src/HOL/Orderings.thy
 author haftmann Thu Jan 28 11:48:49 2010 +0100 (2010-01-28) changeset 34974 18b41bba42b5 parent 34250 3b619abaa67a child 35028 108662d50512 permissions -rw-r--r--
new theory Algebras.thy for generic algebraic structures
```     1 (*  Title:      HOL/Orderings.thy
```
```     2     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
```
```     3 *)
```
```     4
```
```     5 header {* Abstract orderings *}
```
```     6
```
```     7 theory Orderings
```
```     8 imports Algebras
```
```     9 uses
```
```    10   "~~/src/Provers/order.ML"
```
```    11   "~~/src/Provers/quasi.ML"  (* FIXME unused? *)
```
```    12 begin
```
```    13
```
```    14 subsection {* Quasi orders *}
```
```    15
```
```    16 class preorder = ord +
```
```    17   assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
```
```    18   and order_refl [iff]: "x \<le> x"
```
```    19   and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
```
```    20 begin
```
```    21
```
```    22 text {* Reflexivity. *}
```
```    23
```
```    24 lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
```
```    25     -- {* This form is useful with the classical reasoner. *}
```
```    26 by (erule ssubst) (rule order_refl)
```
```    27
```
```    28 lemma less_irrefl [iff]: "\<not> x < x"
```
```    29 by (simp add: less_le_not_le)
```
```    30
```
```    31 lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
```
```    32 unfolding less_le_not_le by blast
```
```    33
```
```    34
```
```    35 text {* Asymmetry. *}
```
```    36
```
```    37 lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
```
```    38 by (simp add: less_le_not_le)
```
```    39
```
```    40 lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
```
```    41 by (drule less_not_sym, erule contrapos_np) simp
```
```    42
```
```    43
```
```    44 text {* Transitivity. *}
```
```    45
```
```    46 lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
```
```    47 by (auto simp add: less_le_not_le intro: order_trans)
```
```    48
```
```    49 lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
```
```    50 by (auto simp add: less_le_not_le intro: order_trans)
```
```    51
```
```    52 lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
```
```    53 by (auto simp add: less_le_not_le intro: order_trans)
```
```    54
```
```    55
```
```    56 text {* Useful for simplification, but too risky to include by default. *}
```
```    57
```
```    58 lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
```
```    59 by (blast elim: less_asym)
```
```    60
```
```    61 lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
```
```    62 by (blast elim: less_asym)
```
```    63
```
```    64
```
```    65 text {* Transitivity rules for calculational reasoning *}
```
```    66
```
```    67 lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
```
```    68 by (rule less_asym)
```
```    69
```
```    70
```
```    71 text {* Dual order *}
```
```    72
```
```    73 lemma dual_preorder:
```
```    74   "preorder (op \<ge>) (op >)"
```
```    75 proof qed (auto simp add: less_le_not_le intro: order_trans)
```
```    76
```
```    77 end
```
```    78
```
```    79
```
```    80 subsection {* Partial orders *}
```
```    81
```
```    82 class order = preorder +
```
```    83   assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
```
```    84 begin
```
```    85
```
```    86 text {* Reflexivity. *}
```
```    87
```
```    88 lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
```
```    89 by (auto simp add: less_le_not_le intro: antisym)
```
```    90
```
```    91 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
```
```    92     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
```
```    93 by (simp add: less_le) blast
```
```    94
```
```    95 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
```
```    96 unfolding less_le by blast
```
```    97
```
```    98
```
```    99 text {* Useful for simplification, but too risky to include by default. *}
```
```   100
```
```   101 lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
```
```   102 by auto
```
```   103
```
```   104 lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
```
```   105 by auto
```
```   106
```
```   107
```
```   108 text {* Transitivity rules for calculational reasoning *}
```
```   109
```
```   110 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
```
```   111 by (simp add: less_le)
```
```   112
```
```   113 lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
```
```   114 by (simp add: less_le)
```
```   115
```
```   116
```
```   117 text {* Asymmetry. *}
```
```   118
```
```   119 lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
```
```   120 by (blast intro: antisym)
```
```   121
```
```   122 lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
```
```   123 by (blast intro: antisym)
```
```   124
```
```   125 lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
```
```   126 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```   127
```
```   128
```
```   129 text {* Least value operator *}
```
```   130
```
```   131 definition (in ord)
```
```   132   Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
```
```   133   "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
```
```   134
```
```   135 lemma Least_equality:
```
```   136   assumes "P x"
```
```   137     and "\<And>y. P y \<Longrightarrow> x \<le> y"
```
```   138   shows "Least P = x"
```
```   139 unfolding Least_def by (rule the_equality)
```
```   140   (blast intro: assms antisym)+
```
```   141
```
```   142 lemma LeastI2_order:
```
```   143   assumes "P x"
```
```   144     and "\<And>y. P y \<Longrightarrow> x \<le> y"
```
```   145     and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
```
```   146   shows "Q (Least P)"
```
```   147 unfolding Least_def by (rule theI2)
```
```   148   (blast intro: assms antisym)+
```
```   149
```
```   150
```
```   151 text {* Dual order *}
```
```   152
```
```   153 lemma dual_order:
```
```   154   "order (op \<ge>) (op >)"
```
```   155 by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym)
```
```   156
```
```   157 end
```
```   158
```
```   159
```
```   160 subsection {* Linear (total) orders *}
```
```   161
```
```   162 class linorder = order +
```
```   163   assumes linear: "x \<le> y \<or> y \<le> x"
```
```   164 begin
```
```   165
```
```   166 lemma less_linear: "x < y \<or> x = y \<or> y < x"
```
```   167 unfolding less_le using less_le linear by blast
```
```   168
```
```   169 lemma le_less_linear: "x \<le> y \<or> y < x"
```
```   170 by (simp add: le_less less_linear)
```
```   171
```
```   172 lemma le_cases [case_names le ge]:
```
```   173   "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   174 using linear by blast
```
```   175
```
```   176 lemma linorder_cases [case_names less equal greater]:
```
```   177   "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   178 using less_linear by blast
```
```   179
```
```   180 lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
```
```   181 apply (simp add: less_le)
```
```   182 using linear apply (blast intro: antisym)
```
```   183 done
```
```   184
```
```   185 lemma not_less_iff_gr_or_eq:
```
```   186  "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"
```
```   187 apply(simp add:not_less le_less)
```
```   188 apply blast
```
```   189 done
```
```   190
```
```   191 lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
```
```   192 apply (simp add: less_le)
```
```   193 using linear apply (blast intro: antisym)
```
```   194 done
```
```   195
```
```   196 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
```
```   197 by (cut_tac x = x and y = y in less_linear, auto)
```
```   198
```
```   199 lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
```
```   200 by (simp add: neq_iff) blast
```
```   201
```
```   202 lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
```
```   203 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   204
```
```   205 lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
```
```   206 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   207
```
```   208 lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
```
```   209 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   210
```
```   211 lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
```
```   212 unfolding not_less .
