src/HOL/HOL.thy
 author wenzelm Fri Nov 09 00:09:47 2001 +0100 (2001-11-09) changeset 12114 a8e860c86252 parent 12023 d982f98e0f0d child 12161 ea4fbf26a945 permissions -rw-r--r--
eliminated old "symbols" syntax, use "xsymbols" instead;
1 (*  Title:      HOL/HOL.thy
2     ID:         \$Id\$
3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
4 *)
6 header {* The basis of Higher-Order Logic *}
8 theory HOL = CPure
9 files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
12 subsection {* Primitive logic *}
14 subsubsection {* Core syntax *}
16 global
18 classes "term" < logic
19 defaultsort "term"
21 typedecl bool
23 arities
24   bool :: "term"
25   fun :: ("term", "term") "term"
27 judgment
28   Trueprop      :: "bool => prop"                   ("(_)" 5)
30 consts
31   Not           :: "bool => bool"                   ("~ _"  40)
32   True          :: bool
33   False         :: bool
34   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
35   arbitrary     :: 'a
37   The           :: "('a => bool) => 'a"
38   All           :: "('a => bool) => bool"           (binder "ALL " 10)
39   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
40   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
41   Let           :: "['a, 'a => 'b] => 'b"
43   "="           :: "['a, 'a] => bool"               (infixl 50)
44   &             :: "[bool, bool] => bool"           (infixr 35)
45   "|"           :: "[bool, bool] => bool"           (infixr 30)
46   -->           :: "[bool, bool] => bool"           (infixr 25)
48 local
51 subsubsection {* Additional concrete syntax *}
53 nonterminals
54   letbinds  letbind
55   case_syn  cases_syn
57 syntax
58   ~=            :: "['a, 'a] => bool"                    (infixl 50)
59   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
61   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
62   ""            :: "letbind => letbinds"                 ("_")
63   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
64   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
66   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
67   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
68   ""            :: "case_syn => cases_syn"               ("_")
69   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
71 translations
72   "x ~= y"                == "~ (x = y)"
73   "THE x. P"              == "The (%x. P)"
74   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
75   "let x = a in e"        == "Let a (%x. e)"
77 syntax ("" output)
78   "="           :: "['a, 'a] => bool"                    (infix 50)
79   "~="          :: "['a, 'a] => bool"                    (infix 50)
81 syntax (xsymbols)
82   Not           :: "bool => bool"                        ("\<not> _"  40)
83   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
84   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
85   "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
86   "op ~="       :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
87   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
88   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
89   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
90   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
91 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \\<orelse> _")*)
93 syntax (xsymbols output)
94   "op ~="       :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
96 syntax (HTML output)
97   Not           :: "bool => bool"                        ("\<not> _"  40)
99 syntax (HOL)
100   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
101   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
102   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
105 subsubsection {* Axioms and basic definitions *}
107 axioms
108   eq_reflection: "(x=y) ==> (x==y)"
110   refl:         "t = (t::'a)"
111   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
113   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
114     -- {* Extensionality is built into the meta-logic, and this rule expresses *}
115     -- {* a related property.  It is an eta-expanded version of the traditional *}
