src/HOL/Orderings.thy
 author wenzelm Tue Oct 09 00:20:13 2007 +0200 (2007-10-09) changeset 24920 2a45e400fdad parent 24867 e5b55d7be9bb child 25062 af5ef0d4d655 permissions -rw-r--r--
generic Syntax.pretty/string_of operations;
```     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 \<^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"
```
```    21
```
```    22 begin
```
```    23
```
```    24 notation (input)
```
```    25   less_eq (infix "\<sqsubseteq>" 50)
```
```    26 and
```
```    27   less    (infix "\<sqsubset>" 50)
```
```    28
```
```    29 text {* Reflexivity. *}
```
```    30
```
```    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)
```
```    34
```
```    35 lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
```
```    36 by (simp add: less_le)
```
```    37
```
```    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
```
```    41
```
```    42 lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
```
```    43 unfolding less_le by blast
```
```    44
```
```    45 lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
```
```    46 unfolding less_le by blast
```
```    47
```
```    48 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
```
```    49 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    50
```
```    51
```
```    52 text {* Useful for simplification, but too risky to include by default. *}
```
```    53
```
```    54 lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
```
```    55 by auto
```
```    56
```
```    57 lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
```
```    58 by auto
```
```    59
```
```    60
```
```    61 text {* Transitivity rules for calculational reasoning *}
```
```    62
```
```    63 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
```
```    64 by (simp add: less_le)
```
```    65
```
```    66 lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
```
```    67 by (simp add: less_le)
```
```    68
```
```    69
```
```    70 text {* Asymmetry. *}
```
```    71
```
```    72 lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
```
```    73 by (simp add: less_le antisym)
```
```    74
```
```    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
```
```    77
```
```    78 lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
```
```    79 by (blast intro: antisym)
```
```    80
```
```    81 lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
```
```    82 by (blast intro: antisym)
```
```    83
```
```    84 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
```
```    85 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    86
```
```    87
```
```    88 text {* Transitivity. *}
```
```    89
```
```    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)
```
```    92
```
```    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)
```
```    95
```
```    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)
```
```    98
```
```    99
```
```   100 text {* Useful for simplification, but too risky to include by default. *}
```
```   101
```
```   102 lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
```
```   103 by (blast elim: less_asym)
```
```   104
```
```   105 lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
```
```   106 by (blast elim: less_asym)
```
```   107
```
```   108
```
```   109 text {* Transitivity rules for calculational reasoning *}
```
```   110
```
```   111 lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
```
```   112 by (rule less_asym)
```
```   113
```
```   114
```
```   115 text {* Reverse order *}
```
```   116
```
```   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)
```
```   121
```
```   122 end
```
```   123
```
```   124
```
```   125 subsection {* Linear (total) orders *}
```
```   126
```
```   127 class linorder = order +
```
```   128   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
```
```   129 begin
```
```   130
```
```   131 lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
```
```   132 unfolding less_le using less_le linear by blast
```
```   133
```
```   134 lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
```
```   135 by (simp add: le_less less_linear)
```
```   136
```
```   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
```
```   140
```
```   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
```
```   144
```
```   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
```
```   149
```
```   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
```
```   155
```
```   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
```
```   160
```
```   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)
```
```   163
```
```   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
```
```   166
```
```   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])
```
```   169
```
```   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])
```
```   172
```
```   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])
```
```   175
```
```   176 text{*Replacing the old Nat.leI*}
```
```   177 lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
```
```   178 unfolding not_less .
```
```   179
```
```   180 lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
```
```   181 unfolding not_less .
```
```   182
```
```   183 (*FIXME inappropriate name (or delete altogether)*)
```
```   184 lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
```
```   185 unfolding not_le .
```
```   186
```
```   187
```
```   188 text {* Reverse order *}
```
```   189
```
```   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)
```
```   194
```
```   195
```
```   196 text {* min/max *}
```
```   197
```
```   198 text {* for historic reasons, definitions are done in context ord *}
```
```   199
```
```   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)"
```
```   203
```
```   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)"
```
```   207
```
```   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)
```
```   211
```
```   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)
```
```   215
```
```   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)
```
```   219
```
```   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)
```
```   223
```
```   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)
```
```   227
```
```   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)
```
```   231
```
```   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)
```
```   235
```
```   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)
```
```   239
```
```   240 end
```
```   241
```
```   242
```
```   243 subsection {* Reasoning tools setup *}
```
```   244
```
```   245 ML {*
```
```   246
```
```   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;
```
```   253
```
```   254 structure Orders: ORDERS =
```
```   255 struct
```
```   256
```
```   257 (** Theory and context data **)
```
```   258
```
```   259 fun struct_eq ((s1: string, ts1), (s2, ts2)) =
```
```   260   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
```
```   261
```
```   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 );
```
```   273
```
```   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;
```
```   287
```
```   288
```
```   289 (** Method **)
```
```   290
```
```   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
```
```   319
```
```   320 fun order_tac prems ctxt =
```
```   321   FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));
```
```   322
```
```   323
```
```   324 (** Attribute **)
```
```   325
```
```   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));
```
```   332
```
```   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)));
```
```   340
```
```   341
```
```   342 (** Diagnostic command **)
```
```   343
```
```   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));
```
```   348
```
```   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));
```
```   353
```
```   354
```
```   355 (** Setup **)
```
```   356
```
```   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")];
```
```   361
```
```   362 end;
```
```   363
```
```   364 *}
```
```   365
```
```   366 setup Orders.setup
```
```   367
```
```   368
```
```   369 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
```
```   370
```
```   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
```
```   418
```
```   419 lemmas (in linorder) [order del: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = _
```
```   420
```
```   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
```
```   478
```
```   479
```
```   480 setup {*
```
```   481 let
```
```   482
```
```   483 fun prp t thm = (#prop (rep_thm thm) = t);
```
```   484
```
```   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;
```
