src/HOL/HOL.thy
author paulson
Thu Mar 04 12:06:07 2004 +0100 (2004-03-04)
changeset 14430 5cb24165a2e1
parent 14398 c5c47703f763
child 14444 24724afce166
permissions -rw-r--r--
new material from Avigad, and simplified treatment of division by 0
     1 (*  Title:      HOL/HOL.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 *)
     6 
     7 header {* The basis of Higher-Order Logic *}
     8 
     9 theory HOL = CPure
    10 files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
    11 
    12 
    13 subsection {* Primitive logic *}
    14 
    15 subsubsection {* Core syntax *}
    16 
    17 classes type < logic
    18 defaultsort type
    19 
    20 global
    21 
    22 typedecl bool
    23 
    24 arities
    25   bool :: type
    26   fun :: (type, type) type
    27 
    28 judgment
    29   Trueprop      :: "bool => prop"                   ("(_)" 5)
    30 
    31 consts
    32   Not           :: "bool => bool"                   ("~ _" [40] 40)
    33   True          :: bool
    34   False         :: bool
    35   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
    36   arbitrary     :: 'a
    37 
    38   The           :: "('a => bool) => 'a"
    39   All           :: "('a => bool) => bool"           (binder "ALL " 10)
    40   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
    41   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
    42   Let           :: "['a, 'a => 'b] => 'b"
    43 
    44   "="           :: "['a, 'a] => bool"               (infixl 50)
    45   &             :: "[bool, bool] => bool"           (infixr 35)
    46   "|"           :: "[bool, bool] => bool"           (infixr 30)
    47   -->           :: "[bool, bool] => bool"           (infixr 25)
    48 
    49 local
    50 
    51 
    52 subsubsection {* Additional concrete syntax *}
    53 
    54 nonterminals
    55   letbinds  letbind
    56   case_syn  cases_syn
    57 
    58 syntax
    59   "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
    60   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
    61 
    62   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
    63   ""            :: "letbind => letbinds"                 ("_")
    64   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
    65   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
    66 
    67   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
    68   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
    69   ""            :: "case_syn => cases_syn"               ("_")
    70   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
    71 
    72 translations
    73   "x ~= y"                == "~ (x = y)"
    74   "THE x. P"              == "The (%x. P)"
    75   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
    76   "let x = a in e"        == "Let a (%x. e)"
    77 
    78 print_translation {*
    79 (* To avoid eta-contraction of body: *)
    80 [("The", fn [Abs abs] =>
    81      let val (x,t) = atomic_abs_tr' abs
    82      in Syntax.const "_The" $ x $ t end)]
    83 *}
    84 
    85 syntax (output)
    86   "="           :: "['a, 'a] => bool"                    (infix 50)
    87   "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
    88 
    89 syntax (xsymbols)
    90   Not           :: "bool => bool"                        ("\<not> _" [40] 40)
    91   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
    92   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
    93   "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
    94   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
    95   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
    96   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
    97   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
    98   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
    99 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
   100 
   101 syntax (xsymbols output)
   102   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
   103 
   104 syntax (HTML output)
   105   Not           :: "bool => bool"                        ("\<not> _" [40] 40)
   106 
   107 syntax (HOL)
   108   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