116     -- {* rule, and similar to the ABS rule of HOL *}
118   the_eq_trivial: "(THE x. x = a) = (a::'a)"
120   impI:         "(P ==> Q) ==> P-->Q"
121   mp:           "[| P-->Q;  P |] ==> Q"
123 defs
124   True_def:     "True      == ((%x::bool. x) = (%x. x))"
125   All_def:      "All(P)    == (P = (%x. True))"
126   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
127   False_def:    "False     == (!P. P)"
128   not_def:      "~ P       == P-->False"
129   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
130   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
131   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
133 axioms
134   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
135   True_or_False:  "(P=True) | (P=False)"
137 defs
138   Let_def:      "Let s f == f(s)"
139   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
141   arbitrary_def:  "False ==> arbitrary == (THE x. False)"
142     -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
143     definition syntactically *}
146 subsubsection {* Generic algebraic operations *}
148 axclass zero < "term"
149 axclass one < "term"
150 axclass plus < "term"
151 axclass minus < "term"
152 axclass times < "term"
153 axclass inverse < "term"
155 global
157 consts
158   "0"           :: "'a::zero"                       ("0")
159   "1"           :: "'a::one"                        ("1")
160   "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
161   -             :: "['a::minus, 'a] => 'a"          (infixl 65)
162   uminus        :: "['a::minus] => 'a"              ("- _"  80)
163   *             :: "['a::times, 'a] => 'a"          (infixl 70)
165 local
167 typed_print_translation {*
168   let
169     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
170       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
171       else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
172   in [tr' "0", tr' "1"] end;
173 *} -- {* show types that are presumably too general *}
176 consts
177   abs           :: "'a::minus => 'a"
178   inverse       :: "'a::inverse => 'a"
179   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
181 syntax (xsymbols)
182   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
183 syntax (HTML output)
184   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
186 axclass plus_ac0 < plus, zero
187   commute: "x + y = y + x"
188   assoc:   "(x + y) + z = x + (y + z)"
189   zero:    "0 + x = x"
192 subsection {* Theory and package setup *}
194 subsubsection {* Basic lemmas *}
196 use "HOL_lemmas.ML"
197 theorems case_split = case_split_thm [case_names True False]
199 declare trans [trans]
200 declare impE [CPure.elim]  iffD1 [CPure.elim]  iffD2 [CPure.elim]
203 subsubsection {* Atomizing meta-level connectives *}
205 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
206 proof
207   assume "!!x. P x"
208   show "ALL x. P x" by (rule allI)
209 next
210   assume "ALL x. P x"
211   thus "!!x. P x" by (rule allE)
212 qed
214 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
215 proof
216   assume r: "A ==> B"
217   show "A --> B" by (rule impI) (rule r)
218 next
219   assume "A --> B" and A
220   thus B by (rule mp)
221 qed
223 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
224 proof
225   assume "x == y"
226   show "x = y" by (unfold prems) (rule refl)
227 next
228   assume "x = y"
229   thus "x == y" by (rule eq_reflection)
230 qed
232 lemma atomize_conj [atomize]:
233   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
234 proof
235   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
236   show "A & B" by (rule conjI)
237 next
238   fix C
239   assume "A & B"
240   assume "A ==> B ==> PROP C"
241   thus "PROP C"
242   proof this
243     show A by (rule conjunct1)
244     show B by (rule conjunct2)
245   qed
246 qed
249 subsubsection {* Classical Reasoner setup *}
252 setup hypsubst_setup
254 declare atomize_all [symmetric, rulify]  atomize_imp [symmetric, rulify]
256 setup Classical.setup
257 setup clasetup
259 declare ext [intro?]
260 declare disjI1 [elim?]  disjI2 [elim?]  ex1_implies_ex [elim?]  sym [elim?]
