src/HOL/Orderings.thy
 author ballarin Tue Sep 25 12:56:27 2007 +0200 (2007-09-25) changeset 24704 9a95634ab135 parent 24641 448edc627ee4 child 24741 a53f5db5acbb permissions -rw-r--r--
Transitivity reasoner gets additional argument of premises to improve integration with simplifier.
```     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 \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
```
```    18   and order_refl [iff]: "x \<sqsubseteq> x"
```
```    19   and order_trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
```
```    20   assumes antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
```
```    21
```
```    22 begin
```
```    23
```
```    24 text {* Reflexivity. *}
```
```    25
```
```    26 lemma eq_refl: "x = y \<Longrightarrow> x \<^loc>\<le> y"
```
```    27     -- {* This form is useful with the classical reasoner. *}
```
```    28 by (erule ssubst) (rule order_refl)
```
```    29
```
```    30 lemma less_irrefl [iff]: "\<not> x \<^loc>< x"
```
```    31 by (simp add: less_le)
```
```    32
```
```    33 lemma le_less: "x \<^loc>\<le> y \<longleftrightarrow> x \<^loc>< y \<or> x = y"
```
```    34     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
```
```    35 by (simp add: less_le) blast
```
```    36
```
```    37 lemma le_imp_less_or_eq: "x \<^loc>\<le> y \<Longrightarrow> x \<^loc>< y \<or> x = y"
```
```    38 unfolding less_le by blast
```
```    39
```
```    40 lemma less_imp_le: "x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y"
```
```    41 unfolding less_le by blast
```
```    42
```
```    43 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
```
```    44 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    45
```
```    46
```
```    47 text {* Useful for simplification, but too risky to include by default. *}
```
```    48
```
```    49 lemma less_imp_not_eq: "x \<^loc>< y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
```
```    50 by auto
```
```    51
```
```    52 lemma less_imp_not_eq2: "x \<^loc>< y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
```
```    53 by auto
```
```    54
```
```    55
```
```    56 text {* Transitivity rules for calculational reasoning *}
```
```    57
```
```    58 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<^loc>\<le> b \<Longrightarrow> a \<^loc>< b"
```
```    59 by (simp add: less_le)
```
```    60
```
```    61 lemma le_neq_trans: "a \<^loc>\<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<^loc>< b"
```
```    62 by (simp add: less_le)
```
```    63
```
```    64
```
```    65 text {* Asymmetry. *}
```
```    66
```
```    67 lemma less_not_sym: "x \<^loc>< y \<Longrightarrow> \<not> (y \<^loc>< x)"
```
```    68 by (simp add: less_le antisym)
```
```    69
```
```    70 lemma less_asym: "x \<^loc>< y \<Longrightarrow> (\<not> P \<Longrightarrow> y \<^loc>< x) \<Longrightarrow> P"
```
```    71 by (drule less_not_sym, erule contrapos_np) simp
```
```    72
```
```    73 lemma eq_iff: "x = y \<longleftrightarrow> x \<^loc>\<le> y \<and> y \<^loc>\<le> x"
```
```    74 by (blast intro: antisym)
```
```    75
```
```    76 lemma antisym_conv: "y \<^loc>\<le> x \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
```
```    77 by (blast intro: antisym)
```
```    78
```
```    79 lemma less_imp_neq: "x \<^loc>< y \<Longrightarrow> x \<noteq> y"
```
```    80 by (erule contrapos_pn, erule subst, rule less_irrefl)
```
```    81
```
```    82
```
```    83 text {* Transitivity. *}
```
```    84
```
```    85 lemma less_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
```
```    86 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    87
```
```    88 lemma le_less_trans: "x \<^loc>\<le> y \<Longrightarrow> y \<^loc>< z \<Longrightarrow> x \<^loc>< z"
```
```    89 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    90
```
```    91 lemma less_le_trans: "x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<^loc>< z"
```
```    92 by (simp add: less_le) (blast intro: order_trans antisym)
```
```    93
```
```    94
```
```    95 text {* Useful for simplification, but too risky to include by default. *}
```
```    96
```
```    97 lemma less_imp_not_less: "x \<^loc>< y \<Longrightarrow> (\<not> y \<^loc>< x) \<longleftrightarrow> True"
```
```    98 by (blast elim: less_asym)
```
```    99
```
```   100 lemma less_imp_triv: "x \<^loc>< y \<Longrightarrow> (y \<^loc>< x \<longrightarrow> P) \<longleftrightarrow> True"
```
```   101 by (blast elim: less_asym)
```
```   102
```
```   103
```
```   104 text {* Transitivity rules for calculational reasoning *}
```
```   105
```
```   106 lemma less_asym': "a \<^loc>< b \<Longrightarrow> b \<^loc>< a \<Longrightarrow> P"
```
```   107 by (rule less_asym)
```
```   108
```
```   109
```
```   110 text {* Reverse order *}
```
```   111
```
```   112 lemma order_reverse:
```
```   113   "order (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
```
```   114 by unfold_locales
```
```   115    (simp add: less_le, auto intro: antisym order_trans)
```
```   116
```
```   117 end
```
```   118
```
```   119
```
```   120 subsection {* Linear (total) orders *}
```
```   121
```
```   122 class linorder = order +
```
```   123   assumes linear: "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
```
```   124 begin
```
```   125
```
```   126 lemma less_linear: "x \<^loc>< y \<or> x = y \<or> y \<^loc>< x"
```
```   127 unfolding less_le using less_le linear by blast
```
```   128
```
```   129 lemma le_less_linear: "x \<^loc>\<le> y \<or> y \<^loc>< x"
```
```   130 by (simp add: le_less less_linear)
```
```   131
```
```   132 lemma le_cases [case_names le ge]:
```
```   133   "(x \<^loc>\<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>\<le> x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   134 using linear by blast
```
```   135
```
```   136 lemma linorder_cases [case_names less equal greater]:
```
```   137   "(x \<^loc>< y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   138 using less_linear by blast
```
```   139
```
```   140 lemma not_less: "\<not> x \<^loc>< y \<longleftrightarrow> y \<^loc>\<le> x"
```
```   141 apply (simp add: less_le)
```
```   142 using linear apply (blast intro: antisym)
```
```   143 done
```