   109   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
   110   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
   111 
   112 
   113 subsubsection {* Axioms and basic definitions *}
   114 
   115 axioms
   116   eq_reflection: "(x=y) ==> (x==y)"
   117 
   118   refl:         "t = (t::'a)"
   119   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
   120 
   121   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
   122     -- {* Extensionality is built into the meta-logic, and this rule expresses *}
   123     -- {* a related property.  It is an eta-expanded version of the traditional *}
   124     -- {* rule, and similar to the ABS rule of HOL *}
   125 
   126   the_eq_trivial: "(THE x. x = a) = (a::'a)"
   127 
   128   impI:         "(P ==> Q) ==> P-->Q"
   129   mp:           "[| P-->Q;  P |] ==> Q"
   130 
   131 defs
   132   True_def:     "True      == ((%x::bool. x) = (%x. x))"
   133   All_def:      "All(P)    == (P = (%x. True))"
   134   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
   135   False_def:    "False     == (!P. P)"
   136   not_def:      "~ P       == P-->False"
   137   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
   138   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
   139   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
   140 
   141 axioms
   142   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
   143   True_or_False:  "(P=True) | (P=False)"
   144 
   145 defs
   146   Let_def:      "Let s f == f(s)"
   147   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
   148 
   149 finalconsts
   150   "op ="
   151   "op -->"
   152   The
   153   arbitrary
   154 
   155 subsubsection {* Generic algebraic operations *}
   156 
   157 axclass zero < type
   158 axclass one < type
   159 axclass plus < type
   160 axclass minus < type
   161 axclass times < type
   162 axclass inverse < type
   163 
   164 global
   165 
   166 consts
   167   "0"           :: "'a::zero"                       ("0")
   168   "1"           :: "'a::one"                        ("1")
   169   "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
   170   -             :: "['a::minus, 'a] => 'a"          (infixl 65)
   171   uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
   172   *             :: "['a::times, 'a] => 'a"          (infixl 70)
   173 
   174 syntax
   175   "_index1"  :: index    ("\<^sub>1")
   176 translations
   177   (index) "\<^sub>1" == "_index 1"
   178 
   179 local
   180 
   181 typed_print_translation {*
   182   let
   183     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
   184       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
   185       else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
   186   in [tr' "0", tr' "1"] end;
   187 *} -- {* show types that are presumably too general *}
   188 
   189 
   190 consts
   191   abs           :: "'a::minus => 'a"
   192   inverse       :: "'a::inverse => 'a"
   193   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
   194 
   195 syntax (xsymbols)
   196   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
   197 syntax (HTML output)
   198   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
   199 
   200 
   201 subsection {* Theory and package setup *}
   202 
   203 subsubsection {* Basic lemmas *}
   204 
   205 use "HOL_lemmas.ML"
   206 theorems case_split = case_split_thm [case_names True False]
   207 
   208 
   209 subsubsection {* Intuitionistic Reasoning *}
   210 
   211 lemma impE':
   212   assumes 1: "P --> Q"
   213     and 2: "Q ==> R"
   214     and 3: "P --> Q ==> P"
   215   shows R
   216 proof -
   217   from 3 and 1 have P .
   218   with 1 have Q by (rule impE)
   219   with 2 show R .
   220 qed
   221 
   222 lemma allE':
   223   assumes 1: "ALL x. P x"
   224     and 2: "P x ==> ALL x. P x ==> Q"
   225   shows Q
   226 proof -
   227   from 1 have "P x" by (rule spec)
   228   from this and 1 show Q by (rule 2)
   229 qed
   230 
   231 lemma notE':
   232   assumes 1: "~ P"
   233     and 2: "~ P ==> P"
   234   shows R
   235 proof -
   236   from 2 and 1 have P .
