src/HOL/HOL.thy
 author krauss Fri Nov 24 13:44:51 2006 +0100 (2006-11-24) changeset 21512 3786eb1b69d6 parent 21504 9c97af4a1567 child 21524 7843e2fd14a9 permissions -rw-r--r--
Lemma "fundef_default_value" uses predicate instead of set.
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 uses ("simpdata.ML") "Tools/res_atpset.ML"
11 begin
13 subsection {* Primitive logic *}
15 subsubsection {* Core syntax *}
17 classes type
18 defaultsort type
20 global
22 typedecl bool
24 arities
25   bool :: type
26   "fun" :: (type, type) type
28 judgment
29   Trueprop      :: "bool => prop"                   ("(_)" 5)
31 consts
32   Not           :: "bool => bool"                   ("~ _"  40)
33   True          :: bool
34   False         :: bool
35   arbitrary     :: 'a
36   undefined     :: 'a
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"
44   "="           :: "['a, 'a] => bool"               (infixl 50)
45   &             :: "[bool, bool] => bool"           (infixr 35)
46   "|"           :: "[bool, bool] => bool"           (infixr 30)
47   -->           :: "[bool, bool] => bool"           (infixr 25)
49 local
51 consts
52   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
55 subsubsection {* Additional concrete syntax *}
57 notation (output)
58   "op ="  (infix "=" 50)
60 abbreviation
61   not_equal :: "['a, 'a] => bool"  (infixl "~=" 50) where
62   "x ~= y == ~ (x = y)"
64 notation (output)
65   not_equal  (infix "~=" 50)
67 notation (xsymbols)
68   Not  ("\<not> _"  40) and
69   "op &"  (infixr "\<and>" 35) and
70   "op |"  (infixr "\<or>" 30) and
71   "op -->"  (infixr "\<longrightarrow>" 25) and
72   not_equal  (infix "\<noteq>" 50)
74 notation (HTML output)
75   Not  ("\<not> _"  40) and
76   "op &"  (infixr "\<and>" 35) and
77   "op |"  (infixr "\<or>" 30) and
78   not_equal  (infix "\<noteq>" 50)
80 abbreviation (iff)
81   iff :: "[bool, bool] => bool"  (infixr "<->" 25) where
82   "A <-> B == A = B"
84 notation (xsymbols)
85   iff  (infixr "\<longleftrightarrow>" 25)
88 nonterminals
89   letbinds  letbind
90   case_syn  cases_syn
92 syntax
93   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
95   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
96   ""            :: "letbind => letbinds"                 ("_")
97   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
98   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
100   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
101   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
102   ""            :: "case_syn => cases_syn"               ("_")
103   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
105 translations
106   "THE x. P"              == "The (%x. P)"
107   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
108   "let x = a in e"        == "Let a (%x. e)"
110 print_translation {*
111 (* To avoid eta-contraction of body: *)
112 [("The", fn [Abs abs] =>
113      let val (x,t) = atomic_abs_tr' abs
114      in Syntax.const "_The" \$ x \$ t end)]
115 *}
117 syntax (xsymbols)
118   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
119   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
120   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
121   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
122 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
124 syntax (HTML output)
125   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
126   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
127   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
129 syntax (HOL)
130   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
131   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
132   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
135 subsubsection {* Axioms and basic definitions *}
137 axioms
138   eq_reflection:  "(x=y) ==> (x==y)"
140   refl:           "t = (t::'a)"
142   ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
143     -- {*Extensionality is built into the meta-logic, and this rule expresses
144          a related property.  It is an eta-expanded version of the traditional
145          rule, and similar to the ABS rule of HOL*}
147   the_eq_trivial: "(THE x. x = a) = (a::'a)"
149   impI:           "(P ==> Q) ==> P-->Q"
150   mp:             "[| P-->Q;  P |] ==> Q"
153 defs
154   True_def:     "True      == ((%x::bool. x) = (%x. x))"
155   All_def:      "All(P)    == (P = (%x. True))"
156   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
157   False_def:    "False     == (!P. P)"
158   not_def:      "~ P       == P-->False"
159   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
160   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
161   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
163 axioms
164   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
165   True_or_False:  "(P=True) | (P=False)"
167 defs
168   Let_def:      "Let s f == f(s)"
169   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
171 finalconsts
172   "op ="
