERS =
```
```   247 sig
```
```   248   val print_structures: Proof.context -> unit
```
```   249   val setup: theory -> theory
```
```   250   val order_tac: thm list -> Proof.context -> int -> tactic
```
```   251 end;
```
```   252
```
```   253 structure Orders: ORDERS =
```
```   254 struct
```
```   255
```
```   256 (** Theory and context data **)
```
```   257
```
```   258 fun struct_eq ((s1: string, ts1), (s2, ts2)) =
```
```   259   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
```
```   260
```
```   261 structure Data = GenericDataFun
```
```   262 (
```
```   263   type T = ((string * term list) * Order_Tac.less_arith) list;
```
```   264     (* Order structures:
```
```   265        identifier of the structure, list of operations and record of theorems
```
```   266        needed to set up the transitivity reasoner,
```
```   267        identifier and operations identify the structure uniquely. *)
```
```   268   val empty = [];
```
```   269   val extend = I;
```
```   270   fun merge _ = AList.join struct_eq (K fst);
```
```   271 );
```
```   272
```
```   273 fun print_structures ctxt =
```
```   274   let
```
```   275     val structs = Data.get (Context.Proof ctxt);
```
```   276     fun pretty_term t = Pretty.block
```
```   277       [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
```
```   278         Pretty.str "::", Pretty.brk 1,
```
```   279         Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
```
```   280     fun pretty_struct ((s, ts), _) = Pretty.block
```
```   281       [Pretty.str s, Pretty.str ":", Pretty.brk 1,
```
```   282        Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
```
```   283   in
```
```   284     Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
```
```   285   end;
```
```   286
```
```   287
```
```   288 (** Method **)
```
```   289
```
```   290 fun struct_tac ((s, [eq, le, less]), thms) prems =
```
```   291   let
```
```   292     fun decomp thy (Trueprop \$ t) =
```
```   293       let
```
```   294         fun excluded t =
```
```   295           (* exclude numeric types: linear arithmetic subsumes transitivity *)
```
```   296           let val T = type_of t
```
```   297           in
```
```   298 	    T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
```
```   299           end;
```
```   300 	fun rel (bin_op \$ t1 \$ t2) =
```
```   301               if excluded t1 then NONE
```
```   302               else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
```
```   303               else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
```
```   304               else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
```
```   305               else NONE
```
```   306 	  | rel _ = NONE;
```
```   307 	fun dec (Const (@{const_name Not}, _) \$ t) = (case rel t
```
```   308 	      of NONE => NONE
```
```   309 	       | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
```
```   310           | dec x = rel x;
```
```   311       in dec t end;
```
```   312   in
```
```   313     case s of
```
```   314       "order" => Order_Tac.partial_tac decomp thms prems
```
```   315     | "linorder" => Order_Tac.linear_tac decomp thms prems
```
```   316     | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
```
```   317   end
```
```   318
```
```   319 fun order_tac prems ctxt =
```
```   320   FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));
```
```   321
```
```   322
```
```   323 (** Attribute **)
```
```   324
```
```   325 fun add_struct_thm s tag =
```
```   326   Thm.declaration_attribute
```
```   327     (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
```
```   328 fun del_struct s =
```
```   329   Thm.declaration_attribute
```
```   330     (fn _ => Data.map (AList.delete struct_eq s));
```
```   331
```
```   332 val attribute = Attrib.syntax
```
```   333      (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) ||
```
```   334           Args.del >> K NONE) --| Args.colon (* FIXME ||
```
```   335         Scan.succeed true *) ) -- Scan.lift Args.name --
```
```   336       Scan.repeat Args.term
```
```   337       >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
```
```   338            | ((NONE, n), ts) => del_struct (n, ts)));
```
```   339
```
```   340
```
```   341 (** Diagnostic command **)
```
```   342
```
```   343 val print = Toplevel.unknown_context o
```
```   344   Toplevel.keep (Toplevel.node_case
```
```   345     (Context.cases (print_structures o ProofContext.init) print_structures)
```
```   346     (print_structures o Proof.context_of));
```
```   347
```
```   348 val _ =
```
```   349   OuterSyntax.improper_command "print_orders"
```
```   350     "print order structures available to transitivity reasoner" OuterKeyword.diag
```
```   351     (Scan.succeed (Toplevel.no_timing o print));
```
```   352
```
```   353
```
```   354 (** Setup **)
```
```   355
```
```   356 val setup =
```
```   357   Method.add_methods
```
```   358     [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #>
```
```   359   Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")];
```
```   360
```
```   361 end;
```
```   362
```
```   363 *}
```
```   364
```
```   365 setup Orders.setup
```
```   366
```
```   367
```
```   368 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
```
```   369
```
```   370 context order
```
```   371 begin
```
```   372
```