```
```   213
```
```   214 lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
```
```   215 unfolding not_less .
```
```   216
```
```   217 (*FIXME inappropriate name (or delete altogether)*)
```
```   218 lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"
```
```   219 unfolding not_le .
```
```   220
```
```   221
```
```   222 text {* Dual order *}
```
```   223
```
```   224 lemma dual_linorder:
```
```   225   "linorder (op \<ge>) (op >)"
```
```   226 by (rule linorder.intro, rule dual_order) (unfold_locales, rule linear)
```
```   227
```
```   228
```
```   229 text {* min/max *}
```
```   230
```
```   231 definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   232   [code del]: "min a b = (if a \<le> b then a else b)"
```
```   233
```
```   234 definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   235   [code del]: "max a b = (if a \<le> b then b else a)"
```
```   236
```
```   237 lemma min_le_iff_disj:
```
```   238   "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
```
```   239 unfolding min_def using linear by (auto intro: order_trans)
```
```   240
```
```   241 lemma le_max_iff_disj:
```
```   242   "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
```
```   243 unfolding max_def using linear by (auto intro: order_trans)
```
```   244
```
```   245 lemma min_less_iff_disj:
```
```   246   "min x y < z \<longleftrightarrow> x < z \<or> y < z"
```
```   247 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   248
```
```   249 lemma less_max_iff_disj:
```
```   250   "z < max x y \<longleftrightarrow> z < x \<or> z < y"
```
```   251 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   252
```
```   253 lemma min_less_iff_conj [simp]:
```
```   254   "z < min x y \<longleftrightarrow> z < x \<and> z < y"
```
```   255 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   256
```
```   257 lemma max_less_iff_conj [simp]:
```
```   258   "max x y < z \<longleftrightarrow> x < z \<and> y < z"
```
```   259 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   260
```
```   261 lemma split_min [noatp]:
```
```   262   "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
```
```   263 by (simp add: min_def)
```
```   264
```
```   265 lemma split_max [noatp]:
```
```   266   "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
```
```   267 by (simp add: max_def)
```
```   268
```
```   269 end
```
```   270
```
```   271 text {* Explicit dictionaries for code generation *}
```
```   272
```
```   273 lemma min_ord_min [code, code_unfold, code_inline del]:
```
```   274   "min = ord.min (op \<le>)"
```
```   275   by (rule ext)+ (simp add: min_def ord.min_def)
```
```   276
```
```   277 declare ord.min_def [code]
```
```   278
```
```   279 lemma max_ord_max [code, code_unfold, code_inline del]:
```
```   280   "max = ord.max (op \<le>)"
```
```   281   by (rule ext)+ (simp add: max_def ord.max_def)
```
```   282
```
```   283 declare ord.max_def [code]
```
```   284
```
```   285
```
```   286 subsection {* Reasoning tools setup *}
```
```   287
```
```   288 ML {*
```
```   289
```
```   290 signature ORDERS =
```
```   291 sig
```
```   292   val print_structures: Proof.context -> unit
```
```   293   val setup: theory -> theory
```
```   294   val order_tac: Proof.context -> thm list -> int -> tactic
```
```   295 end;
```
```   296
```
```   297 structure Orders: ORDERS =
```
```   298 struct
```
```   299
```
```   300 (** Theory and context data **)
```
```   301
```
```   302 fun struct_eq ((s1: string, ts1), (s2, ts2)) =
```
```   303   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
```
```   304
```
```   305 structure Data = Generic_Data
```
```   306 (
```
```   307   type T = ((string * term list) * Order_Tac.less_arith) list;
```
```   308     (* Order structures:
```
```   309        identifier of the structure, list of operations and record of theorems
```
```   310        needed to set up the transitivity reasoner,
```
```   311        identifier and operations identify the structure uniquely. *)
```
```   312   val empty = [];
```
```   313   val extend = I;
```
```   314   fun merge data = AList.join struct_eq (K fst) data;
```
```   315 );
```
```   316
```
```   317 fun print_structures ctxt =
```
```   318   let
```
```   319     val structs = Data.get (Context.Proof ctxt);
```
```   320     fun pretty_term t = Pretty.block
```
```   321       [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
```
```   322         Pretty.str "::", Pretty.brk 1,
```
```   323         Pretty.quote (Syntax.pretty_typ ctxt (type_of t))];
```
```   324     fun pretty_struct ((s, ts), _) = Pretty.block
```
```   325       [Pretty.str s, Pretty.str ":", Pretty.brk 1,
```
```   326        Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
```
```   327   in
```
```   328     Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
```
```   329   end;
```
```   330
```
```   331
```
```   332 (** Method **)
```
```   333
```
```   334 fun struct_tac ((s, [eq, le, less]), thms) ctxt prems =
```
```   335   let
```
```   336     fun decomp thy (@{const Trueprop} \$ t) =
```
```   337       let
```
```   338         fun excluded t =
```
```   339           (* exclude numeric types: linear arithmetic subsumes transitivity *)
```
```   340           let val T = type_of t
```
```   341           in
```
```   342             T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
```
```   343           end;
```
```   344         fun rel (bin_op \$ t1 \$ t2) =
```
```   345               if excluded t1 then NONE
```
```   346               else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
```
```   347               else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
```
```   348               else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
```
```   349               else NONE
```
```   350           | rel _ = NONE;
```
```   351         fun dec (Const (@{const_name Not}, _) \$ t) = (case rel t
```
```   352               of NONE => NONE
```
```   353                | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
```
```   354           | dec x = rel x;
```
```   355       in dec t end
```
```   356       | decomp thy _ = NONE;
```
```   357   in
```
```   358     case s of
```
```   359       "order" => Order_Tac.partial_tac decomp thms ctxt prems
```
```   360     | "linorder" => Order_Tac.linear_tac decomp thms ctxt prems
```
```   361     | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
```
```   362   end
```
```   363
```
```   364 fun order_tac ctxt prems =