```   144
```
```   145 lemma not_less_iff_gr_or_eq:
```
```   146  "\<not>(x \<^loc>< y) \<longleftrightarrow> (x \<^loc>> y | x = y)"
```
```   147 apply(simp add:not_less le_less)
```
```   148 apply blast
```
```   149 done
```
```   150
```
```   151 lemma not_le: "\<not> x \<^loc>\<le> y \<longleftrightarrow> y \<^loc>< x"
```
```   152 apply (simp add: less_le)
```
```   153 using linear apply (blast intro: antisym)
```
```   154 done
```
```   155
```
```   156 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x \<^loc>< y \<or> y \<^loc>< x"
```
```   157 by (cut_tac x = x and y = y in less_linear, auto)
```
```   158
```
```   159 lemma neqE: "x \<noteq> y \<Longrightarrow> (x \<^loc>< y \<Longrightarrow> R) \<Longrightarrow> (y \<^loc>< x \<Longrightarrow> R) \<Longrightarrow> R"
```
```   160 by (simp add: neq_iff) blast
```
```   161
```
```   162 lemma antisym_conv1: "\<not> x \<^loc>< y \<Longrightarrow> x \<^loc>\<le> y \<longleftrightarrow> x = y"
```
```   163 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   164
```
```   165 lemma antisym_conv2: "x \<^loc>\<le> y \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
```
```   166 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   167
```
```   168 lemma antisym_conv3: "\<not> y \<^loc>< x \<Longrightarrow> \<not> x \<^loc>< y \<longleftrightarrow> x = y"
```
```   169 by (blast intro: antisym dest: not_less [THEN iffD1])
```
```   170
```
```   171 text{*Replacing the old Nat.leI*}
```
```   172 lemma leI: "\<not> x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x"
```
```   173 unfolding not_less .
```
```   174
```
```   175 lemma leD: "y \<^loc>\<le> x \<Longrightarrow> \<not> x \<^loc>< y"
```
```   176 unfolding not_less .
```
```   177
```
```   178 (*FIXME inappropriate name (or delete altogether)*)
```
```   179 lemma not_leE: "\<not> y \<^loc>\<le> x \<Longrightarrow> x \<^loc>< y"
```
```   180 unfolding not_le .
```
```   181
```
```   182
```
```   183 text {* Reverse order *}
```
```   184
```
```   185 lemma linorder_reverse:
```
```   186   "linorder (\<lambda>x y. y \<^loc>\<le> x) (\<lambda>x y. y \<^loc>< x)"
```
```   187 by unfold_locales
```
```   188   (simp add: less_le, auto intro: antisym order_trans simp add: linear)
```
```   189
```
```   190
```
```   191 text {* min/max *}
```
```   192
```
```   193 text {* for historic reasons, definitions are done in context ord *}
```
```   194
```
```   195 definition (in ord)
```
```   196   min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   197   [code unfold, code inline del]: "min a b = (if a \<^loc>\<le> b then a else b)"
```
```   198
```
```   199 definition (in ord)
```
```   200   max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
```
```   201   [code unfold, code inline del]: "max a b = (if a \<^loc>\<le> b then b else a)"
```
```   202
```
```   203 lemma min_le_iff_disj:
```
```   204   "min x y \<^loc>\<le> z \<longleftrightarrow> x \<^loc>\<le> z \<or> y \<^loc>\<le> z"
```
```   205 unfolding min_def using linear by (auto intro: order_trans)
```
```   206
```
```   207 lemma le_max_iff_disj:
```
```   208   "z \<^loc>\<le> max x y \<longleftrightarrow> z \<^loc>\<le> x \<or> z \<^loc>\<le> y"
```
```   209 unfolding max_def using linear by (auto intro: order_trans)
```
```   210
```
```   211 lemma min_less_iff_disj:
```
```   212   "min x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<or> y \<^loc>< z"
```
```   213 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   214
```
```   215 lemma less_max_iff_disj:
```
```   216   "z \<^loc>< max x y \<longleftrightarrow> z \<^loc>< x \<or> z \<^loc>< y"
```
```   217 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   218
```
```   219 lemma min_less_iff_conj [simp]:
```
```   220   "z \<^loc>< min x y \<longleftrightarrow> z \<^loc>< x \<and> z \<^loc>< y"
```
```   221 unfolding min_def le_less using less_linear by (auto intro: less_trans)
```
```   222
```
```   223 lemma max_less_iff_conj [simp]:
```
```   224   "max x y \<^loc>< z \<longleftrightarrow> x \<^loc>< z \<and> y \<^loc>< z"
```
```   225 unfolding max_def le_less using less_linear by (auto intro: less_trans)
```
```   226
```
```   227 lemma split_min [noatp]:
```
```   228   "P (min i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P i) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P j)"
```
```   229 by (simp add: min_def)
```
```   230
```
```   231 lemma split_max [noatp]:
```
```   232   "P (max i j) \<longleftrightarrow> (i \<^loc>\<le> j \<longrightarrow> P j) \<and> (\<not> i \<^loc>\<le> j \<longrightarrow> P i)"
```
```   233 by (simp add: max_def)
```
```   234
```
```   235 end
```
```   236
```
```   237
```
```   238 subsection {* Reasoning tools setup *}
```
```   239
```
```   240 ML {*
```
```   241
```
```   242 signature ORDERS =
```
```   243 sig
```
```   244   val print_structures: Proof.context -> unit
```
```   245   val setup: theory -> theory
```
```   246   val order_tac: thm list -> Proof.context -> int -> tactic
```
```   247 end;
```
```   248
```
```   249 structure Orders: ORDERS =
```
```   250 struct
```
```   251
```
```   252 (** Theory and context data **)
```
```   253
```
```   254 fun struct_eq ((s1: string, ts1), (s2, ts2)) =
```
```   255   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);
```
```   256
```
```   257 structure Data = GenericDataFun
```
```   258 (
```
```   259   type T = ((string * term list) * Order_Tac.less_arith) list;
```
```   260     (* Order structures:
```
```   261        identifier of the structure, list of operations and record of theorems
```
```   262        needed to set up the transitivity reasoner,
```
```   263        identifier and operations identify the structure uniquely. *)
```
```   264   val empty = [];
```
```   265   val extend = I;
```
```   266   fun merge _ = AList.join struct_eq (K fst);
```
```   267 );
```
```   268
```
```   269 fun print_structures ctxt =
```
```   270   let
```
```   271     val structs = Data.get (Context.Proof ctxt);
```
```   272     fun pretty_term t = Pretty.block
```
```   273       [Pretty.quote (ProofContext.pretty_term ctxt t), Pretty.brk 1,
```
```   274         Pretty.str "::", Pretty.brk 1,
```