173   "op -->"
174   The
175   arbitrary
176   undefined
179 subsubsection {* Generic algebraic operations *}
181 class zero =
182   fixes zero :: "'a"                       ("\<^loc>0")
184 class one =
185   fixes one  :: "'a"                       ("\<^loc>1")
187 hide (open) const zero one
189 class plus =
190   fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   (infixl "\<^loc>+" 65)
192 class minus =
193   fixes uminus :: "'a \<Rightarrow> 'a"
194   fixes minus  :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>-" 65)
195   fixes abs    :: "'a \<Rightarrow> 'a"
197 class times =
198   fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>*" 70)
200 class inverse =
201   fixes inverse :: "'a \<Rightarrow> 'a"
202   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>'/" 70)
204 syntax
205   "_index1"  :: index    ("\<^sub>1")
206 translations
207   (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
209 typed_print_translation {*
210 let
211   val thy = the_context ();
212   fun tr' c = (c, fn show_sorts => fn T => fn ts =>
213     if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
214     else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
215 in map (tr' o Sign.const_syntax_name thy) ["HOL.one", "HOL.zero"] end;
216 *} -- {* show types that are presumably too general *}
218 notation
219   uminus  ("- _"  80)
221 notation (xsymbols)
222   abs  ("\<bar>_\<bar>")
223 notation (HTML output)
224   abs  ("\<bar>_\<bar>")
227 subsection {* Fundamental rules *}
229 subsubsection {* Equality *}
231 text {* Thanks to Stephan Merz *}
232 lemma subst:
233   assumes eq: "s = t" and p: "P s"
234   shows "P t"
235 proof -
236   from eq have meta: "s \<equiv> t"
237     by (rule eq_reflection)
238   from p show ?thesis
239     by (unfold meta)
240 qed
242 lemma sym: "s = t ==> t = s"
243   by (erule subst) (rule refl)
245 lemma ssubst: "t = s ==> P s ==> P t"
246   by (drule sym) (erule subst)
248 lemma trans: "[| r=s; s=t |] ==> r=t"
249   by (erule subst)
251 lemma def_imp_eq:
252   assumes meq: "A == B"
253   shows "A = B"
254   by (unfold meq) (rule refl)
256 (*a mere copy*)
257 lemma meta_eq_to_obj_eq:
258   assumes meq: "A == B"
259   shows "A = B"
260   by (unfold meq) (rule refl)
262 text {* Useful with @{text erule} for proving equalities from known equalities. *}
263      (* a = b
264         |   |
265         c = d   *)
266 lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
267 apply (rule trans)
268 apply (rule trans)
269 apply (rule sym)
270 apply assumption+
271 done
273 text {* For calculational reasoning: *}
275 lemma forw_subst: "a = b ==> P b ==> P a"
276   by (rule ssubst)
278 lemma back_subst: "P a ==> a = b ==> P b"
279   by (rule subst)
282 subsubsection {*Congruence rules for application*}
284 (*similar to AP_THM in Gordon's HOL*)
285 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
286 apply (erule subst)
287 apply (rule refl)
288 done
290 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
291 lemma arg_cong: "x=y ==> f(x)=f(y)"
292 apply (erule subst)
293 apply (rule refl)
294 done
296 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
297 apply (erule ssubst)+
298 apply (rule refl)
299 done
301 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
302 apply (erule subst)+
303 apply (rule refl)
304 done
307 subsubsection {*Equality of booleans -- iff*}
309 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
310   by (iprover intro: iff [THEN mp, THEN mp] impI assms)
312 lemma iffD2: "[| P=Q; Q |] ==> P"
313   by (erule ssubst)
315 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
316   by (erule iffD2)
318 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
319   by (drule sym) (rule iffD2)
321 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
322   by (drule sym) (rule rev_iffD2)
324 lemma iffE:
325   assumes major: "P=Q"
326     and minor: "[| P --> Q; Q --> P |] ==> R"
327   shows R
328   by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
331 subsubsection {*True*}
333 lemma TrueI: "True"
334   unfolding True_def by (rule refl)
336 lemma eqTrueI: "P ==> P = True"
337   by (iprover intro: iffI TrueI)
339 lemma eqTrueE: "P = True ==> P"
340   by (erule iffD2) (rule TrueI)
343 subsubsection {*Universal quantifier*}
345 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
346   unfolding All_def by (iprover intro: ext eqTrueI assms)
348 lemma spec: "ALL x::'a. P(x) ==> P(x)"
349 apply (unfold All_def)
350 apply (rule eqTrueE)
351 apply (erule fun_cong)
352 done
354 lemma allE:
355   assumes major: "ALL x. P(x)"
356     and minor: "P(x) ==> R"
357   shows R
358   by (iprover intro: minor major [THEN spec])
360 lemma all_dupE:
361   assumes major: "ALL x. P(x)"
362     and minor: "[| P(x); ALL x. P(x) |] ==> R"
363   shows R
364   by (iprover intro: minor major major [THEN spec])
367 subsubsection {* False *}
369 text {*
370   Depends upon @{text spec}; it is impossible to do propositional
371   logic before quantifiers!