   237   with 1 show R by (rule notE)
   238 qed
   239 
   240 lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
   241   and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
   242   and [CPure.elim 2] = allE notE' impE'
   243   and [CPure.intro] = exI disjI2 disjI1
   244 
   245 lemmas [trans] = trans
   246   and [sym] = sym not_sym
   247   and [CPure.elim?] = iffD1 iffD2 impE
   248 
   249 
   250 subsubsection {* Atomizing meta-level connectives *}
   251 
   252 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
   253 proof
   254   assume "!!x. P x"
   255   show "ALL x. P x" by (rule allI)
   256 next
   257   assume "ALL x. P x"
   258   thus "!!x. P x" by (rule allE)
   259 qed
   260 
   261 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
   262 proof
   263   assume r: "A ==> B"
   264   show "A --> B" by (rule impI) (rule r)
   265 next
   266   assume "A --> B" and A
   267   thus B by (rule mp)
   268 qed
   269 
   270 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
   271 proof
   272   assume "x == y"
   273   show "x = y" by (unfold prems) (rule refl)
   274 next
   275   assume "x = y"
   276   thus "x == y" by (rule eq_reflection)
   277 qed
   278 
   279 lemma atomize_conj [atomize]:
   280   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
   281 proof
   282   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
   283   show "A & B" by (rule conjI)
   284 next
   285   fix C
   286   assume "A & B"
   287   assume "A ==> B ==> PROP C"
   288   thus "PROP C"
   289   proof this
   290     show A by (rule conjunct1)
   291     show B by (rule conjunct2)
   292   qed
   293 qed
   294 
   295 lemmas [symmetric, rulify] = atomize_all atomize_imp
   296 
   297 
   298 subsubsection {* Classical Reasoner setup *}
   299 
   300 use "cladata.ML"
   301 setup hypsubst_setup
   302 
   303 ML_setup {*
   304   Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
   305 *}
   306 
   307 setup Classical.setup
   308 setup clasetup
   309 
   310 lemmas [intro?] = ext
   311   and [elim?] = ex1_implies_ex
   312 
   313 use "blastdata.ML"
   314 setup Blast.setup
   315 
   316 
   317 subsubsection {* Simplifier setup *}
   318 
   319 lemma meta_eq_to_obj_eq: "x == y ==> x = y"
   320 proof -
   321   assume r: "x == y"
   322   show "x = y" by (unfold r) (rule refl)
   323 qed
   324 
   325 lemma eta_contract_eq: "(%s. f s) = f" ..
   326 
   327 lemma simp_thms:
   328   shows not_not: "(~ ~ P) = P"
   329   and
   330     "(P ~= Q) = (P = (~Q))"
   331     "(P | ~P) = True"    "(~P | P) = True"
   332     "((~P) = (~Q)) = (P=Q)"
   333     "(x = x) = True"
   334     "(~True) = False"  "(~False) = True"
   335     "(~P) ~= P"  "P ~= (~P)"
   336     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
   337     "(True --> P) = P"  "(False --> P) = True"
   338     "(P --> True) = True"  "(P --> P) = True"
   339     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
   340     "(P & True) = P"  "(True & P) = P"
   341     "(P & False) = False"  "(False & P) = False"
   342     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
   343     "(P & ~P) = False"    "(~P & P) = False"
   344     "(P | True) = True"  "(True | P) = True"
   345     "(P | False) = P"  "(False | P) = P"
   346     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
   347     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
   348     -- {* needed for the one-point-rule quantifier simplification procs *}
   349     -- {* essential for termination!! *} and
   350     "!!P. (EX x. x=t & P(x)) = P(t)"
   351     "!!P. (EX x. t=x & P(x)) = P(t)"
   352     "!!P. (ALL x. x=t --> P(x)) = P(t)"
   353     "!!P. (ALL x. t=x --> P(x)) = P(t)"
   354   by (blast, blast, blast, blast, blast, rules+)
   355 
   356 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
   357   by rules
   358 
   359 lemma ex_simps:
   360   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
   361   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
   362   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
   363   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
   364   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
   365   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
   366   -- {* Miniscoping: pushing in existential quantifiers. *}
   367   by (rules | blast)+
   368 
   369 lemma all_simps:
   370   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
   371   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
   372   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
   373   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
   374   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
   375   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
   376   -- {* Miniscoping: pushing in universal quantifiers. *}
   377   by (rules | blast)+
   378 
   379 lemma disj_absorb: "(A | A) = A"
   380   by blast
   381 
   382 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
   383   by blast
   384 
   385 lemma conj_absorb: "(A & A) = A"
   386   by blast
   387 
   388 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
   389   by blast
   390 
   391 lemma eq_ac:
   392   shows eq_commute: "(a=b) = (b=a)"
   393     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
   394     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
   395 lemma neq_commute: "(a~=b) = (b~=a)" by rules
   396 
   397 lemma conj_comms:
   398   shows conj_commute: "(P&Q) = (Q&P)"
   399     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
   400 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
   401 
   402 lemma disj_comms:
   403   shows disj_commute: "(P|Q) = (Q|P)"
   404     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
   405 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
   406 
   407 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
   408 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
   409 
   410 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
   411 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
   412 
   413 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
   414 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
   415 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
   416 
   417 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
   418 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
   419 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
   420 
   421 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
   422 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
   423 
   424 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
   425 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
   426 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
   427 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
   428 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
   429 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
   430   by blast
   431 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
   432 
   433 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
   434 
   435 
   436 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
   437   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
   438   -- {* cases boil down to the same thing. *}
   439   by blast
   440 
   441 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
   442 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
   443 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
   444 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
   445 
   446 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
   447 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
   448 
   449 text {*
   450   \medskip The @{text "&"} congruence rule: not included by default!