```
```   365   FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt)));
```
```   366
```
```   367
```
```   368 (** Attribute **)
```
```   369
```
```   370 fun add_struct_thm s tag =
```
```   371   Thm.declaration_attribute
```
```   372     (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
```
```   373 fun del_struct s =
```
```   374   Thm.declaration_attribute
```
```   375     (fn _ => Data.map (AList.delete struct_eq s));
```
```   376
```
```   377 val attrib_setup =
```
```   378   Attrib.setup @{binding order}
```
```   379     (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|
```
```   380       Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --
```
```   381       Scan.repeat Args.term
```
```   382       >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
```
```   383            | ((NONE, n), ts) => del_struct (n, ts)))
```
```   384     "theorems controlling transitivity reasoner";
```
```   385
```
```   386
```
```   387 (** Diagnostic command **)
```
```   388
```
```   389 val _ =
```
```   390   OuterSyntax.improper_command "print_orders"
```
```   391     "print order structures available to transitivity reasoner" OuterKeyword.diag
```
```   392     (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o
```
```   393         Toplevel.keep (print_structures o Toplevel.context_of)));
```
```   394
```
```   395
```
```   396 (** Setup **)
```
```   397
```
```   398 val setup =
```
```   399   Method.setup @{binding order} (Scan.succeed (fn ctxt => SIMPLE_METHOD' (order_tac ctxt [])))
```
```   400     "transitivity reasoner" #>
```
```   401   attrib_setup;
```
```   402
```
```   403 end;
```
```   404
```
```   405 *}
```
```   406
```
```   407 setup Orders.setup
```
```   408
```
```   409
```
```   410 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
```
```   411
```
```   412 context order
```
```   413 begin
```
```   414
```
```   415 (* The type constraint on @{term op =} below is necessary since the operation
```
```   416    is not a parameter of the locale. *)
```
```   417
```
```   418 declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]
```
```   419
```
```   420 declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   421
```
```   422 declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   423
```
```   424 declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   425
```
```   426 declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   427
```
```   428 declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   429
```
```   430 declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   431
```
```   432 declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   433
```
```   434 declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   435
```
```   436 declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   437
```
```   438 declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   439
```
```   440 declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   441
```
```   442 declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   443
```
```   444 declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   445
```
```   446 declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   447
```
```   448 end
```
```   449
```
```   450 context linorder
```
```   451 begin
```
```   452
```
```   453 declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]
```
```   454
```
```   455 declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   456
```
```   457 declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   458
```
```   459 declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   460
```
```   461 declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   462
```
```   463 declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   464
```
```   465 declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   466
```
```   467 declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   468
```
```   469 declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   470
```
```   471 declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   472
```
```   473 declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   474
```
```   475 declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   476
```
```   477 declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   478
```
```   479 declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   480
```
```   481 declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   482
```
```   483 declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   484
```
```   485 declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   486
```
```   487 declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   488
```
```   489 declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   490
```
```   491 declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]
```
```   492
```
```   493 end
```
```   494
```
```   495
```
```   496 setup {*
```
```   497 let
```
```   498
```
```   499 fun prp t thm = (#prop (rep_thm thm) = t);
```
```   500
```
```   501 fun prove_antisym_le sg ss ((le as Const(_,T)) \$ r \$ s) =
```
```   502   let val prems = prems_of_ss ss;
```
```   503       val less = Const (@{const_name less}, T);
```
```   504       val t = HOLogic.mk_Trueprop(le \$ s \$ r);
```
```   505   in case find_first (prp t) prems of
```
```   506        NONE =>
```
```   507          let val t = HOLogic.mk_Trueprop(HOLogic.Not \$ (less \$ r \$ s))
```
```   508          in case find_first (prp t) prems of
```
```   509               NONE => NONE
```
```   510             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
```
```   511          end
```
```   512      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
```
```   513   end
```
```   514   handle THM _ => NONE;
```
```   515
```
```   516 fun prove_antisym_less sg ss (NotC \$ ((less as Const(_,T)) \$ r \$ s)) =
```
```   517   let val prems = prems_of_ss ss;
```
```   518       val le = Const (@{const_name less_eq}, T);
```
```   519       val t = HOLogic.mk_Trueprop(le \$ r \$ s);
```
```   520   in case find_first (prp t) prems of
```
```   521        NONE =>
```
```   522          let val t = HOLogic.mk_Trueprop(NotC \$ (less \$ s \$ r))
```
```   523          in case find_first (prp t) prems of
```
```   524               NONE => NONE
```
```   525             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
```
```   526          end
```
```   527      | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
```
