src/HOL/HOL.thy
 author nipkow Thu Dec 02 11:42:01 2004 +0100 (2004-12-02) changeset 15360 300e09825d8b parent 15354 9234f5765d9c child 15362 a000b267be57 permissions -rw-r--r--
Added "ALL x > y" and relatives.
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
9 imports CPure
10 files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML")
11       ("antisym_setup.ML")
12 begin
14 subsection {* Primitive logic *}
16 subsubsection {* Core syntax *}
18 classes type
19 defaultsort type
21 global
23 typedecl bool
25 arities
26   bool :: type
27   fun :: (type, type) type
29 judgment
30   Trueprop      :: "bool => prop"                   ("(_)" 5)
32 consts
33   Not           :: "bool => bool"                   ("~ _"  40)
34   True          :: bool
35   False         :: bool
36   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
37   arbitrary     :: 'a
39   The           :: "('a => bool) => 'a"
40   All           :: "('a => bool) => bool"           (binder "ALL " 10)
41   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
42   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
43   Let           :: "['a, 'a => 'b] => 'b"
45   "="           :: "['a, 'a] => bool"               (infixl 50)
46   &             :: "[bool, bool] => bool"           (infixr 35)
47   "|"           :: "[bool, bool] => bool"           (infixr 30)
48   -->           :: "[bool, bool] => bool"           (infixr 25)
50 local
53 subsubsection {* Additional concrete syntax *}
55 nonterminals
56   letbinds  letbind
57   case_syn  cases_syn
59 syntax
60   "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
61   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
63   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
64   ""            :: "letbind => letbinds"                 ("_")
65   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
66   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
68   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
69   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
70   ""            :: "case_syn => cases_syn"               ("_")
71   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
73 translations
74   "x ~= y"                == "~ (x = y)"
75   "THE x. P"              == "The (%x. P)"
76   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
77   "let x = a in e"        == "Let a (%x. e)"
79 print_translation {*
80 (* To avoid eta-contraction of body: *)
81 [("The", fn [Abs abs] =>
82      let val (x,t) = atomic_abs_tr' abs
83      in Syntax.const "_The" \$ x \$ t end)]
84 *}
86 syntax (output)
87   "="           :: "['a, 'a] => bool"                    (infix 50)
88   "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
90 syntax (xsymbols)
91   Not           :: "bool => bool"                        ("\<not> _"  40)
92   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
93   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
94   "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
95   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
96   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
97   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
98   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
99   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
100 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
102 syntax (xsymbols output)
103   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
105 syntax (HTML output)
106   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
107   Not           :: "bool => bool"                        ("\<not> _"  40)
108   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
109   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
110   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
111   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
112   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
113   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
115 syntax (HOL)
116   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
117   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
118   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
121 subsubsection {* Axioms and basic definitions *}
123 axioms
124   eq_reflection: "(x=y) ==> (x==y)"
126   refl:         "t = (t::'a)"
127   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
129   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
130     -- {* Extensionality is built into the meta-logic, and this rule expresses *}
131     -- {* a related property.  It is an eta-expanded version of the traditional *}
132     -- {* rule, and similar to the ABS rule of HOL *}
134   the_eq_trivial: "(THE x. x = a) = (a::'a)"
136   impI:         "(P ==> Q) ==> P-->Q"
137   mp:           "[| P-->Q;  P |] ==> Q"
139 defs
140   True_def:     "True      == ((%x::bool. x) = (%x. x))"
141   All_def:      "All(P)    == (P = (%x. True))"
142   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
143   False_def:    "False     == (!P. P)"
144   not_def:      "~ P       == P-->False"
145   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
146   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
147   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
149 axioms
150   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
151   True_or_False:  "(P=True) | (P=False)"
153 defs
154   Let_def:      "Let s f == f(s)"
155   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
157 finalconsts
158   "op ="
159   "op -->"
160   The
161   arbitrary
163 subsubsection {* Generic algebraic operations *}
165 axclass zero < type
166 axclass one < type
167 axclass plus < type
168 axclass minus < type
169 axclass times < type
170 axclass inverse < type
172 global
174 consts
175   "0"           :: "'a::zero"                       ("0")
176   "1"           :: "'a::one"                        ("1")
177   "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
178   -             :: "['a::minus, 'a] => 'a"          (infixl 65)
179   uminus        :: "['a::minus] => 'a"              ("- _"  80)
180   *             :: "['a::times, 'a] => 'a"          (infixl 70)
182 syntax
183   "_index1"  :: index    ("\<^sub>1")
184 translations
185   (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
187 local
189 typed_print_translation {*
190   let
191     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
192       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
193       else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
194   in [tr' "0", tr' "1"] end;
195 *} -- {* show types that are presumably too general *}
198 consts
199   abs           :: "'a::minus => 'a"
200   inverse       :: "'a::inverse => 'a"
201   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
203 syntax (xsymbols)
204   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
205 syntax (HTML output)
206   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
209 subsection {* Theory and package setup *}
211 subsubsection {* Basic lemmas *}
213 use "HOL_lemmas.ML"
214 theorems case_split = case_split_thm [case_names True False]
217 subsubsection {* Intuitionistic Reasoning *}
219 lemma impE':
220   assumes 1: "P --> Q"
221     and 2: "Q ==> R"
222     and 3: "P --> Q ==> P"
223   shows R
224 proof -
225   from 3 and 1 have P .