```   275         Pretty.quote (ProofContext.pretty_typ ctxt (type_of t))];
```
```   276     fun pretty_struct ((s, ts), _) = Pretty.block
```
```   277       [Pretty.str s, Pretty.str ":", Pretty.brk 1,
```
```   278        Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];
```
```   279   in
```
```   280     Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))
```
```   281   end;
```
```   282
```
```   283
```
```   284 (** Method **)
```
```   285
```
```   286 fun struct_tac ((s, [eq, le, less]), thms) prems =
```
```   287   let
```
```   288     fun decomp thy (Trueprop \$ t) =
```
```   289       let
```
```   290         fun excluded t =
```
```   291           (* exclude numeric types: linear arithmetic subsumes transitivity *)
```
```   292           let val T = type_of t
```
```   293           in
```
```   294 	    T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT
```
```   295           end;
```
```   296 	fun dec (Const (@{const_name Not}, _) \$ t) = (case dec t
```
```   297 	      of NONE => NONE
```
```   298 	       | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))
```
```   299           | dec (bin_op \$ t1 \$ t2) =
```
```   300               if excluded t1 then NONE
```
```   301               else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)
```
```   302               else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)
```
```   303               else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)
```
```   304               else NONE
```
```   305 	  | dec _ = NONE;
```
```   306       in dec t end;
```
```   307   in
```
```   308     case s of
```
```   309       "order" => Order_Tac.partial_tac decomp thms prems
```
```   310     | "linorder" => Order_Tac.linear_tac decomp thms prems
```
```   311     | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")
```
```   312   end
```
```   313
```
```   314 fun order_tac prems ctxt =
```
```   315   FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt)));
```
```   316
```
```   317
```
```   318 (** Attribute **)
```
```   319
```
```   320 fun add_struct_thm s tag =
```
```   321   Thm.declaration_attribute
```
```   322     (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));
```
```   323 fun del_struct s =
```
```   324   Thm.declaration_attribute
```
```   325     (fn _ => Data.map (AList.delete struct_eq s));
```
```   326
```
```   327 val attribute = Attrib.syntax
```
```   328      (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) ||
```
```   329           Args.del >> K NONE) --| Args.colon (* FIXME ||
```
```   330         Scan.succeed true *) ) -- Scan.lift Args.name --
```
```   331       Scan.repeat Args.term
```
```   332       >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag
```
```   333            | ((NONE, n), ts) => del_struct (n, ts)));
```
```   334
```
```   335
```
```   336 (** Diagnostic command **)
```
```   337
```
```   338 val print = Toplevel.unknown_context o
```
```   339   Toplevel.keep (Toplevel.node_case
```
```   340     (Context.cases (print_structures o ProofContext.init) print_structures)
```
```   341     (print_structures o Proof.context_of));
```
```   342
```
```   343 val printP =
```
```   344   OuterSyntax.improper_command "print_orders"
```
```   345     "print order structures available to transitivity reasoner" OuterKeyword.diag
```
```   346     (Scan.succeed (Toplevel.no_timing o print));
```
```   347
```
```   348
```
```   349 (** Setup **)
```
```   350
```
```   351 val setup = let val _ = OuterSyntax.add_parsers [printP] in
```
```   352     Method.add_methods [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []),
```
```   353       "normalisation of algebraic structure")] #>
```
```   354     Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")]
```
```   355   end;
```
```   356
```
```   357 end;
```
```   358
```
```   359 *}
```
```   360
```
```   361 setup Orders.setup
```
```   362
```
```   363
```
```   364 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}
```
```   365
```
```   366 (* The type constraint on @{term op =} below is necessary since the operation
```
```   367    is not a parameter of the locale. *)
```
```   368 lemmas (in order)
```
```   369   [order add less_reflE: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   370   less_irrefl [THEN notE]
```
```   371 lemmas (in order)
```
```   372   [order add le_refl: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   373   order_refl
```
```   374 lemmas (in order)
```
```   375   [order add less_imp_le: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   376   less_imp_le
```
```   377 lemmas (in order)
```
```   378   [order add eqI: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   379   antisym
```
```   380 lemmas (in order)
```
```   381   [order add eqD1: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   382   eq_refl
```
```   383 lemmas (in order)
```
```   384   [order add eqD2: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   385   sym [THEN eq_refl]
```
```   386 lemmas (in order)
```
```   387   [order add less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   388   less_trans
```
```   389 lemmas (in order)
```
```   390   [order add less_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   391   less_le_trans
```
```   392 lemmas (in order)
```
```   393   [order add le_less_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   394   le_less_trans
```
```   395 lemmas (in order)
```
```   396   [order add le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   397   order_trans
```
```   398 lemmas (in order)
```
```   399   [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   400   le_neq_trans
```
```   401 lemmas (in order)
```
```   402   [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   403   neq_le_trans
```
```   404 lemmas (in order)
```
```   405   [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   406   less_imp_neq
```
```   407 lemmas (in order)
```
```   408   [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   409    eq_neq_eq_imp_neq
```
```   410 lemmas (in order)
```
```   411   [order add not_sym: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   412   not_sym
```