```   528   end
```
```   529   handle THM _ => NONE;
```
```   530
```
```   531 fun add_simprocs procs thy =
```
```   532   Simplifier.map_simpset (fn ss => ss
```
```   533     addsimprocs (map (fn (name, raw_ts, proc) =>
```
```   534       Simplifier.simproc thy name raw_ts proc) procs)) thy;
```
```   535 fun add_solver name tac =
```
```   536   Simplifier.map_simpset (fn ss => ss addSolver
```
```   537     mk_solver' name (fn ss => tac (Simplifier.the_context ss) (Simplifier.prems_of_ss ss)));
```
```   538
```
```   539 in
```
```   540   add_simprocs [
```
```   541        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
```
```   542        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
```
```   543      ]
```
```   544   #> add_solver "Transitivity" Orders.order_tac
```
```   545   (* Adding the transitivity reasoners also as safe solvers showed a slight
```
```   546      speed up, but the reasoning strength appears to be not higher (at least
```
```   547      no breaking of additional proofs in the entire HOL distribution, as
```
```   548      of 5 March 2004, was observed). *)
```
```   549 end
```
```   550 *}
```
```   551
```
```   552
```
```   553 subsection {* Bounded quantifiers *}
```
```   554
```
```   555 syntax
```
```   556   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
```
```   557   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
```
```   558   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
```
```   559   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
```
```   560
```
```   561   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
```
```   562   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
```
```   563   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
```
```   564   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
```
```   565
```
```   566 syntax (xsymbols)
```
```   567   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   568   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   569   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   570   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   571
```
```   572   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   573   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   574   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   575   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   576
```
```   577 syntax (HOL)
```
```   578   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
```
```   579   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
```
```   580   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
```
```   581   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
```
```   582
```
```   583 syntax (HTML output)
```
```   584   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   585   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   586   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   587   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   588
```
```   589   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   590   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   591   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   592   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   593
```
```   594 translations
```
```   595   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
```
```   596   "EX x<y. P"    =>  "EX x. x < y \<and> P"
```
```   597   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
```
```   598   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
```
```   599   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
```
```   600   "EX x>y. P"    =>  "EX x. x > y \<and> P"
```
```   601   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
```
```   602   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
```
```   603
```
```   604 print_translation {*
```
```   605 let
```
```   606   val All_binder = Syntax.binder_name @{const_syntax All};
```
```   607   val Ex_binder = Syntax.binder_name @{const_syntax Ex};
```
```   608   val impl = @{const_syntax "op -->"};
```
```   609   val conj = @{const_syntax "op &"};
```
```   610   val less = @{const_syntax less};
```
```   611   val less_eq = @{const_syntax less_eq};
```
```   612
```
```   613   val trans =
```
```   614    [((All_binder, impl, less), ("_All_less", "_All_greater")),
```
```   615     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
```
```   616     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
```
```   617     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
```
```   618
```
```   619   fun matches_bound v t =
```
```   620      case t of (Const ("_bound", _) \$ Free (v', _)) => (v = v')
```
```   621               | _ => false
```
```   622   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
```
```   623   fun mk v c n P = Syntax.const c \$ Syntax.mark_bound v \$ n \$ P
```
```   624
```
```   625   fun tr' q = (q,
```
```   626     fn [Const ("_bound", _) \$ Free (v, _), Const (c, _) \$ (Const (d, _) \$ t \$ u) \$ P] =>
```
```   627       (case AList.lookup (op =) trans (q, c, d) of
```
```   628         NONE => raise Match
```
```   629       | SOME (l, g) =>
```
```   630           if matches_bound v t andalso not (contains_var v u) then mk v l u P
```
```   631           else if matches_bound v u andalso not (contains_var v t) then mk v g t P
```
```   632           else raise Match)
```
```   633      | _ => raise Match);
```
```   634 in [tr' All_binder, tr' Ex_binder] end
```
```   635 *}
```
```   636
```
```   637
```
```   638 subsection {* Transitivity reasoning *}
```
```   639
```
```   640 context ord
```
```   641 begin
```
```   642
```
```   643 lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
```
```   644   by (rule subst)
```
```   645
```
```   646 lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
```
```   647   by (rule ssubst)
```
```   648
```
```   649 lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
```
```   650   by (rule subst)
```
```   651
```
```   652 lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
```
```   653   by (rule ssubst)
```
```   654
```
```   655 end
```
```   656
```
```   657 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
```
```   658   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   659 proof -
```
```   660   assume r: "!!x y. x < y ==> f x < f y"
```
```   661   assume "a < b" hence "f a < f b" by (rule r)
```
```   662   also assume "f b < c"
```
```   663   finally (less_trans) show ?thesis .