226   with 1 have Q by (rule impE)
227   with 2 show R .
228 qed
230 lemma allE':
231   assumes 1: "ALL x. P x"
232     and 2: "P x ==> ALL x. P x ==> Q"
233   shows Q
234 proof -
235   from 1 have "P x" by (rule spec)
236   from this and 1 show Q by (rule 2)
237 qed
239 lemma notE':
240   assumes 1: "~ P"
241     and 2: "~ P ==> P"
242   shows R
243 proof -
244   from 2 and 1 have P .
245   with 1 show R by (rule notE)
246 qed
248 lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
249   and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
250   and [CPure.elim 2] = allE notE' impE'
251   and [CPure.intro] = exI disjI2 disjI1
253 lemmas [trans] = trans
254   and [sym] = sym not_sym
255   and [CPure.elim?] = iffD1 iffD2 impE
258 subsubsection {* Atomizing meta-level connectives *}
260 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
261 proof
262   assume "!!x. P x"
263   show "ALL x. P x" by (rule allI)
264 next
265   assume "ALL x. P x"
266   thus "!!x. P x" by (rule allE)
267 qed
269 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
270 proof
271   assume r: "A ==> B"
272   show "A --> B" by (rule impI) (rule r)
273 next
274   assume "A --> B" and A
275   thus B by (rule mp)
276 qed
278 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
279 proof
280   assume r: "A ==> False"
281   show "~A" by (rule notI) (rule r)
282 next
283   assume "~A" and A
284   thus False by (rule notE)
285 qed
287 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
288 proof
289   assume "x == y"
290   show "x = y" by (unfold prems) (rule refl)
291 next
292   assume "x = y"
293   thus "x == y" by (rule eq_reflection)
294 qed
296 lemma atomize_conj [atomize]:
297   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
298 proof
299   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
300   show "A & B" by (rule conjI)
301 next
302   fix C
303   assume "A & B"
304   assume "A ==> B ==> PROP C"
305   thus "PROP C"
306   proof this
307     show A by (rule conjunct1)
308     show B by (rule conjunct2)
309   qed
310 qed
312 lemmas [symmetric, rulify] = atomize_all atomize_imp
315 subsubsection {* Classical Reasoner setup *}
318 setup hypsubst_setup
320 ML_setup {*
321   Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
322 *}
324 setup Classical.setup
325 setup clasetup
327 lemmas [intro?] = ext
328   and [elim?] = ex1_implies_ex
330 use "blastdata.ML"
331 setup Blast.setup
334 subsubsection {* Simplifier setup *}
336 lemma meta_eq_to_obj_eq: "x == y ==> x = y"
337 proof -
338   assume r: "x == y"
339   show "x = y" by (unfold r) (rule refl)
340 qed
342 lemma eta_contract_eq: "(%s. f s) = f" ..