372 *}
374 lemma FalseE: "False ==> P"
375   apply (unfold False_def)
376   apply (erule spec)
377   done
379 lemma False_neq_True: "False = True ==> P"
380   by (erule eqTrueE [THEN FalseE])
383 subsubsection {* Negation *}
385 lemma notI:
386   assumes "P ==> False"
387   shows "~P"
388   apply (unfold not_def)
389   apply (iprover intro: impI assms)
390   done
392 lemma False_not_True: "False ~= True"
393   apply (rule notI)
394   apply (erule False_neq_True)
395   done
397 lemma True_not_False: "True ~= False"
398   apply (rule notI)
399   apply (drule sym)
400   apply (erule False_neq_True)
401   done
403 lemma notE: "[| ~P;  P |] ==> R"
404   apply (unfold not_def)
405   apply (erule mp [THEN FalseE])
406   apply assumption
407   done
409 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
410   by (erule notE [THEN notI]) (erule meta_mp)
413 subsubsection {*Implication*}
415 lemma impE:
416   assumes "P-->Q" "P" "Q ==> R"
417   shows "R"
418 by (iprover intro: prems mp)
420 (* Reduces Q to P-->Q, allowing substitution in P. *)
421 lemma rev_mp: "[| P;  P --> Q |] ==> Q"
422 by (iprover intro: mp)
424 lemma contrapos_nn:
425   assumes major: "~Q"
426       and minor: "P==>Q"
427   shows "~P"
428 by (iprover intro: notI minor major [THEN notE])
430 (*not used at all, but we already have the other 3 combinations *)
431 lemma contrapos_pn:
432   assumes major: "Q"
433       and minor: "P ==> ~Q"
434   shows "~P"
435 by (iprover intro: notI minor major notE)
437 lemma not_sym: "t ~= s ==> s ~= t"
438   by (erule contrapos_nn) (erule sym)
440 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
441   by (erule subst, erule ssubst, assumption)
443 (*still used in HOLCF*)
444 lemma rev_contrapos:
445   assumes pq: "P ==> Q"
446       and nq: "~Q"
447   shows "~P"
448 apply (rule nq [THEN contrapos_nn])
449 apply (erule pq)
450 done
452 subsubsection {*Existential quantifier*}
454 lemma exI: "P x ==> EX x::'a. P x"
455 apply (unfold Ex_def)
456 apply (iprover intro: allI allE impI mp)
457 done
459 lemma exE:
460   assumes major: "EX x::'a. P(x)"
461       and minor: "!!x. P(x) ==> Q"
462   shows "Q"
463 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
464 apply (iprover intro: impI [THEN allI] minor)
465 done
468 subsubsection {*Conjunction*}
470 lemma conjI: "[| P; Q |] ==> P&Q"
471 apply (unfold and_def)
472 apply (iprover intro: impI [THEN allI] mp)
473 done
475 lemma conjunct1: "[| P & Q |] ==> P"
476 apply (unfold and_def)
477 apply (iprover intro: impI dest: spec mp)
478 done
480 lemma conjunct2: "[| P & Q |] ==> Q"
481 apply (unfold and_def)
482 apply (iprover intro: impI dest: spec mp)
483 done
485 lemma conjE:
486   assumes major: "P&Q"
487       and minor: "[| P; Q |] ==> R"
488   shows "R"
489 apply (rule minor)
490 apply (rule major [THEN conjunct1])
491 apply (rule major [THEN conjunct2])
492 done
494 lemma context_conjI:
495   assumes prems: "P" "P ==> Q" shows "P & Q"
496 by (iprover intro: conjI prems)
499 subsubsection {*Disjunction*}
501 lemma disjI1: "P ==> P|Q"
502 apply (unfold or_def)
503 apply (iprover intro: allI impI mp)
504 done
506 lemma disjI2: "Q ==> P|Q"
507 apply (unfold or_def)
508 apply (iprover intro: allI impI mp)
509 done
511 lemma disjE:
512   assumes major: "P|Q"
513       and minorP: "P ==> R"
514       and minorQ: "Q ==> R"
515   shows "R"
516 by (iprover intro: minorP minorQ impI
517                  major [unfolded or_def, THEN spec, THEN mp, THEN mp])