262 use "blastdata.ML"
263 setup Blast.setup
266 subsubsection {* Simplifier setup *}
268 use "simpdata.ML"
269 setup Simplifier.setup
270 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
271 setup Splitter.setup setup Clasimp.setup
274 subsubsection {* Generic cases and induction *}
276 constdefs
277   induct_forall :: "('a => bool) => bool"
278   "induct_forall P == \<forall>x. P x"
279   induct_implies :: "bool => bool => bool"
280   "induct_implies A B == A --> B"
281   induct_equal :: "'a => 'a => bool"
282   "induct_equal x y == x = y"
283   induct_conj :: "bool => bool => bool"
284   "induct_conj A B == A & B"
286 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
287   by (simp only: atomize_all induct_forall_def)
289 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
290   by (simp only: atomize_imp induct_implies_def)
292 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
293   by (simp only: atomize_eq induct_equal_def)
295 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
296     induct_conj (induct_forall A) (induct_forall B)"
297   by (unfold induct_forall_def induct_conj_def) blast
299 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
300     induct_conj (induct_implies C A) (induct_implies C B)"
301   by (unfold induct_implies_def induct_conj_def) blast
303 lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
304   by (simp only: atomize_imp atomize_eq induct_conj_def) (rule equal_intr_rule, blast+)
306 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
307   by (simp add: induct_implies_def)
309 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq
310 lemmas induct_rulify1 = induct_atomize [symmetric, standard]
311 lemmas induct_rulify2 =
312   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
313 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
315 hide const induct_forall induct_implies induct_equal induct_conj
318 text {* Method setup. *}
320 ML {*
321   structure InductMethod = InductMethodFun
322   (struct
323     val dest_concls = HOLogic.dest_concls;
324     val cases_default = thm "case_split";
325     val local_impI = thm "induct_impliesI";
326     val conjI = thm "conjI";
327     val atomize = thms "induct_atomize";
328     val rulify1 = thms "induct_rulify1";
329     val rulify2 = thms "induct_rulify2";
330   end);
331 *}
333 setup InductMethod.setup
336 subsection {* Order signatures and orders *}
338 axclass
339   ord < "term"
341 syntax
342   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
343   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
345 global
347 consts
348   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
349   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
351 local
353 syntax (xsymbols)
354   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
355   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
357 (*Tell blast about overloading of < and <= to reduce the risk of
358   its applying a rule for the wrong type*)
359 ML {*
360 Blast.overloaded ("op <" , domain_type);
361 Blast.overloaded ("op <=", domain_type);
362 *}
365 subsubsection {* Monotonicity *}
367 constdefs
368   mono :: "['a::ord => 'b::ord] => bool"
369   "mono f == ALL A B. A <= B --> f A <= f B"
371 lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
372   by (unfold mono_def) blast
374 lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
375   by (unfold mono_def) blast
377 constdefs
378   min :: "['a::ord, 'a] => 'a"
379   "min a b == (if a <= b then a else b)"
380   max :: "['a::ord, 'a] => 'a"
381   "max a b == (if a <= b then b else a)"
383 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
384   by (simp add: min_def)
386 lemma min_of_mono:
387     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
388   by (simp add: min_def)
390 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
391   by (simp add: max_def)
393 lemma max_of_mono:
394     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
395   by (simp add: max_def)
398 subsubsection "Orders"
400 axclass order < ord
401   order_refl [iff]: "x <= x"
402   order_trans: "x <= y ==> y <= z ==> x <= z"
403   order_antisym: "x <= y ==> y <= x ==> x = y"
404   order_less_le: "(x < y) = (x <= y & x ~= y)"
407 text {* Reflexivity. *}
409 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
410     -- {* This form is useful with the classical reasoner. *}
411   apply (erule ssubst)
412   apply (rule order_refl)
413   done
415 lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
416   by (simp add: order_less_le)
418 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
419     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
420   apply (simp add: order_less_le)
421   apply (blast intro!: order_refl)
422   done
424 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
426 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
427   by (simp add: order_less_le)
430 text {* Asymmetry. *}
432 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
433   by (simp add: order_less_le order_antisym)
435 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
436   apply (drule order_less_not_sym)
437   apply (erule contrapos_np)
438   apply simp
439   done
442 text {* Transitivity. *}
444 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
445   apply (simp add: order_less_le)
446   apply (blast intro: order_trans order_antisym)
447   done
449 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
450   apply (simp add: order_less_le)
451   apply (blast intro: order_trans order_antisym)
452   done
454 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
455   apply (simp add: order_less_le)
456   apply (blast intro: order_trans order_antisym)
457   done
460 text {* Useful for simplification, but too risky to include by default. *}
462 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