344 lemma simp_thms:
345   shows not_not: "(~ ~ P) = P"
346   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
347   and
348     "(P ~= Q) = (P = (~Q))"
349     "(P | ~P) = True"    "(~P | P) = True"
350     "(x = x) = True"
351     "(~True) = False"  "(~False) = True"
352     "(~P) ~= P"  "P ~= (~P)"
353     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
354     "(True --> P) = P"  "(False --> P) = True"
355     "(P --> True) = True"  "(P --> P) = True"
356     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
357     "(P & True) = P"  "(True & P) = P"
358     "(P & False) = False"  "(False & P) = False"
359     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
360     "(P & ~P) = False"    "(~P & P) = False"
361     "(P | True) = True"  "(True | P) = True"
362     "(P | False) = P"  "(False | P) = P"
363     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
364     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
365     -- {* needed for the one-point-rule quantifier simplification procs *}
366     -- {* essential for termination!! *} and
367     "!!P. (EX x. x=t & P(x)) = P(t)"
368     "!!P. (EX x. t=x & P(x)) = P(t)"
369     "!!P. (ALL x. x=t --> P(x)) = P(t)"
370     "!!P. (ALL x. t=x --> P(x)) = P(t)"
371   by (blast, blast, blast, blast, blast, rules+)
373 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
374   by rules
376 lemma ex_simps:
377   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
378   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
379   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
380   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
381   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
382   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
383   -- {* Miniscoping: pushing in existential quantifiers. *}
384   by (rules | blast)+
386 lemma all_simps:
387   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
388   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
389   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
390   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
391   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
392   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
393   -- {* Miniscoping: pushing in universal quantifiers. *}
394   by (rules | blast)+
396 lemma disj_absorb: "(A | A) = A"
397   by blast
399 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
400   by blast
402 lemma conj_absorb: "(A & A) = A"
403   by blast
405 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
406   by blast
408 lemma eq_ac:
409   shows eq_commute: "(a=b) = (b=a)"
410     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
411     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
412 lemma neq_commute: "(a~=b) = (b~=a)" by rules
414 lemma conj_comms:
415   shows conj_commute: "(P&Q) = (Q&P)"
416     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
417 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
419 lemma disj_comms:
420   shows disj_commute: "(P|Q) = (Q|P)"
421     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
422 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
424 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
425 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
427 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
428 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
430 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
431 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
432 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
434 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
435 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
436 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
438 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
439 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
441 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
442 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
443 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
444 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
445 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
446 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
447   by blast
448 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
450 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
453 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
454   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
455   -- {* cases boil down to the same thing. *}
456   by blast
458 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
459 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
460 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
461 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
463 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
464 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
466 text {*
467   \medskip The @{text "&"} congruence rule: not included by default!
468   May slow rewrite proofs down by as much as 50\% *}
470 lemma conj_cong:
471     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
472   by rules
474 lemma rev_conj_cong:
475     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
476   by rules
478 text {* The @{text "|"} congruence rule: not included by default! *}
480 lemma disj_cong:
481     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
482   by blast
484 lemma eq_sym_conv: "(x = y) = (y = x)"
485   by rules
488 text {* \medskip if-then-else rules *}
490 lemma if_True: "(if True then x else y) = x"
491   by (unfold if_def) blast
493 lemma if_False: "(if False then x else y) = y"
494   by (unfold if_def) blast