520 subsubsection {*Classical logic*}
522 lemma classical:
523   assumes prem: "~P ==> P"
524   shows "P"
525 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
526 apply assumption
527 apply (rule notI [THEN prem, THEN eqTrueI])
528 apply (erule subst)
529 apply assumption
530 done
532 lemmas ccontr = FalseE [THEN classical, standard]
534 (*notE with premises exchanged; it discharges ~R so that it can be used to
535   make elimination rules*)
536 lemma rev_notE:
537   assumes premp: "P"
538       and premnot: "~R ==> ~P"
539   shows "R"
540 apply (rule ccontr)
541 apply (erule notE [OF premnot premp])
542 done
544 (*Double negation law*)
545 lemma notnotD: "~~P ==> P"
546 apply (rule classical)
547 apply (erule notE)
548 apply assumption
549 done
551 lemma contrapos_pp:
552   assumes p1: "Q"
553       and p2: "~P ==> ~Q"
554   shows "P"
555 by (iprover intro: classical p1 p2 notE)
558 subsubsection {*Unique existence*}
560 lemma ex1I:
561   assumes prems: "P a" "!!x. P(x) ==> x=a"
562   shows "EX! x. P(x)"
563 by (unfold Ex1_def, iprover intro: prems exI conjI allI impI)
565 text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
566 lemma ex_ex1I:
567   assumes ex_prem: "EX x. P(x)"
568       and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
569   shows "EX! x. P(x)"
570 by (iprover intro: ex_prem [THEN exE] ex1I eq)
572 lemma ex1E:
573   assumes major: "EX! x. P(x)"
574       and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
575   shows "R"
576 apply (rule major [unfolded Ex1_def, THEN exE])
577 apply (erule conjE)
578 apply (iprover intro: minor)
579 done
581 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
582 apply (erule ex1E)
583 apply (rule exI)
584 apply assumption
585 done
588 subsubsection {*THE: definite description operator*}
590 lemma the_equality:
591   assumes prema: "P a"
592       and premx: "!!x. P x ==> x=a"
593   shows "(THE x. P x) = a"
594 apply (rule trans [OF _ the_eq_trivial])
595 apply (rule_tac f = "The" in arg_cong)
596 apply (rule ext)
597 apply (rule iffI)
598  apply (erule premx)
599 apply (erule ssubst, rule prema)
600 done
602 lemma theI:
603   assumes "P a" and "!!x. P x ==> x=a"
604   shows "P (THE x. P x)"
605 by (iprover intro: prems the_equality [THEN ssubst])
607 lemma theI': "EX! x. P x ==> P (THE x. P x)"
608 apply (erule ex1E)
609 apply (erule theI)
610 apply (erule allE)
611 apply (erule mp)
612 apply assumption
613 done
615 (*Easier to apply than theI: only one occurrence of P*)
616 lemma theI2:
617   assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
618   shows "Q (THE x. P x)"
619 by (iprover intro: prems theI)
621 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
622 apply (rule the_equality)
623 apply  assumption
624 apply (erule ex1E)
625 apply (erule all_dupE)
626 apply (drule mp)
627 apply  assumption
628 apply (erule ssubst)
629 apply (erule allE)
630 apply (erule mp)
631 apply assumption
632 done
634 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
635 apply (rule the_equality)
636 apply (rule refl)
637 apply (erule sym)
638 done
641 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
643 lemma disjCI:
644   assumes "~Q ==> P" shows "P|Q"
645 apply (rule classical)
646 apply (iprover intro: prems disjI1 disjI2 notI elim: notE)
647 done
649 lemma excluded_middle: "~P | P"
650 by (iprover intro: disjCI)
652 text {*
653   case distinction as a natural deduction rule.