```   413
```
```   414 lemmas (in linorder) [order del: order "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] = _
```
```   415
```
```   416 lemmas (in linorder)
```
```   417   [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   418   less_irrefl [THEN notE]
```
```   419 lemmas (in linorder)
```
```   420   [order add le_refl: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   421   order_refl
```
```   422 lemmas (in linorder)
```
```   423   [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   424   less_imp_le
```
```   425 lemmas (in linorder)
```
```   426   [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   427   not_less [THEN iffD2]
```
```   428 lemmas (in linorder)
```
```   429   [order add not_leI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   430   not_le [THEN iffD2]
```
```   431 lemmas (in linorder)
```
```   432   [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   433   not_less [THEN iffD1]
```
```   434 lemmas (in linorder)
```
```   435   [order add not_leD: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   436   not_le [THEN iffD1]
```
```   437 lemmas (in linorder)
```
```   438   [order add eqI: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   439   antisym
```
```   440 lemmas (in linorder)
```
```   441   [order add eqD1: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   442   eq_refl
```
```   443 lemmas (in linorder)
```
```   444   [order add eqD2: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   445   sym [THEN eq_refl]
```
```   446 lemmas (in linorder)
```
```   447   [order add less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   448   less_trans
```
```   449 lemmas (in linorder)
```
```   450   [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   451   less_le_trans
```
```   452 lemmas (in linorder)
```
```   453   [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   454   le_less_trans
```
```   455 lemmas (in linorder)
```
```   456   [order add le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   457   order_trans
```
```   458 lemmas (in linorder)
```
```   459   [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   460   le_neq_trans
```
```   461 lemmas (in linorder)
```
```   462   [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   463   neq_le_trans
```
```   464 lemmas (in linorder)
```
```   465   [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   466   less_imp_neq
```
```   467 lemmas (in linorder)
```
```   468   [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   469   eq_neq_eq_imp_neq
```
```   470 lemmas (in linorder)
```
```   471   [order add not_sym: linorder "op = :: 'a => 'a => bool" "op \<^loc><=" "op \<^loc><"] =
```
```   472   not_sym
```
```   473
```
```   474
```
```   475 setup {*
```
```   476 let
```
```   477
```
```   478 fun prp t thm = (#prop (rep_thm thm) = t);
```
```   479
```
```   480 fun prove_antisym_le sg ss ((le as Const(_,T)) \$ r \$ s) =
```
```   481   let val prems = prems_of_ss ss;
```
```   482       val less = Const (@{const_name less}, T);
```
```   483       val t = HOLogic.mk_Trueprop(le \$ s \$ r);
```
```   484   in case find_first (prp t) prems of
```
```   485        NONE =>
```
```   486          let val t = HOLogic.mk_Trueprop(HOLogic.Not \$ (less \$ r \$ s))
```
```   487          in case find_first (prp t) prems of
```
```   488               NONE => NONE
```
```   489             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1}))
```
```   490          end
```
```   491      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))
```
```   492   end
```
```   493   handle THM _ => NONE;
```
```   494
```
```   495 fun prove_antisym_less sg ss (NotC \$ ((less as Const(_,T)) \$ r \$ s)) =
```
```   496   let val prems = prems_of_ss ss;
```
```   497       val le = Const (@{const_name less_eq}, T);
```
```   498       val t = HOLogic.mk_Trueprop(le \$ r \$ s);
```
```   499   in case find_first (prp t) prems of
```
```   500        NONE =>
```
```   501          let val t = HOLogic.mk_Trueprop(NotC \$ (less \$ s \$ r))
```
```   502          in case find_first (prp t) prems of
```
```   503               NONE => NONE
```
```   504             | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3}))
```
```   505          end
```
```   506      | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2}))
```
```   507   end
```
```   508   handle THM _ => NONE;
```
```   509
```
```   510 fun add_simprocs procs thy =
```
```   511   (Simplifier.change_simpset_of thy (fn ss => ss
```
```   512     addsimprocs (map (fn (name, raw_ts, proc) =>
```
```   513       Simplifier.simproc thy name raw_ts proc)) procs); thy);
```
```   514 fun add_solver name tac thy =
```
```   515   (Simplifier.change_simpset_of thy (fn ss => ss addSolver
```
```   516     (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy);
```
```   517
```
```   518 in
```
```   519   add_simprocs [
```
```   520        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),
```
```   521        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)
```
```   522      ]
```
```   523   #> add_solver "Transitivity" Orders.order_tac
```
```   524   (* Adding the transitivity reasoners also as safe solvers showed a slight
```
```   525      speed up, but the reasoning strength appears to be not higher (at least
```
```   526      no breaking of additional proofs in the entire HOL distribution, as
```
```   527      of 5 March 2004, was observed). *)
```
```   528 end
```
```   529 *}
```
```   530
```
```   531
```
```   532 subsection {* Dense orders *}
```
```   533
```
```   534 class dense_linear_order = linorder +
```
```   535   assumes gt_ex: "\<exists>y. x \<sqsubset> y"
```
```   536   and lt_ex: "\<exists>y. y \<sqsubset> x"
```
```   537   and dense: "x \<sqsubset> y \<Longrightarrow> (\<exists>z. x \<sqsubset> z \<and> z \<sqsubset> y)"
```
```   538   (*see further theory Dense_Linear_Order*)
```
```   539
```
```   540
```
```   541 lemma interval_empty_iff:
```
```   542   fixes x y z :: "'a\<Colon>dense_linear_order"
```
```   543   shows "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
```
```   544   by (auto dest: dense)
```
```   545
```