496 lemma if_P: "P ==> (if P then x else y) = x"
497   by (unfold if_def) blast
499 lemma if_not_P: "~P ==> (if P then x else y) = y"
500   by (unfold if_def) blast
502 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
503   apply (rule case_split [of Q])
504    apply (subst if_P)
505     prefer 3 apply (subst if_not_P, blast+)
506   done
508 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
509 by (subst split_if, blast)
511 lemmas if_splits = split_if split_if_asm
513 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
514   by (rule split_if)
516 lemma if_cancel: "(if c then x else x) = x"
517 by (subst split_if, blast)
519 lemma if_eq_cancel: "(if x = y then y else x) = x"
520 by (subst split_if, blast)
522 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
523   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
524   by (rule split_if)
526 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
527   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
528   apply (subst split_if, blast)
529   done
531 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
532 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
534 subsubsection {* Actual Installation of the Simplifier *}
536 use "simpdata.ML"
537 setup Simplifier.setup
538 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
539 setup Splitter.setup setup Clasimp.setup
541 declare disj_absorb [simp] conj_absorb [simp]
543 lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
544 by blast+
546 theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
547   apply (rule iffI)
548   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
549   apply (fast dest!: theI')
550   apply (fast intro: ext the1_equality [symmetric])
551   apply (erule ex1E)
552   apply (rule allI)
553   apply (rule ex1I)
554   apply (erule spec)
555   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
556   apply (erule impE)
557   apply (rule allI)
558   apply (rule_tac P = "xa = x" in case_split_thm)
559   apply (drule_tac  x = x in fun_cong, simp_all)
560   done
562 text{*Needs only HOL-lemmas:*}
563 lemma mk_left_commute:
564   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
565           c: "\<And>x y. f x y = f y x"
566   shows "f x (f y z) = f y (f x z)"
567 by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
570 subsubsection {* Generic cases and induction *}
572 constdefs
573   induct_forall :: "('a => bool) => bool"
574   "induct_forall P == \<forall>x. P x"
575   induct_implies :: "bool => bool => bool"
576   "induct_implies A B == A --> B"
577   induct_equal :: "'a => 'a => bool"
578   "induct_equal x y == x = y"
579   induct_conj :: "bool => bool => bool"
580   "induct_conj A B == A & B"
582 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
583   by (simp only: atomize_all induct_forall_def)
585 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
586   by (simp only: atomize_imp induct_implies_def)
588 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
589   by (simp only: atomize_eq induct_equal_def)
591 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
592     induct_conj (induct_forall A) (induct_forall B)"
593   by (unfold induct_forall_def induct_conj_def) rules
595 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
596     induct_conj (induct_implies C A) (induct_implies C B)"
597   by (unfold induct_implies_def induct_conj_def) rules
599 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
600 proof
601   assume r: "induct_conj A B ==> PROP C" and A B
602   show "PROP C" by (rule r) (simp! add: induct_conj_def)
603 next
604   assume r: "A ==> B ==> PROP C" and "induct_conj A B"
605   show "PROP C" by (rule r) (simp! add: induct_conj_def)+
606 qed
608 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
609   by (simp add: induct_implies_def)
611 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
612 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
613 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
614 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
616 hide const induct_forall induct_implies induct_equal induct_conj
619 text {* Method setup. *}
621 ML {*
622   structure InductMethod = InductMethodFun
623   (struct
624     val dest_concls = HOLogic.dest_concls;
625     val cases_default = thm "case_split";
626     val local_impI = thm "induct_impliesI";
627     val conjI = thm "conjI";
628     val atomize = thms "induct_atomize";
629     val rulify1 = thms "induct_rulify1";
630     val rulify2 = thms "induct_rulify2";
631     val localize = [Thm.symmetric (thm "induct_implies_def")];
632   end);
633 *}
635 setup InductMethod.setup
638 subsection {* Order signatures and orders *}
640 axclass
641   ord < type
643 syntax
644   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
645   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
647 global
649 consts
650   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
651   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
653 local
655 syntax (xsymbols)
656   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
657   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
659 syntax (HTML output)
660   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
661   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
663 text{* Syntactic sugar: *}
665 consts
666   "_gt" :: "'a::ord => 'a => bool"             (infixl ">" 50)
667   "_ge" :: "'a::ord => 'a => bool"             (infixl ">=" 50)
668 translations
669   "x > y"  => "y < x"
670   "x >= y" => "y <= x"
672 syntax (xsymbols)
673   "_ge"       :: "'a::ord => 'a => bool"             (infixl "\<ge>" 50)