463   by (blast elim: order_less_asym)
465 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
466   by (blast elim: order_less_asym)
468 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
469   by auto
471 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
472   by auto
475 text {* Other operators. *}
477 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
478   apply (simp add: min_def)
479   apply (blast intro: order_antisym)
480   done
482 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
483   apply (simp add: max_def)
484   apply (blast intro: order_antisym)
485   done
488 subsubsection {* Least value operator *}
490 constdefs
491   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
492   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
493     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
495 lemma LeastI2:
496   "[| P (x::'a::order);
497       !!y. P y ==> x <= y;
498       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
499    ==> Q (Least P)";
500   apply (unfold Least_def)
501   apply (rule theI2)
502     apply (blast intro: order_antisym)+
503   done
505 lemma Least_equality:
506     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k";
507   apply (simp add: Least_def)
508   apply (rule the_equality)
509   apply (auto intro!: order_antisym)
510   done
513 subsubsection "Linear / total orders"
515 axclass linorder < order
516   linorder_linear: "x <= y | y <= x"
518 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
519   apply (simp add: order_less_le)
520   apply (insert linorder_linear)
521   apply blast
522   done
524 lemma linorder_cases [case_names less equal greater]:
525     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
526   apply (insert linorder_less_linear)
527   apply blast
528   done
530 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
531   apply (simp add: order_less_le)
532   apply (insert linorder_linear)
533   apply (blast intro: order_antisym)
534   done
536 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
537   apply (simp add: order_less_le)
538   apply (insert linorder_linear)
539   apply (blast intro: order_antisym)
540   done
542 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
543   apply (cut_tac x = x and y = y in linorder_less_linear)
544   apply auto
545   done
547 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
548   apply (simp add: linorder_neq_iff)
549   apply blast
550   done
553 subsubsection "Min and max on (linear) orders"
555 lemma min_same [simp]: "min (x::'a::order) x = x"
556   by (simp add: min_def)
558 lemma max_same [simp]: "max (x::'a::order) x = x"
559   by (simp add: max_def)
561 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
562   apply (simp add: max_def)
563   apply (insert linorder_linear)
564   apply (blast intro: order_trans)
565   done
567 lemma le_maxI1: "(x::'a::linorder) <= max x y"
568   by (simp add: le_max_iff_disj)
570 lemma le_maxI2: "(y::'a::linorder) <= max x y"
571     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
572   by (simp add: le_max_iff_disj)
574 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
575   apply (simp add: max_def order_le_less)
576   apply (insert linorder_less_linear)
577   apply (blast intro: order_less_trans)
578   done
580 lemma max_le_iff_conj [simp]:
581     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
582   apply (simp add: max_def)
583   apply (insert linorder_linear)
584   apply (blast intro: order_trans)
585   done
587 lemma max_less_iff_conj [simp]:
588     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
589   apply (simp add: order_le_less max_def)
590   apply (insert linorder_less_linear)
591   apply (blast intro: order_less_trans)
592   done
594 lemma le_min_iff_conj [simp]:
595     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
596     -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
597   apply (simp add: min_def)
598   apply (insert linorder_linear)
599   apply (blast intro: order_trans)
600   done
602 lemma min_less_iff_conj [simp]:
603     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
604   apply (simp add: order_le_less min_def)
605   apply (insert linorder_less_linear)
606   apply (blast intro: order_less_trans)
607   done
609 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
610   apply (simp add: min_def)
611   apply (insert linorder_linear)
612   apply (blast intro: order_trans)
613   done
615 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
616   apply (simp add: min_def order_le_less)
617   apply (insert linorder_less_linear)
618   apply (blast intro: order_less_trans)
619   done
621 lemma split_min:
622     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
623   by (simp add: min_def)
625 lemma split_max:
626     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
627   by (simp add: max_def)
630 subsubsection "Bounded quantifiers"
632 syntax
633   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
634   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
635   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
636   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
638 syntax (xsymbols)
639   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
640   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
641   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
642   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
644 syntax (HOL)
645   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
646   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
647   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
648   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
650 translations
651  "ALL x<y. P"   =>  "ALL x. x < y --> P"
652  "EX x<y. P"    =>  "EX x. x < y  & P"
653  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
654  "EX x<=y. P"   =>  "EX x. x <= y & P"
656 end