```
```   664 qed
```
```   665
```
```   666 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
```
```   667   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   668 proof -
```
```   669   assume r: "!!x y. x < y ==> f x < f y"
```
```   670   assume "a < f b"
```
```   671   also assume "b < c" hence "f b < f c" by (rule r)
```
```   672   finally (less_trans) show ?thesis .
```
```   673 qed
```
```   674
```
```   675 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
```
```   676   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
```
```   677 proof -
```
```   678   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   679   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   680   also assume "f b < c"
```
```   681   finally (le_less_trans) show ?thesis .
```
```   682 qed
```
```   683
```
```   684 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
```
```   685   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   686 proof -
```
```   687   assume r: "!!x y. x < y ==> f x < f y"
```
```   688   assume "a <= f b"
```
```   689   also assume "b < c" hence "f b < f c" by (rule r)
```
```   690   finally (le_less_trans) show ?thesis .
```
```   691 qed
```
```   692
```
```   693 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
```
```   694   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   695 proof -
```
```   696   assume r: "!!x y. x < y ==> f x < f y"
```
```   697   assume "a < b" hence "f a < f b" by (rule r)
```
```   698   also assume "f b <= c"
```
```   699   finally (less_le_trans) show ?thesis .
```
```   700 qed
```
```   701
```
```   702 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
```
```   703   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
```
```   704 proof -
```
```   705   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   706   assume "a < f b"
```
```   707   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   708   finally (less_le_trans) show ?thesis .
```
```   709 qed
```
```   710
```
```   711 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
```
```   712   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   713 proof -
```
```   714   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   715   assume "a <= f b"
```
```   716   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   717   finally (order_trans) show ?thesis .
```
```   718 qed
```
```   719
```
```   720 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
```
```   721   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   722 proof -
```
```   723   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   724   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   725   also assume "f b <= c"
```
```   726   finally (order_trans) show ?thesis .
```
```   727 qed
```
```   728
```
```   729 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
```
```   730   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   731 proof -
```
```   732   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   733   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   734   also assume "f b = c"
```
```   735   finally (ord_le_eq_trans) show ?thesis .
```
```   736 qed
```
```   737
```
```   738 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
```
```   739   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   740 proof -
```
```   741   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   742   assume "a = f b"
```
```   743   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   744   finally (ord_eq_le_trans) show ?thesis .
```
```   745 qed
```
```   746
```
```   747 lemma ord_less_eq_subst: "a < b ==> f b = c ==>
```
```   748   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   749 proof -
```
```   750   assume r: "!!x y. x < y ==> f x < f y"
```
```   751   assume "a < b" hence "f a < f b" by (rule r)
```
```   752   also assume "f b = c"
```
```   753   finally (ord_less_eq_trans) show ?thesis .
```
```   754 qed
```
```   755
```
```   756 lemma ord_eq_less_subst: "a = f b ==> b < c ==>
```
```   757   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   758 proof -
```
```   759   assume r: "!!x y. x < y ==> f x < f y"
```
```   760   assume "a = f b"
```
```   761   also assume "b < c" hence "f b < f c" by (rule r)
```
```   762   finally (ord_eq_less_trans) show ?thesis .
```
```   763 qed
```
```   764
```
```   765 text {*
```
```   766   Note that this list of rules is in reverse order of priorities.
```
```   767 *}
```
```   768
```
```   769 lemmas [trans] =
```
```   770   order_less_subst2
```
```   771   order_less_subst1
```
```   772   order_le_less_subst2
```
```   773   order_le_less_subst1
```
```   774   order_less_le_subst2
```
```   775   order_less_le_subst1
```
```   776   order_subst2
```
```   777   order_subst1
```
```   778   ord_le_eq_subst
```
```   779   ord_eq_le_subst
```
```   780   ord_less_eq_subst
```
```   781   ord_eq_less_subst
```
```   782   forw_subst
```
```   783   back_subst
```
```   784   rev_mp
```
```   785   mp
```
```   786
```
```   787 lemmas (in order) [trans] =
```
```   788   neq_le_trans
```
```   789   le_neq_trans
```
```   790
```
```   791 lemmas (in preorder) [trans] =
```
```   792   less_trans
```
```   793   less_asym'
```
```   794   le_less_trans
```
```   795   less_le_trans
```
```   796   order_trans
```
```   797
```
```   798 lemmas (in order) [trans] =
```
```   799   antisym
```
```   800
```
```   801 lemmas (in ord) [trans] =
```
```   802   ord_le_eq_trans
```
```   803   ord_eq_le_trans
```
```   804   ord_less_eq_trans
```
```   805   ord_eq_less_trans
```
```   806
```
```   807 lemmas [trans] =
```
```   808   trans
```
```   809
```
```   810 lemmas order_trans_rules =
```
```   811   order_less_subst2
```
```   812   order_less_subst1
```
```   813   order_le_less_subst2
```
```   814   order_le_less_subst1
```
```   815   order_less_le_subst2
```
```   816   order_less_le_subst1
```
```   817   order_subst2
```
```   818   order_subst1
```
```   819   ord_le_eq_subst
```
```   820   ord_eq_le_subst
```
```   821   ord_less_eq_subst
```
```   822   ord_eq_less_subst
```
```   823   forw_subst
```
```   824   back_subst
```
```   825   rev_mp
```
```   826   mp
```
```   827   neq_le_trans
```
```   828   le_neq_trans
```
```   829   less_trans
```
```   830   less_asym'
```
```   831   le_less_trans
```
```   832   less_le_trans
```
```   833   order_trans