```   546 subsection {* Name duplicates *}
```
```   547
```
```   548 lemmas order_less_le = less_le
```
```   549 lemmas order_eq_refl = order_class.eq_refl
```
```   550 lemmas order_less_irrefl = order_class.less_irrefl
```
```   551 lemmas order_le_less = order_class.le_less
```
```   552 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
```
```   553 lemmas order_less_imp_le = order_class.less_imp_le
```
```   554 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
```
```   555 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
```
```   556 lemmas order_neq_le_trans = order_class.neq_le_trans
```
```   557 lemmas order_le_neq_trans = order_class.le_neq_trans
```
```   558
```
```   559 lemmas order_antisym = antisym
```
```   560 lemmas order_less_not_sym = order_class.less_not_sym
```
```   561 lemmas order_less_asym = order_class.less_asym
```
```   562 lemmas order_eq_iff = order_class.eq_iff
```
```   563 lemmas order_antisym_conv = order_class.antisym_conv
```
```   564 lemmas order_less_trans = order_class.less_trans
```
```   565 lemmas order_le_less_trans = order_class.le_less_trans
```
```   566 lemmas order_less_le_trans = order_class.less_le_trans
```
```   567 lemmas order_less_imp_not_less = order_class.less_imp_not_less
```
```   568 lemmas order_less_imp_triv = order_class.less_imp_triv
```
```   569 lemmas order_less_asym' = order_class.less_asym'
```
```   570
```
```   571 lemmas linorder_linear = linear
```
```   572 lemmas linorder_less_linear = linorder_class.less_linear
```
```   573 lemmas linorder_le_less_linear = linorder_class.le_less_linear
```
```   574 lemmas linorder_le_cases = linorder_class.le_cases
```
```   575 lemmas linorder_not_less = linorder_class.not_less
```
```   576 lemmas linorder_not_le = linorder_class.not_le
```
```   577 lemmas linorder_neq_iff = linorder_class.neq_iff
```
```   578 lemmas linorder_neqE = linorder_class.neqE
```
```   579 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1
```
```   580 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2
```
```   581 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3
```
```   582
```
```   583 lemmas min_le_iff_disj = linorder_class.min_le_iff_disj
```
```   584 lemmas le_max_iff_disj = linorder_class.le_max_iff_disj
```
```   585 lemmas min_less_iff_disj = linorder_class.min_less_iff_disj
```
```   586 lemmas less_max_iff_disj = linorder_class.less_max_iff_disj
```
```   587 lemmas min_less_iff_conj [simp] = linorder_class.min_less_iff_conj
```
```   588 lemmas max_less_iff_conj [simp] = linorder_class.max_less_iff_conj
```
```   589 lemmas split_min = linorder_class.split_min
```
```   590 lemmas split_max = linorder_class.split_max
```
```   591
```
```   592
```
```   593 subsection {* Bounded quantifiers *}
```
```   594
```
```   595 syntax
```
```   596   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
```
```   597   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
```
```   598   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
```
```   599   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
```
```   600
```
```   601   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
```
```   602   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
```
```   603   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
```
```   604   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
```
```   605
```
```   606 syntax (xsymbols)
```
```   607   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   608   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   609   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   610   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   611
```
```   612   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   613   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   614   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   615   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   616
```
```   617 syntax (HOL)
```
```   618   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
```
```   619   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
```
```   620   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
```
```   621   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
```
```   622
```
```   623 syntax (HTML output)
```
```   624   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
```
```   625   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
```
```   626   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
```
```   627   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
```
```   628
```
```   629   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
```
```   630   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
```
```   631   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   632   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
```
```   633
```
```   634 translations
```
```   635   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"
```
```   636   "EX x<y. P"    =>  "EX x. x < y \<and> P"
```
```   637   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"
```
```   638   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"
```
```   639   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"
```
```   640   "EX x>y. P"    =>  "EX x. x > y \<and> P"
```
```   641   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"
```
```   642   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"
```
```   643
```
```   644 print_translation {*
```
```   645 let
```
```   646   val All_binder = Syntax.binder_name @{const_syntax All};
```
```   647   val Ex_binder = Syntax.binder_name @{const_syntax Ex};
```
```   648   val impl = @{const_syntax "op -->"};
```
```   649   val conj = @{const_syntax "op &"};
```
```   650   val less = @{const_syntax less};
```
```   651   val less_eq = @{const_syntax less_eq};
```
```   652
```
```   653   val trans =
```
```   654    [((All_binder, impl, less), ("_All_less", "_All_greater")),
```
```   655     ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")),
```
```   656     ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")),
```
```   657     ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))];
```
```   658
```