```   373 (* The type constraint on @{term op =} below is necessary since the operation
```
```   374    is not a parameter of the locale. *)
```
```   375
```
```   376 lemmas
```
```   377   [order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"] =
```
```   378   less_irrefl [THEN notE]
```
```   379 lemmas
```
```   380   [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   381   order_refl
```
```   382 lemmas
```
```   383   [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   384   less_imp_le
```
```   385 lemmas
```
```   386   [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   387   antisym
```
```   388 lemmas
```
```   389   [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   390   eq_refl
```
```   391 lemmas
```
```   392   [order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   393   sym [THEN eq_refl]
```
```   394 lemmas
```
```   395   [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   396   less_trans
```
```   397 lemmas
```
```   398   [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   399   less_le_trans
```
```   400 lemmas
```
```   401   [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   402   le_less_trans
```
```   403 lemmas
```
```   404   [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   405   order_trans
```
```   406 lemmas
```
```   407   [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   408   le_neq_trans
```
```   409 lemmas
```
```   410   [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   411   neq_le_trans
```
```   412 lemmas
```
```   413   [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   414   less_imp_neq
```
```   415 lemmas
```
```   416   [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   417    eq_neq_eq_imp_neq
```
```   418 lemmas
```
```   419   [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   420   not_sym
```
```   421
```
```   422 end
```
```   423
```
```   424 context linorder
```
```   425 begin
```
```   426
```
```   427 lemmas
```
```   428   [order del: order "op = :: 'a => 'a => bool" "op <=" "op <"] = _
```
```   429
```
```   430 lemmas
```
```   431   [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   432   less_irrefl [THEN notE]
```
```   433 lemmas
```
```   434   [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   435   order_refl
```
```   436 lemmas
```
```   437   [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   438   less_imp_le
```
```   439 lemmas
```
```   440   [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   441   not_less [THEN iffD2]
```
```   442 lemmas
```
```   443   [order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   444   not_le [THEN iffD2]
```
```   445 lemmas
```
```   446   [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   447   not_less [THEN iffD1]
```
```   448 lemmas
```
```   449   [order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   450   not_le [THEN iffD1]
```
```   451 lemmas
```
```   452   [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   453   antisym
```
```   454 lemmas
```
```   455   [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   456   eq_refl
```
```   457 lemmas
```
```   458   [order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   459   sym [THEN eq_refl]
```
```   460 lemmas
```
```   461   [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   462   less_trans
```
```   463 lemmas
```
```   464   [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   465   less_le_trans
```
```   466 lemmas
```
```   467   [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   468   le_less_trans
```
```   469 lemmas
```
```   470   [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   471   order_trans
```
```   472 lemmas
```
```   473   [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   474   le_neq_trans
```
```   475 lemmas
```
```   476   [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   477   neq_le_trans
```
```   478 lemmas
```
```   479   [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   480   less_imp_neq
```
```   481 lemmas
```
```   482   [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   483   eq_neq_eq_imp_neq
```
```   484 lemmas
```
```   485   [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] =
```
```   486   not_sym
```
```   487
```
```   488 end
```
```   489
```
```   490
```
```   491 setup {*
```
```   492 let
```
```   493
```
```   494 fun prp t thm = (#prop (rep_thm thm) = t);
```
```   495
```
```   496 fun prove_antisym_le sg ss ((le as Const(_,T)) \$ r \$ s) =
```
```   497   let val prems = prems_of_ss ss;
```
```   498       val less = Const (@{const_name less}, T);
```
```   499       val t = HOLogic.mk_Trueprop(le \$ s \$ r);
```
```   500   in case find_first (prp t) prems of
```
```   501        NONE =>
```
```   502          let val t = HOLogic.mk_Trueprop(HOLogic.Not \$ (less \$ r \$ s))
```
```   503          in case find_first (prp t) prems of
```
```   504               NONE => NONE
```
```   505             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
```
```   506          end
```
```   507      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
```
```   508   end
```
```   509   handle THM _ => NONE;
```
```   510
```
```   511 fun prove_antisym_less sg ss (NotC \$ ((less as Const(_,T)) \$ r \$ s)) =
```
```   512   let val prems = prems_of_ss ss;
```
```   513       val le = Const (@{const_name less_eq}, T);
```