675 syntax (HTML output)
676   "_ge"       :: "['a::ord, 'a] => bool"             (infixl "\<ge>" 50)
679 subsubsection {* Monotonicity *}
681 locale mono =
682   fixes f
683   assumes mono: "A <= B ==> f A <= f B"
685 lemmas monoI [intro?] = mono.intro
686   and monoD [dest?] = mono.mono
688 constdefs
689   min :: "['a::ord, 'a] => 'a"
690   "min a b == (if a <= b then a else b)"
691   max :: "['a::ord, 'a] => 'a"
692   "max a b == (if a <= b then b else a)"
694 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
695   by (simp add: min_def)
697 lemma min_of_mono:
698     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
699   by (simp add: min_def)
701 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
702   by (simp add: max_def)
704 lemma max_of_mono:
705     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
706   by (simp add: max_def)
709 subsubsection "Orders"
711 axclass order < ord
712   order_refl [iff]: "x <= x"
713   order_trans: "x <= y ==> y <= z ==> x <= z"
714   order_antisym: "x <= y ==> y <= x ==> x = y"
715   order_less_le: "(x < y) = (x <= y & x ~= y)"
718 text {* Reflexivity. *}
720 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
721     -- {* This form is useful with the classical reasoner. *}
722   apply (erule ssubst)
723   apply (rule order_refl)
724   done
726 lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
727   by (simp add: order_less_le)
729 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
730     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
731   apply (simp add: order_less_le, blast)
732   done
734 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
736 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
737   by (simp add: order_less_le)
740 text {* Asymmetry. *}
742 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
743   by (simp add: order_less_le order_antisym)
745 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
746   apply (drule order_less_not_sym)
747   apply (erule contrapos_np, simp)
748   done
750 lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
751 by (blast intro: order_antisym)
753 lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
754 by(blast intro:order_antisym)
756 text {* Transitivity. *}
758 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
759   apply (simp add: order_less_le)
760   apply (blast intro: order_trans order_antisym)
761   done
763 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
764   apply (simp add: order_less_le)
765   apply (blast intro: order_trans order_antisym)
766   done
768 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
769   apply (simp add: order_less_le)
770   apply (blast intro: order_trans order_antisym)
771   done
774 text {* Useful for simplification, but too risky to include by default. *}
776 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
777   by (blast elim: order_less_asym)
779 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
780   by (blast elim: order_less_asym)
782 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
783   by auto
785 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
786   by auto
789 text {* Other operators. *}
791 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
792   apply (simp add: min_def)
793   apply (blast intro: order_antisym)
794   done
796 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
797   apply (simp add: max_def)
798   apply (blast intro: order_antisym)
799   done
802 subsubsection {* Least value operator *}
804 constdefs
805   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
806   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
807     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
809 lemma LeastI2:
810   "[| P (x::'a::order);
811       !!y. P y ==> x <= y;
812       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
813    ==> Q (Least P)"
814   apply (unfold Least_def)
815   apply (rule theI2)
816     apply (blast intro: order_antisym)+
817   done
819 lemma Least_equality:
820     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
821   apply (simp add: Least_def)
822   apply (rule the_equality)
823   apply (auto intro!: order_antisym)
824   done
827 subsubsection "Linear / total orders"
829 axclass linorder < order
830   linorder_linear: "x <= y | y <= x"
832 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
833   apply (simp add: order_less_le)
834   apply (insert linorder_linear, blast)
835   done
837 lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
838   by (simp add: order_le_less linorder_less_linear)
840 lemma linorder_le_cases [case_names le ge]:
841     "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
842   by (insert linorder_linear, blast)
844 lemma linorder_cases [case_names less equal greater]:
845     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
846   by (insert linorder_less_linear, blast)
848 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
849   apply (simp add: order_less_le)
850   apply (insert linorder_linear)
851   apply (blast intro: order_antisym)
852   done
854 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
855   apply (simp add: order_less_le)
856   apply (insert linorder_linear)
857   apply (blast intro: order_antisym)
858   done
860 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
861 by (cut_tac x = x and y = y in linorder_less_linear, auto)
863 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
864 by (simp add: linorder_neq_iff, blast)
866 lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
867 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
869 lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