```
```   834   antisym
```
```   835   ord_le_eq_trans
```
```   836   ord_eq_le_trans
```
```   837   ord_less_eq_trans
```
```   838   ord_eq_less_trans
```
```   839   trans
```
```   840
```
```   841 text {* These support proving chains of decreasing inequalities
```
```   842     a >= b >= c ... in Isar proofs. *}
```
```   843
```
```   844 lemma xt1:
```
```   845   "a = b ==> b > c ==> a > c"
```
```   846   "a > b ==> b = c ==> a > c"
```
```   847   "a = b ==> b >= c ==> a >= c"
```
```   848   "a >= b ==> b = c ==> a >= c"
```
```   849   "(x::'a::order) >= y ==> y >= x ==> x = y"
```
```   850   "(x::'a::order) >= y ==> y >= z ==> x >= z"
```
```   851   "(x::'a::order) > y ==> y >= z ==> x > z"
```
```   852   "(x::'a::order) >= y ==> y > z ==> x > z"
```
```   853   "(a::'a::order) > b ==> b > a ==> P"
```
```   854   "(x::'a::order) > y ==> y > z ==> x > z"
```
```   855   "(a::'a::order) >= b ==> a ~= b ==> a > b"
```
```   856   "(a::'a::order) ~= b ==> a >= b ==> a > b"
```
```   857   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   858   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   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   by auto
```
```   862
```
```   863 lemma xt2:
```
```   864   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
```
```   865 by (subgoal_tac "f b >= f c", force, force)
```
```   866
```
```   867 lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
```
```   868     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
```
```   869 by (subgoal_tac "f a >= f b", force, force)
```
```   870
```
```   871 lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
```
```   872   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
```
```   873 by (subgoal_tac "f b >= f c", force, force)
```
```   874
```
```   875 lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
```
```   876     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   877 by (subgoal_tac "f a > f b", force, force)
```
```   878
```
```   879 lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
```
```   880     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   881 by (subgoal_tac "f b > f c", force, force)
```
```   882
```
```   883 lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
```
```   884     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
```
```   885 by (subgoal_tac "f a >= f b", force, force)
```
```   886
```
```   887 lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
```
```   888     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   889 by (subgoal_tac "f b > f c", force, force)
```
```   890
```
```   891 lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
```
```   892     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   893 by (subgoal_tac "f a > f b", force, force)
```
```   894
```
```   895 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
```
```   896
```
```   897 (*
```
```   898   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
```
```   899   for the wrong thing in an Isar proof.
```
```   900
```
```   901   The extra transitivity rules can be used as follows:
```
```   902
```
```   903 lemma "(a::'a::order) > z"
```
```   904 proof -
```
```   905   have "a >= b" (is "_ >= ?rhs")
```
```   906     sorry
```
```   907   also have "?rhs >= c" (is "_ >= ?rhs")
```
```   908     sorry
```
```   909   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
```
```   910     sorry
```
```   911   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
```
```   912     sorry
```
```   913   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
```
```   914     sorry
```
```   915   also (xtrans) have "?rhs > z"
```
```   916     sorry
```
```   917   finally (xtrans) show ?thesis .
```
```   918 qed
```
```   919
```
```   920   Alternatively, one can use "declare xtrans [trans]" and then
```
```   921   leave out the "(xtrans)" above.
```
```   922 *)
```
```   923
```
```   924
```
```   925 subsection {* Monotonicity, least value operator and min/max *}
```
```   926
```
```   927 context order
```
```   928 begin
```
```   929
```
```   930 definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
```
```   931   "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
```
```   932
```
```   933 lemma monoI [intro?]:
```
```   934   fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
```
```   935   shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
```
```   936   unfolding mono_def by iprover
```
```   937
```
```   938 lemma monoD [dest?]:
```
```   939   fixes f :: "'a \<Rightarrow> 'b\<Colon>order"
```
```   940   shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```   941   unfolding mono_def by iprover
```
```   942
```
```   943 definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
```
```   944   "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
```
```   945
```
```   946 lemma strict_monoI [intro?]:
```
```   947   assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
```
```   948   shows "strict_mono f"
```
```   949   using assms unfolding strict_mono_def by auto
```
```   950
```
```   951 lemma strict_monoD [dest?]:
```
```   952   "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
```
```   953   unfolding strict_mono_def by auto
```
```   954
```
```   955 lemma strict_mono_mono [dest?]:
```
```   956   assumes "strict_mono f"
```
```   957   shows "mono f"
```
```   958 proof (rule monoI)
```
```   959   fix x y
```
```   960   assume "x \<le> y"
```
```   961   show "f x \<le> f y"
```
```   962   proof (cases "x = y")
```
```   963     case True then show ?thesis by simp
```
```   964   next
```
```   965     case False with `x \<le> y` have "x < y" by simp
```
```   966     with assms strict_monoD have "f x < f y" by auto
```
```   967     then show ?thesis by simp
```
```   968   qed
```
```   969 qed
```
```   970
```
```   971 end
```
```   972
```
```   973 context linorder
```
```   974 begin
```
```   975
```
```   976 lemma strict_mono_eq:
```
```   977   assumes "strict_mono f"
```
```   978   shows "f x = f y \<longleftrightarrow> x = y"
```
```   979 proof
```
```   980   assume "f x = f y"
```
```   981   show "x = y" proof (cases x y rule: linorder_cases)
```
```   982     case less with assms strict_monoD have "f x < f y" by auto
```
```   983     with `f x = f y` show ?thesis by simp
```
```   984   next
```
```   985     case equal then show ?thesis .