654   Note that @{term "~P"} is the second case, not the first
655 *}
656 lemma case_split_thm:
657   assumes prem1: "P ==> Q"
658       and prem2: "~P ==> Q"
659   shows "Q"
660 apply (rule excluded_middle [THEN disjE])
661 apply (erule prem2)
662 apply (erule prem1)
663 done
664 lemmas case_split = case_split_thm [case_names True False]
666 (*Classical implies (-->) elimination. *)
667 lemma impCE:
668   assumes major: "P-->Q"
669       and minor: "~P ==> R" "Q ==> R"
670   shows "R"
671 apply (rule excluded_middle [of P, THEN disjE])
672 apply (iprover intro: minor major [THEN mp])+
673 done
675 (*This version of --> elimination works on Q before P.  It works best for
676   those cases in which P holds "almost everywhere".  Can't install as
677   default: would break old proofs.*)
678 lemma impCE':
679   assumes major: "P-->Q"
680       and minor: "Q ==> R" "~P ==> R"
681   shows "R"
682 apply (rule excluded_middle [of P, THEN disjE])
683 apply (iprover intro: minor major [THEN mp])+
684 done
686 (*Classical <-> elimination. *)
687 lemma iffCE:
688   assumes major: "P=Q"
689       and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
690   shows "R"
691 apply (rule major [THEN iffE])
692 apply (iprover intro: minor elim: impCE notE)
693 done
695 lemma exCI:
696   assumes "ALL x. ~P(x) ==> P(a)"
697   shows "EX x. P(x)"
698 apply (rule ccontr)
699 apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"])
700 done
703 subsubsection {* Intuitionistic Reasoning *}
705 lemma impE':
706   assumes 1: "P --> Q"
707     and 2: "Q ==> R"
708     and 3: "P --> Q ==> P"
709   shows R
710 proof -
711   from 3 and 1 have P .
712   with 1 have Q by (rule impE)
713   with 2 show R .
714 qed
716 lemma allE':
717   assumes 1: "ALL x. P x"
718     and 2: "P x ==> ALL x. P x ==> Q"
719   shows Q
720 proof -
721   from 1 have "P x" by (rule spec)
722   from this and 1 show Q by (rule 2)
723 qed
725 lemma notE':
726   assumes 1: "~ P"
727     and 2: "~ P ==> P"
728   shows R
729 proof -
730   from 2 and 1 have P .
731   with 1 show R by (rule notE)
732 qed
734 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
735   and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
736   and [Pure.elim 2] = allE notE' impE'
737   and [Pure.intro] = exI disjI2 disjI1
739 lemmas [trans] = trans
740   and [sym] = sym not_sym
741   and [Pure.elim?] = iffD1 iffD2 impE
744 subsubsection {* Atomizing meta-level connectives *}
746 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
747 proof
748   assume "!!x. P x"
749   show "ALL x. P x" by (rule allI)
750 next
751   assume "ALL x. P x"
752   thus "!!x. P x" by (rule allE)
753 qed
755 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
756 proof
757   assume r: "A ==> B"
758   show "A --> B" by (rule impI) (rule r)
759 next
760   assume "A --> B" and A
761   thus B by (rule mp)
762 qed
764 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
765 proof
766   assume r: "A ==> False"
767   show "~A" by (rule notI) (rule r)
768 next
769   assume "~A" and A
770   thus False by (rule notE)
771 qed
773 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
774 proof
775   assume "x == y"
776   show "x = y" by (unfold prems) (rule refl)
777 next
778   assume "x = y"
779   thus "x == y" by (rule eq_reflection)
780 qed
782 lemma atomize_conj [atomize]:
783   includes meta_conjunction_syntax
784   shows "(A && B) == Trueprop (A & B)"
785 proof
786   assume conj: "A && B"
787   show "A & B"
788   proof (rule conjI)
789     from conj show A by (rule conjunctionD1)
790     from conj show B by (rule conjunctionD2)
791   qed
792 next
793   assume conj: "A & B"
794   show "A && B"
795   proof -
796     from conj show A ..
797     from conj show B ..