   451   May slow rewrite proofs down by as much as 50\% *}
   452 
   453 lemma conj_cong:
   454     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
   455   by rules
   456 
   457 lemma rev_conj_cong:
   458     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
   459   by rules
   460 
   461 text {* The @{text "|"} congruence rule: not included by default! *}
   462 
   463 lemma disj_cong:
   464     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
   465   by blast
   466 
   467 lemma eq_sym_conv: "(x = y) = (y = x)"
   468   by rules
   469 
   470 
   471 text {* \medskip if-then-else rules *}
   472 
   473 lemma if_True: "(if True then x else y) = x"
   474   by (unfold if_def) blast
   475 
   476 lemma if_False: "(if False then x else y) = y"
   477   by (unfold if_def) blast
   478 
   479 lemma if_P: "P ==> (if P then x else y) = x"
   480   by (unfold if_def) blast
   481 
   482 lemma if_not_P: "~P ==> (if P then x else y) = y"
   483   by (unfold if_def) blast
   484 
   485 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
   486   apply (rule case_split [of Q])
   487    apply (subst if_P)
   488     prefer 3 apply (subst if_not_P, blast+)
   489   done
   490 
   491 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
   492 by (subst split_if, blast)
   493 
   494 lemmas if_splits = split_if split_if_asm
   495 
   496 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
   497   by (rule split_if)
   498 
   499 lemma if_cancel: "(if c then x else x) = x"
   500 by (subst split_if, blast)
   501 
   502 lemma if_eq_cancel: "(if x = y then y else x) = x"
   503 by (subst split_if, blast)
   504 
   505 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
   506   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
   507   by (rule split_if)
   508 
   509 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
   510   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
   511   apply (subst split_if, blast)
   512   done
   513 
   514 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
   515 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
   516 
   517 subsubsection {* Actual Installation of the Simplifier *}
   518 
   519 use "simpdata.ML"
   520 setup Simplifier.setup
   521 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
   522 setup Splitter.setup setup Clasimp.setup
   523 
   524 declare disj_absorb [simp] conj_absorb [simp] 
   525 
   526 lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
   527 by blast+
   528 
   529 theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
   530   apply (rule iffI)
   531   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
   532   apply (fast dest!: theI')
   533   apply (fast intro: ext the1_equality [symmetric])
   534   apply (erule ex1E)
   535   apply (rule allI)
   536   apply (rule ex1I)
   537   apply (erule spec)
   538   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
   539   apply (erule impE)
   540   apply (rule allI)
   541   apply (rule_tac P = "xa = x" in case_split_thm)
   542   apply (drule_tac [3] x = x in fun_cong, simp_all)
   543   done
   544 
   545 text{*Needs only HOL-lemmas:*}
   546 lemma mk_left_commute:
   547   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
   548           c: "\<And>x y. f x y = f y x"
   549   shows "f x (f y z) = f y (f x z)"
   550 by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
   551 
   552 
   553 subsubsection {* Generic cases and induction *}
   554 
   555 constdefs
   556   induct_forall :: "('a => bool) => bool"
   557   "induct_forall P == \<forall>x. P x"
   558   induct_implies :: "bool => bool => bool"
   559   "induct_implies A B == A --> B"
   560   induct_equal :: "'a => 'a => bool"
   561   "induct_equal x y == x = y"
   562   induct_conj :: "bool => bool => bool"
   563   "induct_conj A B == A & B"
   564 
   565 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
   566   by (simp only: atomize_all induct_forall_def)
   567 
   568 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
   569   by (simp only: atomize_imp induct_implies_def)
   570 
   571 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
   572   by (simp only: atomize_eq induct_equal_def)
   573 
   574 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
   575     induct_conj (induct_forall A) (induct_forall B)"
   576   by (unfold induct_forall_def induct_conj_def) rules
   577 
   578 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