```   514       val t = HOLogic.mk_Trueprop(le \$ r \$ s);
```
```   515   in case find_first (prp t) prems of
```
```   516        NONE =>
```
```   517          let val t = HOLogic.mk_Trueprop(NotC \$ (less \$ s \$ r))
```
```   518          in case find_first (prp t) prems of
```
```   519               NONE => NONE
```
```   520             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
```
```   521          end
```
```   522      | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
```
```   523   end
```
```   524   handle THM _ => NONE;
```
```   525
```
```   526 fun add_simprocs procs thy =
```
```   527   (Simplifier.change_simpset_of thy (fn ss => ss
```
```   528     addsimprocs (map (fn (name, raw_ts, proc) =>
```
```   529       Simplifier.simproc thy name raw_ts proc)) procs); thy);
```
```   530 fun add_solver name tac thy =
```
```   531   (Simplifier.change_simpset_of thy (fn ss => ss addSolver
```
```   532     (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy);
```
```   533
```
```   534 in
```
```   535   add_simprocs [
```
```   536        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
```
```   537        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
```
```   538      ]
```
```   539   #> add_solver "Transitivity" Orders.order_tac
```
```   540   (* Adding the transitivity reasoners also as safe solvers showed a slight
```
```   541      speed up, but the reasoning strength appears to be not higher (at least
```
```   542      no breaking of additional proofs in the entire HOL distribution, as
```
```   543      of 5 March 2004, was observed). *)
```
```   544 end
```
```   545 *}
```
```   546
```
```   547
```
```   548 subsection {* Dense orders *}
```
```   549
```
```   550 class dense_linear_order = linorder +
```
```   551   assumes gt_ex: "\<exists>y. x < y"
```
```   552   and lt_ex: "\<exists>y. y < x"
```
```   553   and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
```
```   554   (*see further theory Dense_Linear_Order*)
```
```   555 begin
```
```   556
```
```   557 lemma interval_empty_iff:
```
```   558   "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
```
```   559   by (auto dest: dense)
```
```   560
```
```   561 end
```
```   562
```
```   563 subsection {* Name duplicates *}
```
```   564
```
```   565 lemmas order_less_le = less_le
```
```   566 lemmas order_eq_refl = order_class.eq_refl
```
```   567 lemmas order_less_irrefl = order_class.less_irrefl
```
```   568 lemmas order_le_less = order_class.le_less
```
```   569 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
```
```   570 lemmas order_less_imp_le = order_class.less_imp_le
```
```   571 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
```
```   572 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
```
```   573 lemmas order_neq_le_trans = order_class.neq_le_trans
```
```   574 lemmas order_le_neq_trans = order_class.le_neq_trans
```
```   575
```
```   576 lemmas order_antisym = antisym
```
```   577 lemmas order_less_not_sym = order_class.less_not_sym
```
```   578 lemmas order_less_asym = order_class.less_asym
```
```   579 lemmas order_eq_iff = order_class.eq_iff
```
```   580 lemmas order_antisym_conv = order_class.antisym_conv
```
```   581 lemmas order_less_trans = order_class.less_trans
```
```   582 lemmas order_le_less_trans = order_class.le_less_trans
```
```   583 lemmas order_less_le_trans = order_class.less_le_trans
```
```   584 lemmas order_less_imp_not_less = order_class.less_imp_not_less
```
```   585 lemmas order_less_imp_triv = order_class.less_imp_triv
```
```   586 lemmas order_less_asym' = order_class.less_asym'
```
```   587
```
```   588 lemmas linorder_linear = linear
```
```   589 lemmas linorder_less_linear = linorder_class.less_linear
```
```   590 lemmas linorder_le_less_linear = linorder_class.le_less_linear
```
```   591 lemmas linorder_le_cases = linorder_class.le_cases
```
```   592 lemmas linorder_not_less = linorder_class.not_less
```
```   593 lemmas linorder_not_le = linorder_class.not_le
```
```   594 lemmas linorder_neq_iff = linorder_class.neq_iff
```
```   595 lemmas linorder_neqE = linorder_class.neqE
```
```   596 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
```
```   597 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
```
```   598 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
```
```   599
```
```   600
```
```   601 subsection {* Bounded quantifiers *}
```
```   602
```
```   603 syntax
```
```   604   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
```
```   605   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
```
```   606   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
```
```   607   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
```
```   608
```
```   609   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
```
```   610   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
```
```   611   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
```
```   612   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
```
```   613
```
```   614 syntax (xsymbols)
```
```   615   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   616   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   617   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   618   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   619
```