```
```   986   next
```
```   987     case greater with assms strict_monoD have "f y < f x" by auto
```
```   988     with `f x = f y` show ?thesis by simp
```
```   989   qed
```
```   990 qed simp
```
```   991
```
```   992 lemma strict_mono_less_eq:
```
```   993   assumes "strict_mono f"
```
```   994   shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
```
```   995 proof
```
```   996   assume "x \<le> y"
```
```   997   with assms strict_mono_mono monoD show "f x \<le> f y" by auto
```
```   998 next
```
```   999   assume "f x \<le> f y"
```
```  1000   show "x \<le> y" proof (rule ccontr)
```
```  1001     assume "\<not> x \<le> y" then have "y < x" by simp
```
```  1002     with assms strict_monoD have "f y < f x" by auto
```
```  1003     with `f x \<le> f y` show False by simp
```
```  1004   qed
```
```  1005 qed
```
```  1006
```
```  1007 lemma strict_mono_less:
```
```  1008   assumes "strict_mono f"
```
```  1009   shows "f x < f y \<longleftrightarrow> x < y"
```
```  1010   using assms
```
```  1011     by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
```
```  1012
```
```  1013 lemma min_of_mono:
```
```  1014   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
```
```  1015   shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
```
```  1016   by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
```
```  1017
```
```  1018 lemma max_of_mono:
```
```  1019   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
```
```  1020   shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
```
```  1021   by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
```
```  1022
```
```  1023 end
```
```  1024
```
```  1025 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
```
```  1026 by (simp add: min_def)
```
```  1027
```
```  1028 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
```
```  1029 by (simp add: max_def)
```
```  1030
```
```  1031 lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
```
```  1032 apply (simp add: min_def)
```
```  1033 apply (blast intro: antisym)
```
```  1034 done
```
```  1035
```
```  1036 lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
```
```  1037 apply (simp add: max_def)
```
```  1038 apply (blast intro: antisym)
```
```  1039 done
```
```  1040
```
```  1041
```
```  1042 subsection {* Top and bottom elements *}
```
```  1043
```
```  1044 class top = preorder +
```
```  1045   fixes top :: 'a
```
```  1046   assumes top_greatest [simp]: "x \<le> top"
```
```  1047
```
```  1048 class bot = preorder +
```
```  1049   fixes bot :: 'a
```
```  1050   assumes bot_least [simp]: "bot \<le> x"
```
```  1051
```
```  1052
```
```  1053 subsection {* Dense orders *}
```
```  1054
```
```  1055 class dense_linear_order = linorder +
```
```  1056   assumes gt_ex: "\<exists>y. x < y"
```
```  1057   and lt_ex: "\<exists>y. y < x"
```
```  1058   and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
```
```  1059
```
```  1060
```
```  1061 subsection {* Wellorders *}
```
```  1062
```
```  1063 class wellorder = linorder +
```
```  1064   assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
```
```  1065 begin
```
```  1066
```
```  1067 lemma wellorder_Least_lemma:
```
```  1068   fixes k :: 'a
```
```  1069   assumes "P k"
```
```  1070   shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
```
```  1071 proof -
```
```  1072   have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
```
```  1073   using assms proof (induct k rule: less_induct)
```
```  1074     case (less x) then have "P x" by simp
```
```  1075     show ?case proof (rule classical)
```
```  1076       assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
```
```  1077       have "\<And>y. P y \<Longrightarrow> x \<le> y"
```
```  1078       proof (rule classical)
```
```  1079         fix y
```
```  1080         assume "P y" and "\<not> x \<le> y"
```
```  1081         with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
```
```  1082           by (auto simp add: not_le)
```
```  1083         with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
```
```  1084           by auto
```
```  1085         then show "x \<le> y" by auto
```
```  1086       qed
```
```  1087       with `P x` have Least: "(LEAST a. P a) = x"
```
```  1088         by (rule Least_equality)
```
```  1089       with `P x` show ?thesis by simp
```
```  1090     qed
```
```  1091   qed
```
```  1092   then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
```
```  1093 qed
```
```  1094
```
```  1095 -- "The following 3 lemmas are due to Brian Huffman"
```
```  1096 lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
```
```  1097   by (erule exE) (erule LeastI)
```
```  1098
```
```  1099 lemma LeastI2:
```
```  1100   "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
```
```  1101   by (blast intro: LeastI)
```
```  1102
```
```  1103 lemma LeastI2_ex:
```
```  1104   "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
```
```  1105   by (blast intro: LeastI_ex)
```
```  1106
```
```  1107 lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
```
```  1108 apply (simp (no_asm_use) add: not_le [symmetric])
```
```  1109 apply (erule contrapos_nn)
```
```  1110 apply (erule Least_le)
```
```  1111 done
```
```  1112
```
```  1113 end
```
```  1114
```
```  1115
```
```  1116 subsection {* Order on bool *}
```
```  1117
```
```  1118 instantiation bool :: "{order, top, bot}"
```
```  1119 begin
```
```  1120
```
```  1121 definition
```
```  1122   le_bool_def [code del]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
```
```  1123