870 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
872 lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
873 by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
875 use "antisym_setup.ML";
876 setup antisym_setup
878 subsubsection "Min and max on (linear) orders"
880 lemma min_same [simp]: "min (x::'a::order) x = x"
881   by (simp add: min_def)
883 lemma max_same [simp]: "max (x::'a::order) x = x"
884   by (simp add: max_def)
886 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
887   apply (simp add: max_def)
888   apply (insert linorder_linear)
889   apply (blast intro: order_trans)
890   done
892 lemma le_maxI1: "(x::'a::linorder) <= max x y"
893   by (simp add: le_max_iff_disj)
895 lemma le_maxI2: "(y::'a::linorder) <= max x y"
896     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
897   by (simp add: le_max_iff_disj)
899 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
900   apply (simp add: max_def order_le_less)
901   apply (insert linorder_less_linear)
902   apply (blast intro: order_less_trans)
903   done
905 lemma max_le_iff_conj [simp]:
906     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
907   apply (simp add: max_def)
908   apply (insert linorder_linear)
909   apply (blast intro: order_trans)
910   done
912 lemma max_less_iff_conj [simp]:
913     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
914   apply (simp add: order_le_less max_def)
915   apply (insert linorder_less_linear)
916   apply (blast intro: order_less_trans)
917   done
919 lemma le_min_iff_conj [simp]:
920     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
921     -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
922   apply (simp add: min_def)
923   apply (insert linorder_linear)
924   apply (blast intro: order_trans)
925   done
927 lemma min_less_iff_conj [simp]:
928     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
929   apply (simp add: order_le_less min_def)
930   apply (insert linorder_less_linear)
931   apply (blast intro: order_less_trans)
932   done
934 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
935   apply (simp add: min_def)
936   apply (insert linorder_linear)
937   apply (blast intro: order_trans)
938   done
940 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
941   apply (simp add: min_def order_le_less)
942   apply (insert linorder_less_linear)
943   apply (blast intro: order_less_trans)
944   done
946 lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
948 apply(rule conjI)
949 apply(blast intro:order_trans)
951 apply(blast dest: order_less_trans order_le_less_trans)
952 done
954 lemma max_commute: "!!x::'a::linorder. max x y = max y x"
957 apply(blast dest: order_less_trans)
958 done
960 lemmas max_ac = max_assoc max_commute
961                 mk_left_commute[of max,OF max_assoc max_commute]
963 lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
965 apply(rule conjI)
966 apply(blast intro:order_trans)
968 apply(blast dest: order_less_trans order_le_less_trans)
969 done
971 lemma min_commute: "!!x::'a::linorder. min x y = min y x"
974 apply(blast dest: order_less_trans)
975 done
977 lemmas min_ac = min_assoc min_commute
978                 mk_left_commute[of min,OF min_assoc min_commute]
980 lemma split_min:
981     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
982   by (simp add: min_def)
984 lemma split_max:
985     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
986   by (simp add: max_def)
989 subsubsection {* Transitivity rules for calculational reasoning *}
992 lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
993   by (simp add: order_less_le)
995 lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
996   by (simp add: order_less_le)
998 lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
999   by (rule order_less_asym)
1002 subsubsection {* Setup of transitivity reasoner as Solver *}
1004 lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
1005   by (erule contrapos_pn, erule subst, rule order_less_irrefl)
1007 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
1008   by (erule subst, erule ssubst, assumption)
1010 ML_setup {*
1012 (* The setting up of Quasi_Tac serves as a demo.  Since there is no
1013    class for quasi orders, the tactics Quasi_Tac.trans_tac and
1014    Quasi_Tac.quasi_tac are not of much use. *)
1016 fun decomp_gen sort sign (Trueprop \$ t) =
1017   let fun of_sort t = Sign.of_sort sign (type_of t, sort)
1018   fun dec (Const ("Not", _) \$ t) = (
1019 	  case dec t of
1020 	    None => None
1021 	  | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
1022 	| dec (Const ("op =",  _) \$ t1 \$ t2) =
1023 	    if of_sort t1
1024 	    then Some (t1, "=", t2)
1025 	    else None
1026 	| dec (Const ("op <=",  _) \$ t1 \$ t2) =
1027 	    if of_sort t1
1028 	    then Some (t1, "<=", t2)
1029 	    else None
1030 	| dec (Const ("op <",  _) \$ t1 \$ t2) =
1031 	    if of_sort t1
1032 	    then Some (t1, "<", t2)
1033 	    else None
1034 	| dec _ = None
1035   in dec t end;
1037 structure Quasi_Tac = Quasi_Tac_Fun (
1038   struct
1039     val le_trans = thm "order_trans";
1040     val le_refl = thm "order_refl";
1041     val eqD1 = thm "order_eq_refl";
1042     val eqD2 = thm "sym" RS thm "order_eq_refl";
1043     val less_reflE = thm "order_less_irrefl" RS thm "notE";
1044     val less_imp_le = thm "order_less_imp_le";
1045     val le_neq_trans = thm "order_le_neq_trans";
1046     val neq_le_trans = thm "order_neq_le_trans";
1047     val less_imp_neq = thm "less_imp_neq";
1048     val decomp_trans = decomp_gen ["HOL.order"];
1049     val decomp_quasi = decomp_gen ["HOL.order"];
1051   end);  (* struct *)