```   620   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   621   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   622   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   623   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   624
```
```   625 syntax (HOL)
```
```   626   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
```
```   627   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
```
```   628   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
```
```   629   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
```
```   630
```
```   631 syntax (HTML output)
```
```   632   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   633   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   634   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   635   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   636
```
```   637   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   638   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   639   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   640   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   641
```
```   642 translations
```
```   643   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
```
```   644   "EX x<y. P"    =>  "EX x. x < y \<and> P"
```
```   645   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
```
```   646   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
```
```   647   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
```
```   648   "EX x>y. P"    =>  "EX x. x > y \<and> P"
```
```   649   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
```
```   650   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
```
```   651
```
```   652 print_translation {*
```
```   653 let
```
```   654   val All_binder = Syntax.binder_name @{const_syntax All};
```
```   655   val Ex_binder = Syntax.binder_name @{const_syntax Ex};
```
```   656   val impl = @{const_syntax "op -->"};
```
```   657   val conj = @{const_syntax "op &"};
```
```   658   val less = @{const_syntax less};
```
```   659   val less_eq = @{const_syntax less_eq};
```
```   660
```
```   661   val trans =
```
```   662    [((All_binder, impl, less), ("_All_less", "_All_greater")),
```
```   663     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
```
```   664     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
```
```   665     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
```
```   666
```
```   667   fun matches_bound v t =
```
```   668      case t of (Const ("_bound", _) \$ Free (v', _)) => (v = v')
```
```   669               | _ => false
```
```   670   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
```
```   671   fun mk v c n P = Syntax.const c \$ Syntax.mark_bound v \$ n \$ P
```
```   672
```
```   673   fun tr' q = (q,
```
```   674     fn [Const ("_bound", _) \$ Free (v, _), Const (c, _) \$ (Const (d, _) \$ t \$ u) \$ P] =>
```
```   675       (case AList.lookup (op =) trans (q, c, d) of
```
```   676         NONE => raise Match
```
```   677       | SOME (l, g) =>
```
```   678           if matches_bound v t andalso not (contains_var v u) then mk v l u P
```
```   679           else if matches_bound v u andalso not (contains_var v t) then mk v g t P
```
```   680           else raise Match)
```
```   681      | _ => raise Match);
```
```   682 in [tr' All_binder, tr' Ex_binder] end
```
```   683 *}
```
```   684
```
```   685
```
```   686 subsection {* Transitivity reasoning *}
```
```   687
```
```   688 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
```
```   689 by (rule subst)
```
```   690
```
```   691 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
```
```   692 by (rule ssubst)
```
```   693
```
```   694 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
```
```   695 by (rule subst)
```
```   696
```
```   697 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
```
```   698 by (rule ssubst)
```
```   699
```
```   700 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
```
```   701   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   702 proof -
```
```   703   assume r: "!!x y. x < y ==> f x < f y"
```
```   704   assume "a < b" hence "f a < f b" by (rule r)
```
```   705   also assume "f b < c"
```
```   706   finally (order_less_trans) show ?thesis .
```
```   707 qed
```
```   708
```
```   709 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
```
```   710   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   711 proof -
```
```   712   assume r: "!!x y. x < y ==> f x < f y"
```
```   713   assume "a < f b"
```
```   714   also assume "b < c" hence "f b < f c" by (rule r)
```
```   715   finally (order_less_trans) show ?thesis .
```
```   716 qed
```
```   717
```
```   718 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
```
```   719   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
```
```   720 proof -
```
```   721   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   722   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   723   also assume "f b < c"
```
```   724   finally (order_le_less_trans) show ?thesis .
```
```   725 qed
```
```   726
```
```   727 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
```
```   728   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   729 proof -
```
```   730   assume r: "!!x y. x < y ==> f x < f y"
```
```   731   assume "a <= f b"
```
```   732   also assume "b < c" hence "f b < f c" by (rule r)
```
```   733   finally (order_le_less_trans) show ?thesis .