798   qed
799 qed
801 lemmas [symmetric, rulify] = atomize_all atomize_imp
802   and [symmetric, defn] = atomize_all atomize_imp atomize_eq
805 subsection {* Package setup *}
807 subsubsection {* Fundamental ML bindings *}
809 ML {*
810 structure HOL =
811 struct
812   (*FIXME reduce this to a minimum*)
813   val eq_reflection = thm "eq_reflection";
814   val def_imp_eq = thm "def_imp_eq";
815   val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
816   val ccontr = thm "ccontr";
817   val impI = thm "impI";
818   val impCE = thm "impCE";
819   val notI = thm "notI";
820   val notE = thm "notE";
821   val iffI = thm "iffI";
822   val iffCE = thm "iffCE";
823   val conjI = thm "conjI";
824   val conjE = thm "conjE";
825   val disjCI = thm "disjCI";
826   val disjE = thm "disjE";
827   val TrueI = thm "TrueI";
828   val FalseE = thm "FalseE";
829   val allI = thm "allI";
830   val allE = thm "allE";
831   val exI = thm "exI";
832   val exE = thm "exE";
833   val ex_ex1I = thm "ex_ex1I";
834   val the_equality = thm "the_equality";
835   val mp = thm "mp";
836   val rev_mp = thm "rev_mp"
837   val classical = thm "classical";
838   val subst = thm "subst";
839   val refl = thm "refl";
840   val sym = thm "sym";
841   val trans = thm "trans";
842   val arg_cong = thm "arg_cong";
843   val iffD1 = thm "iffD1";
844   val iffD2 = thm "iffD2";
845   val disjE = thm "disjE";
846   val conjE = thm "conjE";
847   val exE = thm "exE";
848   val contrapos_nn = thm "contrapos_nn";
849   val contrapos_pp = thm "contrapos_pp";
850   val notnotD = thm "notnotD";
851   val conjunct1 = thm "conjunct1";
852   val conjunct2 = thm "conjunct2";
853   val spec = thm "spec";
854   val imp_cong = thm "imp_cong";
855   val the_sym_eq_trivial = thm "the_sym_eq_trivial";
856   val triv_forall_equality = thm "triv_forall_equality";
857   val case_split = thm "case_split_thm";
858 end
859 *}
862 subsubsection {* Classical Reasoner setup *}
864 lemma thin_refl:
865   "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
867 ML {*
868 structure Hypsubst = HypsubstFun(
869 struct
870   structure Simplifier = Simplifier
871   val dest_eq = HOLogic.dest_eq
872   val dest_Trueprop = HOLogic.dest_Trueprop
873   val dest_imp = HOLogic.dest_imp
874   val eq_reflection = HOL.eq_reflection
875   val rev_eq_reflection = HOL.def_imp_eq
876   val imp_intr = HOL.impI
877   val rev_mp = HOL.rev_mp
878   val subst = HOL.subst
879   val sym = HOL.sym
880   val thin_refl = thm "thin_refl";
881 end);
883 structure Classical = ClassicalFun(
884 struct
885   val mp = HOL.mp
886   val not_elim = HOL.notE
887   val classical = HOL.classical
888   val sizef = Drule.size_of_thm
889   val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
890 end);
892 structure BasicClassical: BASIC_CLASSICAL = Classical;
893 *}
895 setup {*
896 let
897   (*prevent substitution on bool*)
898   fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
899     Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
900       (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
901 in
902   Hypsubst.hypsubst_setup
903   #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
904   #> Classical.setup
905   #> ResAtpset.setup
906 end
907 *}
909 declare iffI [intro!]
910   and notI [intro!]
911   and impI [intro!]
912   and disjCI [intro!]
913   and conjI [intro!]
914   and TrueI [intro!]
915   and refl [intro!]
917 declare iffCE [elim!]
918   and FalseE [elim!]
919   and impCE [elim!]
920   and disjE [elim!]
921   and conjE [elim!]
922   and conjE [elim!]
924 declare ex_ex1I [intro!]
925   and allI [intro!]
926   and the_equality [intro]
927   and exI [intro]
929 declare exE [elim!]
930   allE [elim]
932 ML {*
933 structure HOL =
934 struct
936 open HOL;
938 val claset = Classical.claset_of (the_context ());
940 end;
941 *}
943 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
944   apply (erule swap)
945   apply (erule (1) meta_mp)
946   done
948 declare ex_ex1I [rule del, intro! 2]
949   and ex1I [intro]
951 lemmas [intro?] = ext
952   and [elim?] = ex1_implies_ex
954 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
955 lemma alt_ex1E [elim!]:
956   assumes major: "\<exists>!x. P x"
957       and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
958   shows R
959 apply (rule ex1E [OF major])
960 apply (rule prem)
961 apply (tactic "ares_tac [HOL.allI] 1")+
962 apply (tactic "etac (Classical.dup_elim HOL.allE) 1")
963 by iprover
965 ML {*
966 structure Blast = BlastFun(
967 struct
968   type claset = Classical.claset
969   val equality_name = "op ="
970   val not_name = "Not"
971   val notE = HOL.notE
972   val ccontr = HOL.ccontr
973   val contr_tac = Classical.contr_tac
974   val dup_intr = Classical.dup_intr
975   val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
976   val claset	= Classical.claset
977   val rep_cs = Classical.rep_cs
978   val cla_modifiers = Classical.cla_modifiers
979   val cla_meth' = Classical.cla_meth'
980 end);
982 structure HOL =
983 struct
985 open HOL;
987 val Blast_tac = Blast.Blast_tac;
988 val blast_tac = Blast.blast_tac;
990 fun case_tac a = res_inst_tac [("P", a)] case_split;
992 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
993 local
994   fun wrong_prem (Const ("All", _) \$ (Abs (_, _, t))) = wrong_prem t
995     | wrong_prem (Bound _) = true
996     | wrong_prem _ = false;
997   val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
998 in
999   fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
1000   fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
1001 end;
1003 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