   579     induct_conj (induct_implies C A) (induct_implies C B)"
   580   by (unfold induct_implies_def induct_conj_def) rules
   581 
   582 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
   583 proof
   584   assume r: "induct_conj A B ==> PROP C" and A B
   585   show "PROP C" by (rule r) (simp! add: induct_conj_def)
   586 next
   587   assume r: "A ==> B ==> PROP C" and "induct_conj A B"
   588   show "PROP C" by (rule r) (simp! add: induct_conj_def)+
   589 qed
   590 
   591 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
   592   by (simp add: induct_implies_def)
   593 
   594 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
   595 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
   596 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
   597 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
   598 
   599 hide const induct_forall induct_implies induct_equal induct_conj
   600 
   601 
   602 text {* Method setup. *}
   603 
   604 ML {*
   605   structure InductMethod = InductMethodFun
   606   (struct
   607     val dest_concls = HOLogic.dest_concls;
   608     val cases_default = thm "case_split";
   609     val local_impI = thm "induct_impliesI";
   610     val conjI = thm "conjI";
   611     val atomize = thms "induct_atomize";
   612     val rulify1 = thms "induct_rulify1";
   613     val rulify2 = thms "induct_rulify2";
   614     val localize = [Thm.symmetric (thm "induct_implies_def")];
   615   end);
   616 *}
   617 
   618 setup InductMethod.setup
   619 
   620 
   621 subsection {* Order signatures and orders *}
   622 
   623 axclass
   624   ord < type
   625 
   626 syntax
   627   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
   628   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
   629 
   630 global
   631 
   632 consts
   633   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
   634   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
   635 
   636 local
   637 
   638 syntax (xsymbols)
   639   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
   640   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
   641 
   642 
   643 lemma Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
   644 by blast
   645 
   646 subsubsection {* Monotonicity *}
   647 
   648 locale mono =
   649   fixes f
   650   assumes mono: "A <= B ==> f A <= f B"
   651 
   652 lemmas monoI [intro?] = mono.intro
   653   and monoD [dest?] = mono.mono
   654 
   655 constdefs
   656   min :: "['a::ord, 'a] => 'a"
   657   "min a b == (if a <= b then a else b)"
   658   max :: "['a::ord, 'a] => 'a"
   659   "max a b == (if a <= b then b else a)"
   660 
   661 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
   662   by (simp add: min_def)
   663 
   664 lemma min_of_mono:
   665     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
   666   by (simp add: min_def)
   667 
   668 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
   669   by (simp add: max_def)
   670 
   671 lemma max_of_mono:
   672     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
   673   by (simp add: max_def)
   674 
   675 
   676 subsubsection "Orders"
   677 
   678 axclass order < ord
   679   order_refl [iff]: "x <= x"
   680   order_trans: "x <= y ==> y <= z ==> x <= z"
   681   order_antisym: "x <= y ==> y <= x ==> x = y"
   682   order_less_le: "(x < y) = (x <= y & x ~= y)"
   683 
   684 
   685 text {* Reflexivity. *}
   686 
   687 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
   688     -- {* This form is useful with the classical reasoner. *}
   689   apply (erule ssubst)
   690   apply (rule order_refl)
   691   done
   692 
   693 lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
   694   by (simp add: order_less_le)
   695 
   696 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
   697     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
   698   apply (simp add: order_less_le, blast)
   699   done
   700 
   701 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
   702 
   703 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
   704   by (simp add: order_less_le)
   705 
   706 
   707 text {* Asymmetry. *}
   708 
   709 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
   710   by (simp add: order_less_le order_antisym)
   711 
   712 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
   713   apply (drule order_less_not_sym)
   714   apply (erule contrapos_np, simp)
   715   done
   716 