```
```   734 qed
```
```   735
```
```   736 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
```
```   737   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   738 proof -
```
```   739   assume r: "!!x y. x < y ==> f x < f y"
```
```   740   assume "a < b" hence "f a < f b" by (rule r)
```
```   741   also assume "f b <= c"
```
```   742   finally (order_less_le_trans) show ?thesis .
```
```   743 qed
```
```   744
```
```   745 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
```
```   746   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
```
```   747 proof -
```
```   748   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   749   assume "a < f b"
```
```   750   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   751   finally (order_less_le_trans) show ?thesis .
```
```   752 qed
```
```   753
```
```   754 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
```
```   755   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   756 proof -
```
```   757   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   758   assume "a <= f b"
```
```   759   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   760   finally (order_trans) show ?thesis .
```
```   761 qed
```
```   762
```
```   763 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
```
```   764   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   765 proof -
```
```   766   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   767   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   768   also assume "f b <= c"
```
```   769   finally (order_trans) show ?thesis .
```
```   770 qed
```
```   771
```
```   772 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
```
```   773   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   774 proof -
```
```   775   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   776   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   777   also assume "f b = c"
```
```   778   finally (ord_le_eq_trans) show ?thesis .
```
```   779 qed
```
```   780
```
```   781 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
```
```   782   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   783 proof -
```
```   784   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   785   assume "a = f b"
```
```   786   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   787   finally (ord_eq_le_trans) show ?thesis .
```
```   788 qed
```
```   789
```
```   790 lemma ord_less_eq_subst: "a < b ==> f b = c ==>
```
```   791   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   792 proof -
```
```   793   assume r: "!!x y. x < y ==> f x < f y"
```
```   794   assume "a < b" hence "f a < f b" by (rule r)
```
```   795   also assume "f b = c"
```
```   796   finally (ord_less_eq_trans) show ?thesis .
```
```   797 qed
```
```   798
```
```   799 lemma ord_eq_less_subst: "a = f b ==> b < c ==>
```
```   800   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   801 proof -
```
```   802   assume r: "!!x y. x < y ==> f x < f y"
```
```   803   assume "a = f b"
```
```   804   also assume "b < c" hence "f b < f c" by (rule r)
```
```   805   finally (ord_eq_less_trans) show ?thesis .
```
```   806 qed
```
```   807
```
```   808 text {*
```
```   809   Note that this list of rules is in reverse order of priorities.
```
```   810 *}
```
```   811
```
```   812 lemmas order_trans_rules [trans] =
```
```   813   order_less_subst2
```
```   814   order_less_subst1
```
```   815   order_le_less_subst2
```
```   816   order_le_less_subst1
```
```   817   order_less_le_subst2
```
```   818   order_less_le_subst1
```
```   819   order_subst2
```
```   820   order_subst1
```
```   821   ord_le_eq_subst
```
```   822   ord_eq_le_subst
```
```   823   ord_less_eq_subst
```
```   824   ord_eq_less_subst
```
```   825   forw_subst
```
```   826   back_subst
```
```   827   rev_mp
```
```   828   mp
```
```   829   order_neq_le_trans
```
```   830   order_le_neq_trans
```
```   831   order_less_trans
```
```   832   order_less_asym'
```
```   833   order_le_less_trans
```
```   834   order_less_le_trans
```
```   835   order_trans
```
```   836   order_antisym
```
```   837   ord_le_eq_trans
```
```   838   ord_eq_le_trans
```
```   839   ord_less_eq_trans
```
```   840   ord_eq_less_trans
```
```   841   trans
```
```   842
```
```   843
```
```   844 (* FIXME cleanup *)
```
```   845
```
```   846 text {* These support proving chains of decreasing inequalities
```
```   847     a >= b >= c ... in Isar proofs. *}
```
```   848
```
```   849 lemma xt1:
```
```   850   "a = b ==> b > c ==> a > c"
```
```   851   "a > b ==> b = c ==> a > c"
```
```   852   "a = b ==> b >= c ==> a >= c"
```
```   853   "a >= b ==> b = c ==> a >= c"
```
```   854   "(x::'a::order) >= y ==> y >= x ==> x = y"
```
```   855   "(x::'a::order) >= y ==> y >= z ==> x >= z"
```
```   856   "(x::'a::order) > y ==> y >= z ==> x > z"
```
```   857   "(x::'a::order) >= y ==> y > z ==> x > z"
```
```   858   "(a::'a::order) > b ==> b > a ==> P"
```
```   859   "(x::'a::order) > y ==> y > z ==> x > z"
```
```   860   "(a::'a::order) >= b ==> a ~= b ==> a > b"
```
```   861   "(a::'a::order) ~= b ==> a >= b ==> a > b"
```