```   659   fun matches_bound v t =
```
```   660      case t of (Const ("_bound", _) \$ Free (v', _)) => (v = v')
```
```   661               | _ => false
```
```   662   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false)
```
```   663   fun mk v c n P = Syntax.const c \$ Syntax.mark_bound v \$ n \$ P
```
```   664
```
```   665   fun tr' q = (q,
```
```   666     fn [Const ("_bound", _) \$ Free (v, _), Const (c, _) \$ (Const (d, _) \$ t \$ u) \$ P] =>
```
```   667       (case AList.lookup (op =) trans (q, c, d) of
```
```   668         NONE => raise Match
```
```   669       | SOME (l, g) =>
```
```   670           if matches_bound v t andalso not (contains_var v u) then mk v l u P
```
```   671           else if matches_bound v u andalso not (contains_var v t) then mk v g t P
```
```   672           else raise Match)
```
```   673      | _ => raise Match);
```
```   674 in [tr' All_binder, tr' Ex_binder] end
```
```   675 *}
```
```   676
```
```   677
```
```   678 subsection {* Transitivity reasoning *}
```
```   679
```
```   680 lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
```
```   681 by (rule subst)
```
```   682
```
```   683 lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
```
```   684 by (rule ssubst)
```
```   685
```
```   686 lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
```
```   687 by (rule subst)
```
```   688
```
```   689 lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
```
```   690 by (rule ssubst)
```
```   691
```
```   692 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
```
```   693   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   694 proof -
```
```   695   assume r: "!!x y. x < y ==> f x < f y"
```
```   696   assume "a < b" hence "f a < f b" by (rule r)
```
```   697   also assume "f b < c"
```
```   698   finally (order_less_trans) show ?thesis .
```
```   699 qed
```
```   700
```
```   701 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
```
```   702   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   703 proof -
```
```   704   assume r: "!!x y. x < y ==> f x < f y"
```
```   705   assume "a < f b"
```
```   706   also assume "b < c" hence "f b < f c" by (rule r)
```
```   707   finally (order_less_trans) show ?thesis .
```
```   708 qed
```
```   709
```
```   710 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
```
```   711   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
```
```   712 proof -
```
```   713   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   714   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   715   also assume "f b < c"
```
```   716   finally (order_le_less_trans) show ?thesis .
```
```   717 qed
```
```   718
```
```   719 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
```
```   720   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   721 proof -
```
```   722   assume r: "!!x y. x < y ==> f x < f y"
```
```   723   assume "a <= f b"
```
```   724   also assume "b < c" hence "f b < f c" by (rule r)
```
```   725   finally (order_le_less_trans) show ?thesis .
```
```   726 qed
```
```   727
```
```   728 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
```
```   729   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   730 proof -
```
```   731   assume r: "!!x y. x < y ==> f x < f y"
```
```   732   assume "a < b" hence "f a < f b" by (rule r)
```
```   733   also assume "f b <= c"
```
```   734   finally (order_less_le_trans) show ?thesis .
```
```   735 qed
```
```   736
```
```   737 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
```
```   738   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
```
```   739 proof -
```
```   740   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   741   assume "a < f b"
```
```   742   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   743   finally (order_less_le_trans) show ?thesis .
```
```   744 qed
```
```   745
```
```   746 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
```
```   747   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   748 proof -
```
```   749   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   750   assume "a <= f b"
```
```   751   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   752   finally (order_trans) show ?thesis .
```
```   753 qed
```
```   754
```
```   755 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
```
```   756   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   757 proof -
```
```   758   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   759   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   760   also assume "f b <= c"
```
```   761   finally (order_trans) show ?thesis .
```
```   762 qed
```
```   763
```
```   764 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
```
```   765   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
```
```   766 proof -
```
```   767   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   768   assume "a <= b" hence "f a <= f b" by (rule r)
```
```   769   also assume "f b = c"
```
```   770   finally (ord_le_eq_trans) show ?thesis .
```
```   771 qed
```
```   772
```
```   773 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
```
```   774   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
```
```   775 proof -
```
```   776   assume r: "!!x y. x <= y ==> f x <= f y"
```
```   777   assume "a = f b"
```
```   778   also assume "b <= c" hence "f b <= f c" by (rule r)
```
```   779   finally (ord_eq_le_trans) show ?thesis .
```
```   780 qed
```
```   781
```
```   782 lemma ord_less_eq_subst: "a < b ==> f b = c ==>
```
```   783   (!!x y. x < y ==> f x < f y) ==> f a < c"
```
```   784 proof -
```
```   785   assume r: "!!x y. x < y ==> f x < f y"
```
```   786   assume "a < b" hence "f a < f b" by (rule r)
```
```   787   also assume "f b = c"
```
```   788   finally (ord_less_eq_trans) show ?thesis .
```
```   789 qed
```
```   790
```
```   791 lemma ord_eq_less_subst: "a = f b ==> b < c ==>
```
```   792   (!!x y. x < y ==> f x < f y) ==> a < f c"
```
```   793 proof -
```
```   794   assume r: "!!x y. x < y ==> f x < f y"
```
```   795   assume "a = f b"
```
```   796   also assume "b < c" hence "f b < f c" by (rule r)
```
```   797   finally (ord_eq_less_trans) show ?thesis .