   717 lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"  
   718 by (blast intro: order_antisym)
   719 
   720 
   721 text {* Transitivity. *}
   722 
   723 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
   724   apply (simp add: order_less_le)
   725   apply (blast intro: order_trans order_antisym)
   726   done
   727 
   728 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
   729   apply (simp add: order_less_le)
   730   apply (blast intro: order_trans order_antisym)
   731   done
   732 
   733 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
   734   apply (simp add: order_less_le)
   735   apply (blast intro: order_trans order_antisym)
   736   done
   737 
   738 
   739 text {* Useful for simplification, but too risky to include by default. *}
   740 
   741 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
   742   by (blast elim: order_less_asym)
   743 
   744 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
   745   by (blast elim: order_less_asym)
   746 
   747 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
   748   by auto
   749 
   750 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
   751   by auto
   752 
   753 
   754 text {* Other operators. *}
   755 
   756 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
   757   apply (simp add: min_def)
   758   apply (blast intro: order_antisym)
   759   done
   760 
   761 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
   762   apply (simp add: max_def)
   763   apply (blast intro: order_antisym)
   764   done
   765 
   766 
   767 subsubsection {* Least value operator *}
   768 
   769 constdefs
   770   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
   771   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
   772     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
   773 
   774 lemma LeastI2:
   775   "[| P (x::'a::order);
   776       !!y. P y ==> x <= y;
   777       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
   778    ==> Q (Least P)"
   779   apply (unfold Least_def)
   780   apply (rule theI2)
   781     apply (blast intro: order_antisym)+
   782   done
   783 
   784 lemma Least_equality:
   785     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
   786   apply (simp add: Least_def)
   787   apply (rule the_equality)
   788   apply (auto intro!: order_antisym)
   789   done
   790 
   791 
   792 subsubsection "Linear / total orders"
   793 
   794 axclass linorder < order
   795   linorder_linear: "x <= y | y <= x"
   796 
   797 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
   798   apply (simp add: order_less_le)
   799   apply (insert linorder_linear, blast)
   800   done
   801 
   802 lemma linorder_le_cases [case_names le ge]:
   803     "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
   804   by (insert linorder_linear, blast)
   805 
   806 lemma linorder_cases [case_names less equal greater]:
   807     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
   808   by (insert linorder_less_linear, blast)
   809 
   810 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
   811   apply (simp add: order_less_le)
   812   apply (insert linorder_linear)
   813   apply (blast intro: order_antisym)
   814   done
   815 
   816 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
   817   apply (simp add: order_less_le)
   818   apply (insert linorder_linear)
   819   apply (blast intro: order_antisym)
   820   done
   821 
   822 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
   823 by (cut_tac x = x and y = y in linorder_less_linear, auto)
   824 
   825 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
   826 by (simp add: linorder_neq_iff, blast)
   827 
   828 
   829 subsubsection "Min and max on (linear) orders"
   830 
   831 lemma min_same [simp]: "min (x::'a::order) x = x"
   832   by (simp add: min_def)
   833 
   834 lemma max_same [simp]: "max (x::'a::order) x = x"
   835   by (simp add: max_def)
   836 
   837 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
   838   apply (simp add: max_def)
   839   apply (insert linorder_linear)
   840   apply (blast intro: order_trans)
   841   done
   842 
   843 lemma le_maxI1: "(x::'a::linorder) <= max x y"
   844   by (simp add: le_max_iff_disj)
   845 
   846 lemma le_maxI2: "(y::'a::linorder) <= max x y"
   847     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
   848   by (simp add: le_max_iff_disj)
   849 
   850 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