```   862   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   863   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   864   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
```
```   865   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
```
```   866   by auto
```
```   867
```
```   868 lemma xt2:
```
```   869   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
```
```   870 by (subgoal_tac "f b >= f c", force, force)
```
```   871
```
```   872 lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
```
```   873     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
```
```   874 by (subgoal_tac "f a >= f b", force, force)
```
```   875
```
```   876 lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
```
```   877   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
```
```   878 by (subgoal_tac "f b >= f c", force, force)
```
```   879
```
```   880 lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
```
```   881     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   882 by (subgoal_tac "f a > f b", force, force)
```
```   883
```
```   884 lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
```
```   885     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   886 by (subgoal_tac "f b > f c", force, force)
```
```   887
```
```   888 lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
```
```   889     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
```
```   890 by (subgoal_tac "f a >= f b", force, force)
```
```   891
```
```   892 lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
```
```   893     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   894 by (subgoal_tac "f b > f c", force, force)
```
```   895
```
```   896 lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
```
```   897     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   898 by (subgoal_tac "f a > f b", force, force)
```
```   899
```
```   900 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
```
```   901
```
```   902 (*
```
```   903   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
```
```   904   for the wrong thing in an Isar proof.
```
```   905
```
```   906   The extra transitivity rules can be used as follows:
```
```   907
```
```   908 lemma "(a::'a::order) > z"
```
```   909 proof -
```
```   910   have "a >= b" (is "_ >= ?rhs")
```
```   911     sorry
```
```   912   also have "?rhs >= c" (is "_ >= ?rhs")
```
```   913     sorry
```
```   914   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
```
```   915     sorry
```
```   916   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
```
```   917     sorry
```
```   918   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
```
```   919     sorry
```
```   920   also (xtrans) have "?rhs > z"
```
```   921     sorry
```
```   922   finally (xtrans) show ?thesis .
```
```   923 qed
```
```   924
```
```   925   Alternatively, one can use "declare xtrans [trans]" and then
```
```   926   leave out the "(xtrans)" above.
```
```   927 *)
```
```   928
```
```   929 subsection {* Order on bool *}
```
```   930
```
```   931 instance bool :: order
```
```   932   le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
```
```   933   less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
```
```   934   by intro_classes (auto simp add: le_bool_def less_bool_def)
```
```   935 lemmas [code func del] = le_bool_def less_bool_def
```
```   936
```
```   937 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
```
```   938 by (simp add: le_bool_def)
```
```   939
```
```   940 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
```
```   941 by (simp add: le_bool_def)
```
```   942
```
```   943 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```   944 by (simp add: le_bool_def)
```
```   945
```
```   946 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
```
```   947 by (simp add: le_bool_def)
```
```   948
```
```   949 lemma [code func]:
```
```   950   "False \<le> b \<longleftrightarrow> True"
```
```   951   "True \<le> b \<longleftrightarrow> b"
```
```   952   "False < b \<longleftrightarrow> b"
```
```   953   "True < b \<longleftrightarrow> False"
```
```   954   unfolding le_bool_def less_bool_def by simp_all
```
```   955
```
```   956
```
```   957 subsection {* Order on sets *}
```
```   958
```
```   959 instance set :: (type) order
```
```   960   by (intro_classes,
```
```   961       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
```
```   962
```
```   963 lemmas basic_trans_rules [trans] =
```
```   964   order_trans_rules set_rev_mp set_mp
```
```   965
```
```   966
```
```   967 subsection {* Order on functions *}
```
```   968
```
```   969 instance "fun" :: (type, ord) ord
```
```   970   le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
```
```   971   less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
```
```   972
```
```   973 lemmas [code func del] = le_fun_def less_fun_def
```
```   974
```
```   975 instance "fun" :: (type, order) order
```
```   976   by default
```
```   977     (auto simp add: le_fun_def less_fun_def expand_fun_eq
```
```   978        intro: order_trans order_antisym)
```
```   979
```
```   980 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
```