```
```  1124 definition
```
```  1125   less_bool_def [code del]: "(P\<Colon>bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
```
```  1126
```
```  1127 definition
```
```  1128   top_bool_eq: "top = True"
```
```  1129
```
```  1130 definition
```
```  1131   bot_bool_eq: "bot = False"
```
```  1132
```
```  1133 instance proof
```
```  1134 qed (auto simp add: bot_bool_eq top_bool_eq less_bool_def, auto simp add: le_bool_def)
```
```  1135
```
```  1136 end
```
```  1137
```
```  1138 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
```
```  1139   by (simp add: le_bool_def)
```
```  1140
```
```  1141 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
```
```  1142   by (simp add: le_bool_def)
```
```  1143
```
```  1144 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```  1145   by (simp add: le_bool_def)
```
```  1146
```
```  1147 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
```
```  1148   by (simp add: le_bool_def)
```
```  1149
```
```  1150 lemma bot_boolE: "bot \<Longrightarrow> P"
```
```  1151   by (simp add: bot_bool_eq)
```
```  1152
```
```  1153 lemma top_boolI: top
```
```  1154   by (simp add: top_bool_eq)
```
```  1155
```
```  1156 lemma [code]:
```
```  1157   "False \<le> b \<longleftrightarrow> True"
```
```  1158   "True \<le> b \<longleftrightarrow> b"
```
```  1159   "False < b \<longleftrightarrow> b"
```
```  1160   "True < b \<longleftrightarrow> False"
```
```  1161   unfolding le_bool_def less_bool_def by simp_all
```
```  1162
```
```  1163
```
```  1164 subsection {* Order on functions *}
```
```  1165
```
```  1166 instantiation "fun" :: (type, ord) ord
```
```  1167 begin
```
```  1168
```
```  1169 definition
```
```  1170   le_fun_def [code del]: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
```
```  1171
```
```  1172 definition
```
```  1173   less_fun_def [code del]: "(f\<Colon>'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
```
```  1174
```
```  1175 instance ..
```
```  1176
```
```  1177 end
```
```  1178
```
```  1179 instance "fun" :: (type, preorder) preorder proof
```
```  1180 qed (auto simp add: le_fun_def less_fun_def
```
```  1181   intro: order_trans antisym intro!: ext)
```
```  1182
```
```  1183 instance "fun" :: (type, order) order proof
```
```  1184 qed (auto simp add: le_fun_def intro: antisym ext)
```
```  1185
```
```  1186 instantiation "fun" :: (type, top) top
```
```  1187 begin
```
```  1188
```
```  1189 definition
```
```  1190   top_fun_eq: "top = (\<lambda>x. top)"
```
```  1191
```
```  1192 instance proof
```
```  1193 qed (simp add: top_fun_eq le_fun_def)
```
```  1194
```
```  1195 end
```
```  1196
```
```  1197 instantiation "fun" :: (type, bot) bot
```
```  1198 begin
```
```  1199
```
```  1200 definition
```
```  1201   bot_fun_eq: "bot = (\<lambda>x. bot)"
```
```  1202
```
```  1203 instance proof
```
```  1204 qed (simp add: bot_fun_eq le_fun_def)
```
```  1205
```
```  1206 end
```
```  1207
```
```  1208 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
```
```  1209   unfolding le_fun_def by simp
```
```  1210
```
```  1211 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
```
```  1212   unfolding le_fun_def by simp
```
```  1213
```
```  1214 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
```
```  1215   unfolding le_fun_def by simp
```
```  1216
```
```  1217
```
```  1218 subsection {* Name duplicates *}
```
```  1219
```
```  1220 lemmas order_eq_refl = preorder_class.eq_refl
```
```  1221 lemmas order_less_irrefl = preorder_class.less_irrefl
```
```  1222 lemmas order_less_imp_le = preorder_class.less_imp_le
```
```  1223 lemmas order_less_not_sym = preorder_class.less_not_sym
```
```  1224 lemmas order_less_asym = preorder_class.less_asym
```
```  1225 lemmas order_less_trans = preorder_class.less_trans
```
```  1226 lemmas order_le_less_trans = preorder_class.le_less_trans
```
```  1227 lemmas order_less_le_trans = preorder_class.less_le_trans
```
```  1228 lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
```
```  1229 lemmas order_less_imp_triv = preorder_class.less_imp_triv
```
```  1230 lemmas order_less_asym' = preorder_class.less_asym'
```
```  1231
```
```  1232 lemmas order_less_le = order_class.less_le
```
```  1233 lemmas order_le_less = order_class.le_less
```
```  1234 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
```
```  1235 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
```
```  1236 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
```
```  1237 lemmas order_neq_le_trans = order_class.neq_le_trans
```
```  1238 lemmas order_le_neq_trans = order_class.le_neq_trans
```
```  1239 lemmas order_antisym = order_class.antisym
```
```  1240 lemmas order_eq_iff = order_class.eq_iff
```
```  1241 lemmas order_antisym_conv = order_class.antisym_conv
```
```  1242
```
```  1243 lemmas linorder_linear = linorder_class.linear
```
```  1244 lemmas linorder_less_linear = linorder_class.less_linear
```
```  1245 lemmas linorder_le_less_linear = linorder_class.le_less_linear
```
```  1246 lemmas linorder_le_cases = linorder_class.le_cases
```
```  1247 lemmas linorder_not_less = linorder_class.not_less
```
```  1248 lemmas linorder_not_le = linorder_class.not_le
```
```  1249 lemmas linorder_neq_iff = linorder_class.neq_iff
```
```  1250 lemmas linorder_neqE = linorder_class.neqE
```
```  1251 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
```
```  1252 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
```
```  1253 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
```
```  1254
```
```  1255 end
```