1005 fun Trueprop_conv conv ct = (case term_of ct of
1006     Const ("Trueprop", _) \$ _ =>
1007       let val (ct1, ct2) = Thm.dest_comb ct
1008       in Thm.combination (Thm.reflexive ct1) (conv ct2) end
1009   | _ => raise TERM ("Trueprop_conv", []));
1011 fun Equals_conv conv ct = (case term_of ct of
1012     Const ("op =", _) \$ _ \$ _ =>
1013       let
1014         val ((ct1, ct2), ct3) = (apfst Thm.dest_comb o Thm.dest_comb) ct;
1015       in Thm.combination (Thm.combination (Thm.reflexive ct1) (conv ct2)) (conv ct3) end
1016   | _ => conv ct);
1018 fun constrain_op_eq_thms thy thms =
1019   let
1020     fun add_eq (Const ("op =", ty)) =
1021           fold (insert (eq_fst (op =)))
1022             (Term.add_tvarsT ty [])
1023       | add_eq _ =
1024           I
1025     val eqs = fold (fold_aterms add_eq o Thm.prop_of) thms [];
1026     val instT = map (fn (v_i, sort) =>
1027       (Thm.ctyp_of thy (TVar (v_i, sort)),
1028          Thm.ctyp_of thy (TVar (v_i, Sorts.inter_sort (Sign.classes_of thy) (sort, [HOLogic.class_eq]))))) eqs;
1029   in
1030     thms
1031     |> map (Thm.instantiate (instT, []))
1032   end;
1034 end;
1035 *}
1037 setup Blast.setup
1040 subsubsection {* Simplifier *}
1042 lemma eta_contract_eq: "(%s. f s) = f" ..
1044 lemma simp_thms:
1045   shows not_not: "(~ ~ P) = P"
1046   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
1047   and
1048     "(P ~= Q) = (P = (~Q))"
1049     "(P | ~P) = True"    "(~P | P) = True"
1050     "(x = x) = True"
1051   and not_True_eq_False: "(\<not> True) = False"
1052   and not_False_eq_True: "(\<not> False) = True"
1053   and
1054     "(~P) ~= P"  "P ~= (~P)"
1055     "(True=P) = P"
1056   and eq_True: "(P = True) = P"
1057   and "(False=P) = (~P)"
1058   and eq_False: "(P = False) = (\<not> P)"
1059   and
1060     "(True --> P) = P"  "(False --> P) = True"
1061     "(P --> True) = True"  "(P --> P) = True"
1062     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
1063     "(P & True) = P"  "(True & P) = P"
1064     "(P & False) = False"  "(False & P) = False"
1065     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
1066     "(P & ~P) = False"    "(~P & P) = False"
1067     "(P | True) = True"  "(True | P) = True"
1068     "(P | False) = P"  "(False | P) = P"
1069     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
1070     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
1071     -- {* needed for the one-point-rule quantifier simplification procs *}
1072     -- {* essential for termination!! *} and
1073     "!!P. (EX x. x=t & P(x)) = P(t)"
1074     "!!P. (EX x. t=x & P(x)) = P(t)"
1075     "!!P. (ALL x. x=t --> P(x)) = P(t)"
1076     "!!P. (ALL x. t=x --> P(x)) = P(t)"
1077   by (blast, blast, blast, blast, blast, iprover+)
1079 lemma disj_absorb: "(A | A) = A"
1080   by blast
1082 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
1083   by blast
1085 lemma conj_absorb: "(A & A) = A"
1086   by blast
1088 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
1089   by blast
1091 lemma eq_ac:
1092   shows eq_commute: "(a=b) = (b=a)"
1093     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
1094     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
1095 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
1097 lemma conj_comms:
1098   shows conj_commute: "(P&Q) = (Q&P)"
1099     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
1100 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
1102 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
1104 lemma disj_comms:
1105   shows disj_commute: "(P|Q) = (Q|P)"
1106     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
1107 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
1109 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
1111 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
1112 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
1114 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
1115 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
1117 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
1118 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
1119 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
1121 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
1122 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
1123 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
1125 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
1126 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
1128 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
1129   by iprover
1131 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
1132 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
1133 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
1134 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
1135 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
1136 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
1137   by blast
1138 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
1140 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
1143 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
1144   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
1145   -- {* cases boil down to the same thing. *}
1146   by blast
1148 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
1149 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
1150 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
1151 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
1153 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
1154 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
1156 text {*
1157   \medskip The @{text "&"} congruence rule: not included by default!