```
```   798 qed
```
```   799
```
```   800 text {*
```
```   801   Note that this list of rules is in reverse order of priorities.
```
```   802 *}
```
```   803
```
```   804 lemmas order_trans_rules [trans] =
```
```   805   order_less_subst2
```
```   806   order_less_subst1
```
```   807   order_le_less_subst2
```
```   808   order_le_less_subst1
```
```   809   order_less_le_subst2
```
```   810   order_less_le_subst1
```
```   811   order_subst2
```
```   812   order_subst1
```
```   813   ord_le_eq_subst
```
```   814   ord_eq_le_subst
```
```   815   ord_less_eq_subst
```
```   816   ord_eq_less_subst
```
```   817   forw_subst
```
```   818   back_subst
```
```   819   rev_mp
```
```   820   mp
```
```   821   order_neq_le_trans
```
```   822   order_le_neq_trans
```
```   823   order_less_trans
```
```   824   order_less_asym'
```
```   825   order_le_less_trans
```
```   826   order_less_le_trans
```
```   827   order_trans
```
```   828   order_antisym
```
```   829   ord_le_eq_trans
```
```   830   ord_eq_le_trans
```
```   831   ord_less_eq_trans
```
```   832   ord_eq_less_trans
```
```   833   trans
```
```   834
```
```   835
```
```   836 (* FIXME cleanup *)
```
```   837
```
```   838 text {* These support proving chains of decreasing inequalities
```
```   839     a >= b >= c ... in Isar proofs. *}
```
```   840
```
```   841 lemma xt1:
```
```   842   "a = b ==> b > c ==> a > c"
```
```   843   "a > b ==> b = c ==> a > c"
```
```   844   "a = b ==> b >= c ==> a >= c"
```
```   845   "a >= b ==> b = c ==> a >= c"
```
```   846   "(x::'a::order) >= y ==> y >= x ==> x = y"
```
```   847   "(x::'a::order) >= y ==> y >= z ==> x >= z"
```
```   848   "(x::'a::order) > y ==> y >= z ==> x > z"
```
```   849   "(x::'a::order) >= y ==> y > z ==> x > z"
```
```   850   "(a::'a::order) > b ==> b > a ==> P"
```
```   851   "(x::'a::order) > y ==> y > z ==> x > z"
```
```   852   "(a::'a::order) >= b ==> a ~= b ==> a > b"
```
```   853   "(a::'a::order) ~= b ==> a >= b ==> a > b"
```
```   854   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   855   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   856   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
```
```   857   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c"
```
```   858 by auto
```
```   859
```
```   860 lemma xt2:
```
```   861   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
```
```   862 by (subgoal_tac "f b >= f c", force, force)
```
```   863
```
```   864 lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
```
```   865     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
```
```   866 by (subgoal_tac "f a >= f b", force, force)
```
```   867
```
```   868 lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
```
```   869   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
```
```   870 by (subgoal_tac "f b >= f c", force, force)
```
```   871
```
```   872 lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
```
```   873     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   874 by (subgoal_tac "f a > f b", force, force)
```
```   875
```
```   876 lemma xt6: "(a::'a::order) >= f b ==> b > c ==>
```
```   877     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   878 by (subgoal_tac "f b > f c", force, force)
```
```   879
```
```   880 lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
```
```   881     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
```
```   882 by (subgoal_tac "f a >= f b", force, force)
```
```   883
```
```   884 lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
```
```   885     (!!x y. x > y ==> f x > f y) ==> a > f c"
```
```   886 by (subgoal_tac "f b > f c", force, force)
```
```   887
```
```   888 lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
```
```   889     (!!x y. x > y ==> f x > f y) ==> f a > c"
```
```   890 by (subgoal_tac "f a > f b", force, force)
```
```   891
```
```   892 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
```
```   893
```
```   894 (*
```
```   895   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
```
```   896   for the wrong thing in an Isar proof.
```
```   897
```
```   898   The extra transitivity rules can be used as follows:
```
```   899
```
```   900 lemma "(a::'a::order) > z"
```
```   901 proof -
```
```   902   have "a >= b" (is "_ >= ?rhs")
```
```   903     sorry
```
```   904   also have "?rhs >= c" (is "_ >= ?rhs")
```
```   905     sorry
```
```   906   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
```
```   907     sorry
```
```   908   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
```
```   909     sorry
```
```   910   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
```
```   911     sorry
```
```   912   also (xtrans) have "?rhs > z"
```
```   913     sorry
```
```   914   finally (xtrans) show ?thesis .
```
```   915 qed
```
```   916
```
```   917   Alternatively, one can use "declare xtrans [trans]" and then
```
```   918   leave out the "(xtrans)" above.