   851   apply (simp add: max_def order_le_less)
   852   apply (insert linorder_less_linear)
   853   apply (blast intro: order_less_trans)
   854   done
   855 
   856 lemma max_le_iff_conj [simp]:
   857     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
   858   apply (simp add: max_def)
   859   apply (insert linorder_linear)
   860   apply (blast intro: order_trans)
   861   done
   862 
   863 lemma max_less_iff_conj [simp]:
   864     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
   865   apply (simp add: order_le_less max_def)
   866   apply (insert linorder_less_linear)
   867   apply (blast intro: order_less_trans)
   868   done
   869 
   870 lemma le_min_iff_conj [simp]:
   871     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
   872     -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
   873   apply (simp add: min_def)
   874   apply (insert linorder_linear)
   875   apply (blast intro: order_trans)
   876   done
   877 
   878 lemma min_less_iff_conj [simp]:
   879     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
   880   apply (simp add: order_le_less min_def)
   881   apply (insert linorder_less_linear)
   882   apply (blast intro: order_less_trans)
   883   done
   884 
   885 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
   886   apply (simp add: min_def)
   887   apply (insert linorder_linear)
   888   apply (blast intro: order_trans)
   889   done
   890 
   891 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
   892   apply (simp add: min_def order_le_less)
   893   apply (insert linorder_less_linear)
   894   apply (blast intro: order_less_trans)
   895   done
   896 
   897 lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
   898 apply(simp add:max_def)
   899 apply(rule conjI)
   900 apply(blast intro:order_trans)
   901 apply(simp add:linorder_not_le)
   902 apply(blast dest: order_less_trans order_le_less_trans)
   903 done
   904 
   905 lemma max_commute: "!!x::'a::linorder. max x y = max y x"
   906 apply(simp add:max_def)
   907 apply(rule conjI)
   908 apply(blast intro:order_antisym)
   909 apply(simp add:linorder_not_le)
   910 apply(blast dest: order_less_trans)
   911 done
   912 
   913 lemmas max_ac = max_assoc max_commute
   914                 mk_left_commute[of max,OF max_assoc max_commute]
   915 
   916 lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
   917 apply(simp add:min_def)
   918 apply(rule conjI)
   919 apply(blast intro:order_trans)
   920 apply(simp add:linorder_not_le)
   921 apply(blast dest: order_less_trans order_le_less_trans)
   922 done
   923 
   924 lemma min_commute: "!!x::'a::linorder. min x y = min y x"
   925 apply(simp add:min_def)
   926 apply(rule conjI)
   927 apply(blast intro:order_antisym)
   928 apply(simp add:linorder_not_le)
   929 apply(blast dest: order_less_trans)
   930 done
   931 
   932 lemmas min_ac = min_assoc min_commute
   933                 mk_left_commute[of min,OF min_assoc min_commute]
   934 
   935 lemma split_min:
   936     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
   937   by (simp add: min_def)
   938 
   939 lemma split_max:
   940     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
   941   by (simp add: max_def)
   942 
   943 
   944 subsubsection {* Transitivity rules for calculational reasoning *}
   945 
   946 
   947 lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
   948   by (simp add: order_less_le)
   949 
   950 lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
   951   by (simp add: order_less_le)
   952 
   953 lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
   954   by (rule order_less_asym)
   955 
   956 
   957 subsubsection {* Setup of transitivity reasoner *}
   958 
   959 lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
   960   by (erule contrapos_pn, erule subst, rule order_less_irrefl)
   961 
   962 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
   963   by (erule subst, erule ssubst, assumption)
   964 
   965 ML_setup {*
   966 
   967 structure Trans_Tac = Trans_Tac_Fun (
   968   struct
   969     val less_reflE = thm "order_less_irrefl" RS thm "notE";
   970     val le_refl = thm "order_refl";
   971     val less_imp_le = thm "order_less_imp_le";
   972     val not_lessI = thm "linorder_not_less" RS thm "iffD2";
   973     val not_leI = thm "linorder_not_le" RS thm "iffD2";
   974     val not_lessD = thm "linorder_not_less" RS thm "iffD1";
   975     val not_leD = thm "linorder_not_le" RS thm "iffD1";