```   981   unfolding le_fun_def by simp
```
```   982
```
```   983 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   984   unfolding le_fun_def by simp
```
```   985
```
```   986 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
```
```   987   unfolding le_fun_def by simp
```
```   988
```
```   989 text {*
```
```   990   Handy introduction and elimination rules for @{text "\<le>"}
```
```   991   on unary and binary predicates
```
```   992 *}
```
```   993
```
```   994 lemma predicate1I [Pure.intro!, intro!]:
```
```   995   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
```
```   996   shows "P \<le> Q"
```
```   997   apply (rule le_funI)
```
```   998   apply (rule le_boolI)
```
```   999   apply (rule PQ)
```
```  1000   apply assumption
```
```  1001   done
```
```  1002
```
```  1003 lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
```
```  1004   apply (erule le_funE)
```
```  1005   apply (erule le_boolE)
```
```  1006   apply assumption+
```
```  1007   done
```
```  1008
```
```  1009 lemma predicate2I [Pure.intro!, intro!]:
```
```  1010   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
```
```  1011   shows "P \<le> Q"
```
```  1012   apply (rule le_funI)+
```
```  1013   apply (rule le_boolI)
```
```  1014   apply (rule PQ)
```
```  1015   apply assumption
```
```  1016   done
```
```  1017
```
```  1018 lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
```
```  1019   apply (erule le_funE)+
```
```  1020   apply (erule le_boolE)
```
```  1021   apply assumption+
```
```  1022   done
```
```  1023
```
```  1024 lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
```
```  1025   by (rule predicate1D)
```
```  1026
```
```  1027 lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
```
```  1028   by (rule predicate2D)
```
```  1029
```
```  1030
```
```  1031 subsection {* Monotonicity, least value operator and min/max *}
```
```  1032
```
```  1033 context order
```
```  1034 begin
```
```  1035
```
```  1036 definition
```
```  1037   mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool"
```
```  1038 where
```
```  1039   "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
```
```  1040
```
```  1041 lemma monoI [intro?]:
```
```  1042   fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
```
```  1043   shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
```
```  1044   unfolding mono_def by iprover
```
```  1045
```
```  1046 lemma monoD [dest?]:
```
```  1047   fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
```
```  1048   shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1049   unfolding mono_def by iprover
```
```  1050
```
```  1051 end
```
```  1052
```
```  1053 context linorder
```
```  1054 begin
```
```  1055
```
```  1056 lemma min_of_mono:
```
```  1057   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
```
```  1058   shows "mono f \<Longrightarrow> Orderings.min (f m) (f n) = f (min m n)"
```
```  1059   by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
```
```  1060
```
```  1061 lemma max_of_mono:
```
```  1062   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
```
```  1063   shows "mono f \<Longrightarrow> Orderings.max (f m) (f n) = f (max m n)"
```
```  1064   by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
```
```  1065
```
```  1066 end
```
```  1067
```
```  1068 lemma LeastI2_order:
```
```  1069   "[| P (x::'a::order);
```
```  1070       !!y. P y ==> x <= y;
```
```  1071       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
```
```  1072    ==> Q (Least P)"
```
```  1073 apply (unfold Least_def)
```
```  1074 apply (rule theI2)
```
```  1075   apply (blast intro: order_antisym)+
```
```  1076 done
```
```  1077
```
```  1078 lemma Least_mono:
```
```  1079   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
```
```  1080     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
```
```  1081     -- {* Courtesy of Stephan Merz *}
```
```  1082   apply clarify
```
```  1083   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
```
```  1084   apply (rule LeastI2_order)
```
```  1085   apply (auto elim: monoD intro!: order_antisym)
```
```  1086   done
```
```  1087
```
```  1088 lemma Least_equality:
```
```  1089   "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
```
```  1090 apply (simp add: Least_def)
```
```  1091 apply (rule the_equality)
```
```  1092 apply (auto intro!: order_antisym)
```
```  1093 done
```
```  1094
```
```  1095 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
```
```  1096 by (simp add: min_def)
```
```  1097
```
```  1098 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
```
```  1099 by (simp add: max_def)
```
```  1100
```
```  1101 lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
```
```  1102 apply (simp add: min_def)
```
```  1103 apply (blast intro: order_antisym)
```
```  1104 done
```
```  1105
```
```  1106 lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
```
```  1107 apply (simp add: max_def)
```
```  1108 apply (blast intro: order_antisym)
```
```  1109 done
```
```  1110
```
```  1111 subsection {* legacy ML bindings *}
```
```  1112
```
```  1113 ML {*
```
```  1114 val monoI = @{thm monoI};
```
```  1115 *}
```
```  1116
```
```  1117 end
```