```
```   919 *)
```
```   920
```
```   921 subsection {* Order on bool *}
```
```   922
```
```   923 instance bool :: order
```
```   924   le_bool_def: "P \<le> Q \<equiv> P \<longrightarrow> Q"
```
```   925   less_bool_def: "P < Q \<equiv> P \<le> Q \<and> P \<noteq> Q"
```
```   926   by intro_classes (auto simp add: le_bool_def less_bool_def)
```
```   927 lemmas [code func del] = le_bool_def less_bool_def
```
```   928
```
```   929 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
```
```   930 by (simp add: le_bool_def)
```
```   931
```
```   932 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
```
```   933 by (simp add: le_bool_def)
```
```   934
```
```   935 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
```
```   936 by (simp add: le_bool_def)
```
```   937
```
```   938 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
```
```   939 by (simp add: le_bool_def)
```
```   940
```
```   941 lemma [code func]:
```
```   942   "False \<le> b \<longleftrightarrow> True"
```
```   943   "True \<le> b \<longleftrightarrow> b"
```
```   944   "False < b \<longleftrightarrow> b"
```
```   945   "True < b \<longleftrightarrow> False"
```
```   946   unfolding le_bool_def less_bool_def by simp_all
```
```   947
```
```   948
```
```   949 subsection {* Order on sets *}
```
```   950
```
```   951 instance set :: (type) order
```
```   952   by (intro_classes,
```
```   953       (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
```
```   954
```
```   955 lemmas basic_trans_rules [trans] =
```
```   956   order_trans_rules set_rev_mp set_mp
```
```   957
```
```   958
```
```   959 subsection {* Order on functions *}
```
```   960
```
```   961 instance "fun" :: (type, ord) ord
```
```   962   le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
```
```   963   less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
```
```   964
```
```   965 lemmas [code func del] = le_fun_def less_fun_def
```
```   966
```
```   967 instance "fun" :: (type, order) order
```
```   968   by default
```
```   969     (auto simp add: le_fun_def less_fun_def expand_fun_eq
```
```   970        intro: order_trans order_antisym)
```
```   971
```
```   972 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
```
```   973   unfolding le_fun_def by simp
```
```   974
```
```   975 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
```
```   976   unfolding le_fun_def by simp
```
```   977
```
```   978 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
```
```   979   unfolding le_fun_def by simp
```
```   980
```
```   981 text {*
```
```   982   Handy introduction and elimination rules for @{text "\<le>"}
```
```   983   on unary and binary predicates
```
```   984 *}
```
```   985
```
```   986 lemma predicate1I [Pure.intro!, intro!]:
```
```   987   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
```
```   988   shows "P \<le> Q"
```
```   989   apply (rule le_funI)
```
```   990   apply (rule le_boolI)
```
```   991   apply (rule PQ)
```
```   992   apply assumption
```
```   993   done
```
```   994
```
```   995 lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
```
```   996   apply (erule le_funE)
```
```   997   apply (erule le_boolE)
```
```   998   apply assumption+
```
```   999   done
```
```  1000
```
```  1001 lemma predicate2I [Pure.intro!, intro!]:
```
```  1002   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
```
```  1003   shows "P \<le> Q"
```
```  1004   apply (rule le_funI)+
```
```  1005   apply (rule le_boolI)
```
```  1006   apply (rule PQ)
```
```  1007   apply assumption
```
```  1008   done
```
```  1009
```
```  1010 lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
```
```  1011   apply (erule le_funE)+
```
```  1012   apply (erule le_boolE)
```
```  1013   apply assumption+
```
```  1014   done
```
```  1015
```
```  1016 lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
```
```  1017   by (rule predicate1D)
```
```  1018
```
```  1019 lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
```
```  1020   by (rule predicate2D)
```
```  1021
```
```  1022
```
```  1023 subsection {* Monotonicity, least value operator and min/max *}
```
```  1024
```
```  1025 locale mono =
```
```  1026   fixes f
```
```  1027   assumes mono: "A \<le> B \<Longrightarrow> f A \<le> f B"
```
```  1028
```
```  1029 lemmas monoI [intro?] = mono.intro
```
```  1030   and monoD [dest?] = mono.mono
```
```  1031
```
```  1032 lemma LeastI2_order:
```
```  1033   "[| P (x::'a::order);
```
```  1034       !!y. P y ==> x <= y;
```
```  1035       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
```
```  1036    ==> Q (Least P)"
```
```  1037 apply (unfold Least_def)
```
```  1038 apply (rule theI2)
```
```  1039   apply (blast intro: order_antisym)+
```
```  1040 done
```
```  1041
```
```  1042 lemma Least_mono:
```
```  1043   "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
```
```  1044     ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
```
```  1045     -- {* Courtesy of Stephan Merz *}
```
```  1046   apply clarify
```
```  1047   apply (erule_tac P = "%x. x : S" in LeastI2_order, fast)
```
```  1048   apply (rule LeastI2_order)
```
```  1049   apply (auto elim: monoD intro!: order_antisym)
```
```  1050   done
```
```  1051
```
```  1052 lemma Least_equality:
```
```  1053   "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
```
```  1054 apply (simp add: Least_def)
```
```  1055 apply (rule the_equality)
```
```  1056 apply (auto intro!: order_antisym)
```
```  1057 done
```
```  1058
```
```  1059 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
```
```  1060 by (simp add: min_def)
```
```  1061
```
```  1062 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
```
```  1063 by (simp add: max_def)
```
```  1064
```
```  1065 lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
```
```  1066 apply (simp add: min_def)
```
```  1067 apply (blast intro: order_antisym)
```
```  1068 done
```
```  1069
```
```  1070 lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
```
```  1071 apply (simp add: max_def)
```
```  1072 apply (blast intro: order_antisym)
```
```  1073 done
```
```  1074
```
```  1075 lemma min_of_mono:
```
```  1076   "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
```
```  1077 by (simp add: min_def)
```
```  1078
```
```  1079 lemma max_of_mono:
```
```  1080   "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
```
```  1081 by (simp add: max_def)
```
```  1082
```
```  1083
```
```  1084 subsection {* legacy ML bindings *}
```
```  1085
```
```  1086 ML {*
```
```  1087 val monoI = @{thm monoI};
```
```  1088 *}
```
```  1089
```
```  1090 end
```