```   499
```
```   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;
```
```   514
```
```   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);
```
```   522
```
```   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 *}
```
```   535
```
```   536
```
```   537 subsection {* Dense orders *}
```
```   538
```
```   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*)
```
```   544
```
```   545
```
```   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)
```
```   550
```
```   551 subsection {* Name duplicates *}
```
```   552
```
```   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
```
```   563
```
```   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'
```
```   575
```
```   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
```
```   587
```
```   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
```
```   596
```
```   597
```
```   598 subsection {* Bounded quantifiers *}
```
```   599
```
```   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)
```
```   605
```
```   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)
```
```   610
```
```   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)
```
```   616
```
```   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)
```
```   621
```
```   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)
```
```   627
```
```   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)
```
```   633
```
```   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)
```
```   638
```
```   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"
```
```   648
```
```   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};
```
```   657
```
```   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"))];
```
```   663
```
```   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
```
```   669
```
```   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 *}
```
```   681
```
```   682
```
```   683 subsection {* Transitivity reasoning *}
```
```   684
```
```   685 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
```
```   686 by (rule subst)
```
```   687
```
```   688 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
```
```   689 by (rule ssubst)
```
```   690
```
```   691 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
```
```   692 by (rule subst)
```
```   693
```
```   694 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
```
```   695 by (rule ssubst)
```
```   696
```
```   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
```
```   705
```
```   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
```
```   714
```
```   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
```
```   723
```
```   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
```
```   732
```
```   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
```
```   741
```
```   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
```
```   750
```
```   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
```
```   759
```
```   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
```
```   768
```
```   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
```
```   777
```
```   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
```
```   786
```
```   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
```
```   795
```
```   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
```
```   804
```
```   805 text {*
```
```   806   Note that this list of rules is in reverse order of priorities.
```
```   807 *}
```
```   808
```
```   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
```
```   839
```
```   840
```
```   841 (* FIXME cleanup *)
```
```   842
```
```   843 text {* These support proving chains of decreasing inequalities
```
```   844     a >= b >= c ... in Isar proofs. *}
```
```   845
```
```   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
```
```   864
```
```   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)
```
```   868
```
```   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)
```
```   872
```
```   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)
```
```   876
```
```   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)
```
```   880
```
```   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)
```
```   884
```
```   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)
```
```   888
```
```   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)
```
```   892
```
```   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)
```
```   896
```
```   897 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
```
```   898
```
```   899 (*
```
```   900   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
```
```   901   for the wrong thing in an Isar proof.
```
```   902
```
```   903   The extra transitivity rules can be used as follows:
```
```   904
```
```   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
```
```   921
```
```   922   Alternatively, one can use "declare xtrans [trans]" and then
```
```   923   leave out the "(xtrans)" above.
```
```   924 *)
```
```   925
```
```   926 subsection {* Order on bool *}
```
```   927
```
```   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
```
```   933
```
```   934 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
```
```   935 by (simp add: le_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_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```   941 by (simp add: le_bool_def)
```
```   942
```
```   943 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
```
```   944 by (simp add: le_bool_def)
```
```   945
```
```   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
```
```   952
```
```   953
```
```   954 subsection {* Order on sets *}
```
```   955
```
```   956 instance set :: (type) order
```
```   957   by (intro_classes,
```
```   958       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
```
```   959
```
```   960 lemmas basic_trans_rules [trans] =
```
```   961   order_trans_rules set_rev_mp set_mp
```
```   962
```
```   963
```
```   964 subsection {* Order on functions *}
```
```   965
```
```   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" ..
```
```   969
```
```   970 lemmas [code func del] = le_fun_def less_fun_def
```
```   971
```
```   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)
```
```   976
```
```   977 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
```
```   978   unfolding le_fun_def by simp
```
```   979
```
```   980 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   981   unfolding le_fun_def by simp
```
```   982
```
```   983 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
```
```   984   unfolding le_fun_def by simp
```
```   985
```
```   986 text {*
```
```   987   Handy introduction and elimination rules for @{text "\<le>"}
```
```   988   on unary and binary predicates
```
```   989 *}
```
```   990
```
```   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
```
```   999
```
```  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
```
```  1005
```
```  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
```
```  1014
```
```  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
```
```  1020
```
```  1021 lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
```
```  1022   by (rule predicate1D)
```
```  1023
```
```  1024 lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
```
```  1025   by (rule predicate2D)
```
```  1026
```
```  1027
```
```  1028 subsection {* Monotonicity, least value operator and min/max *}
```
```  1029
```
```  1030 locale mono =
```
```  1031   fixes f
```
```  1032   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
```
```  1033
```
```  1034 lemmas monoI [intro?] = mono.intro
```
```  1035   and monoD [dest?] = mono.mono
```
```  1036
```
```  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
```
```  1046
```
```  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
```
```  1056
```
```  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
```
```  1063
```
```  1064 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
```
```  1065 by (simp add: min_def)
```
```  1066
```
```  1067 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
```
```  1068 by (simp add: max_def)
```
```  1069
```
```  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
```
```  1074
```
```  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
```
```  1079
```
```  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)
```
```  1083
```
```  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)
```
```  1087
```
```  1088
```
```  1089 subsection {* legacy ML bindings *}
```
```  1090
```
```  1091 ML {*
```
```  1092 val monoI = @{thm monoI};
```
```  1093 *}
```
```  1094
```
```  1095 end
```