1053 structure Order_Tac = Order_Tac_Fun (
1054   struct
1055     val less_reflE = thm "order_less_irrefl" RS thm "notE";
1056     val le_refl = thm "order_refl";
1057     val less_imp_le = thm "order_less_imp_le";
1058     val not_lessI = thm "linorder_not_less" RS thm "iffD2";
1059     val not_leI = thm "linorder_not_le" RS thm "iffD2";
1060     val not_lessD = thm "linorder_not_less" RS thm "iffD1";
1061     val not_leD = thm "linorder_not_le" RS thm "iffD1";
1062     val eqI = thm "order_antisym";
1063     val eqD1 = thm "order_eq_refl";
1064     val eqD2 = thm "sym" RS thm "order_eq_refl";
1065     val less_trans = thm "order_less_trans";
1066     val less_le_trans = thm "order_less_le_trans";
1067     val le_less_trans = thm "order_le_less_trans";
1068     val le_trans = thm "order_trans";
1069     val le_neq_trans = thm "order_le_neq_trans";
1070     val neq_le_trans = thm "order_neq_le_trans";
1071     val less_imp_neq = thm "less_imp_neq";
1072     val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
1073     val decomp_part = decomp_gen ["HOL.order"];
1074     val decomp_lin = decomp_gen ["HOL.linorder"];
1076   end);  (* struct *)
1078 simpset_ref() := simpset ()
1079     addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
1080     addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac));
1081   (* Adding the transitivity reasoners also as safe solvers showed a slight
1082      speed up, but the reasoning strength appears to be not higher (at least
1083      no breaking of additional proofs in the entire HOL distribution, as
1084      of 5 March 2004, was observed). *)
1085 *}
1087 (* Optional setup of methods *)
1089 (*
1090 method_setup trans_partial =
1091   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.partial_tac)) *}
1092   {* transitivity reasoner for partial orders *}
1093 method_setup trans_linear =
1094   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.linear_tac)) *}
1095   {* transitivity reasoner for linear orders *}
1096 *)
1098 (*
1099 declare order.order_refl [simp del] order_less_irrefl [simp del]
1101 can currently not be removed, abel_cancel relies on it.
1102 *)
1104 subsubsection "Bounded quantifiers"
1106 syntax
1107   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
1108   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
1109   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
1110   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
1112   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _>_./ _)"  [0, 0, 10] 10)
1113   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _>_./ _)"  [0, 0, 10] 10)
1114   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _>=_./ _)" [0, 0, 10] 10)
1115   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _>=_./ _)" [0, 0, 10] 10)
1117 syntax (xsymbols)
1118   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
1119   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
1120   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
1121   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
1123   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
1124   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
1125   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
1126   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
1128 syntax (HOL)
1129   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
1130   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
1131   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
1132   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
1134 syntax (HTML output)
1135   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
1136   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
1137   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
1138   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
1140   "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
1141   "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
1142   "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
1143   "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
1145 translations
1146  "ALL x<y. P"   =>  "ALL x. x < y --> P"
1147  "EX x<y. P"    =>  "EX x. x < y  & P"
1148  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
1149  "EX x<=y. P"   =>  "EX x. x <= y & P"
1150  "ALL x>y. P"   =>  "ALL x. x > y --> P"
1151  "EX x>y. P"    =>  "EX x. x > y  & P"
1152  "ALL x>=y. P"  =>  "ALL x. x >= y --> P"
1153  "EX x>=y. P"   =>  "EX x. x >= y & P"
1155 print_translation {*
1156 let
1157   fun all_tr' [Const ("_bound",_) \$ Free (v,_),
1158                Const("op -->",_) \$ (Const ("op <",_) \$ (Const ("_bound",_) \$ Free (v',_)) \$ n ) \$ P] =
1159   (if v=v' then Syntax.const "_lessAll" \$ Syntax.mark_bound v' \$ n \$ P else raise Match)
1161   | all_tr' [Const ("_bound",_) \$ Free (v,_),
1162                Const("op -->",_) \$ (Const ("op <=",_) \$ (Const ("_bound",_) \$ Free (v',_)) \$ n ) \$ P] =
1163   (if v=v' then Syntax.const "_leAll" \$ Syntax.mark_bound v' \$ n \$ P else raise Match);
1165   fun ex_tr' [Const ("_bound",_) \$ Free (v,_),
1166                Const("op &",_) \$ (Const ("op <",_) \$ (Const ("_bound",_) \$ Free (v',_)) \$ n ) \$ P] =
1167   (if v=v' then Syntax.const "_lessEx" \$ Syntax.mark_bound v' \$ n \$ P else raise Match)
1169   | ex_tr' [Const ("_bound",_) \$ Free (v,_),
1170                Const("op &",_) \$ (Const ("op <=",_) \$ (Const ("_bound",_) \$ Free (v',_)) \$ n ) \$ P] =
1171   (if v=v' then Syntax.const "_leEx" \$ Syntax.mark_bound v' \$ n \$ P else raise Match)
1172 in
1173 [("ALL ", all_tr'), ("EX ", ex_tr')]
1174 end
1175 *}
1177 end