   976     val eqI = thm "order_antisym";
   977     val eqD1 = thm "order_eq_refl";
   978     val eqD2 = thm "sym" RS thm "order_eq_refl";
   979     val less_trans = thm "order_less_trans";
   980     val less_le_trans = thm "order_less_le_trans";
   981     val le_less_trans = thm "order_le_less_trans";
   982     val le_trans = thm "order_trans";
   983     val le_neq_trans = thm "order_le_neq_trans";
   984     val neq_le_trans = thm "order_neq_le_trans";
   985     val less_imp_neq = thm "less_imp_neq";
   986     val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
   987 
   988     fun decomp_gen sort sign (Trueprop $ t) =
   989       let fun of_sort t = Sign.of_sort sign (type_of t, sort)
   990       fun dec (Const ("Not", _) $ t) = (
   991               case dec t of
   992                 None => None
   993               | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
   994             | dec (Const ("op =",  _) $ t1 $ t2) = 
   995                 if of_sort t1
   996                 then Some (t1, "=", t2)
   997                 else None
   998             | dec (Const ("op <=",  _) $ t1 $ t2) = 
   999                 if of_sort t1
  1000                 then Some (t1, "<=", t2)
  1001                 else None
  1002             | dec (Const ("op <",  _) $ t1 $ t2) = 
  1003                 if of_sort t1
  1004                 then Some (t1, "<", t2)
  1005                 else None
  1006             | dec _ = None
  1007       in dec t end;
  1008 
  1009     val decomp_part = decomp_gen ["HOL.order"];
  1010     val decomp_lin = decomp_gen ["HOL.linorder"];
  1011 
  1012   end);  (* struct *)
  1013 
  1014 Context.>> (fn thy => (simpset_ref_of thy :=
  1015   simpset_of thy
  1016     addSolver (mk_solver "Trans_linear" (fn _ => Trans_Tac.linear_tac))
  1017     addSolver (mk_solver "Trans_partial" (fn _ => Trans_Tac.partial_tac));
  1018   thy))
  1019 *}
  1020 
  1021 (* Optional methods
  1022 
  1023 method_setup trans_partial =
  1024   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trans_tac_partial)) *}
  1025   {* simple transitivity reasoner *}	    
  1026 method_setup trans_linear =
  1027   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trans_tac_linear)) *}
  1028   {* simple transitivity reasoner *}
  1029 *)
  1030 
  1031 subsubsection "Bounded quantifiers"
  1032 
  1033 syntax
  1034   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
  1035   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
  1036   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
  1037   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
  1038 
  1039 syntax (xsymbols)
  1040   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
  1041   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
  1042   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
  1043   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
  1044 
  1045 syntax (HOL)
  1046   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
  1047   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
  1048   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
  1049   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
  1050 
  1051 translations
  1052  "ALL x<y. P"   =>  "ALL x. x < y --> P"
  1053  "EX x<y. P"    =>  "EX x. x < y  & P"
  1054  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
  1055  "EX x<=y. P"   =>  "EX x. x <= y & P"
  1056 
  1057 print_translation {*
  1058 let
  1059   fun all_tr' [Const ("_bound",_) $ Free (v,_), 
  1060                Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
  1061   (if v=v' then Syntax.const "_lessAll" $ Syntax.mark_bound v' $ n $ P else raise Match)
  1062 
  1063   | all_tr' [Const ("_bound",_) $ Free (v,_), 
  1064                Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
  1065   (if v=v' then Syntax.const "_leAll" $ Syntax.mark_bound v' $ n $ P else raise Match);
  1066 
  1067   fun ex_tr' [Const ("_bound",_) $ Free (v,_), 
  1068                Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
  1069   (if v=v' then Syntax.const "_lessEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
  1070 
  1071   | ex_tr' [Const ("_bound",_) $ Free (v,_), 
  1072                Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
  1073   (if v=v' then Syntax.const "_leEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
  1074 in
  1075 [("ALL ", all_tr'), ("EX ", ex_tr')]
  1076 end
  1077 *}
  1078 
  1079 end