1158   May slow rewrite proofs down by as much as 50\% *}
1160 lemma conj_cong:
1161     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
1162   by iprover
1164 lemma rev_conj_cong:
1165     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
1166   by iprover
1168 text {* The @{text "|"} congruence rule: not included by default! *}
1170 lemma disj_cong:
1171     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
1172   by blast
1175 text {* \medskip if-then-else rules *}
1177 lemma if_True: "(if True then x else y) = x"
1178   by (unfold if_def) blast
1180 lemma if_False: "(if False then x else y) = y"
1181   by (unfold if_def) blast
1183 lemma if_P: "P ==> (if P then x else y) = x"
1184   by (unfold if_def) blast
1186 lemma if_not_P: "~P ==> (if P then x else y) = y"
1187   by (unfold if_def) blast
1189 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
1190   apply (rule case_split [of Q])
1191    apply (simplesubst if_P)
1192     prefer 3 apply (simplesubst if_not_P, blast+)
1193   done
1195 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
1196 by (simplesubst split_if, blast)
1198 lemmas if_splits = split_if split_if_asm
1200 lemma if_cancel: "(if c then x else x) = x"
1201 by (simplesubst split_if, blast)
1203 lemma if_eq_cancel: "(if x = y then y else x) = x"
1204 by (simplesubst split_if, blast)
1206 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
1207   -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
1208   by (rule split_if)
1210 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
1211   -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
1212   apply (simplesubst split_if, blast)
1213   done
1215 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
1216 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
1218 text {* \medskip let rules for simproc *}
1220 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
1221   by (unfold Let_def)
1223 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
1224   by (unfold Let_def)
1226 text {*
1227   The following copy of the implication operator is useful for
1228   fine-tuning congruence rules.  It instructs the simplifier to simplify
1229   its premise.
1230 *}
1232 constdefs
1233   simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
1234   "simp_implies \<equiv> op ==>"
1236 lemma simp_impliesI:
1237   assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1238   shows "PROP P =simp=> PROP Q"
1239   apply (unfold simp_implies_def)
1240   apply (rule PQ)
1241   apply assumption
1242   done
1244 lemma simp_impliesE:
1245   assumes PQ:"PROP P =simp=> PROP Q"
1246   and P: "PROP P"
1247   and QR: "PROP Q \<Longrightarrow> PROP R"
1248   shows "PROP R"
1249   apply (rule QR)
1250   apply (rule PQ [unfolded simp_implies_def])
1251   apply (rule P)
1252   done
1254 lemma simp_implies_cong:
1255   assumes PP' :"PROP P == PROP P'"
1256   and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
1257   shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
1258 proof (unfold simp_implies_def, rule equal_intr_rule)
1259   assume PQ: "PROP P \<Longrightarrow> PROP Q"
1260   and P': "PROP P'"
1261   from PP' [symmetric] and P' have "PROP P"
1262     by (rule equal_elim_rule1)
1263   hence "PROP Q" by (rule PQ)
1264   with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1265 next
1266   assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1267   and P: "PROP P"
1268   from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1269   hence "PROP Q'" by (rule P'Q')
1270   with P'QQ' [OF P', symmetric] show "PROP Q"
1271     by (rule equal_elim_rule1)
1272 qed
1274 lemma uncurry:
1275   assumes "P \<longrightarrow> Q \<longrightarrow> R"
1276   shows "P \<and> Q \<longrightarrow> R"
1277   using prems by blast
1279 lemma iff_allI:
1280   assumes "\<And>x. P x = Q x"
1281   shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1282   using prems by blast
1284 lemma iff_exI:
1285   assumes "\<And>x. P x = Q x"
1286   shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1287   using prems by blast
1289 lemma all_comm:
1290   "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1291   by blast
1293 lemma ex_comm:
1294   "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1295   by blast
1297 use "simpdata.ML"
1298 setup {*
1299   Simplifier.method_setup Splitter.split_modifiers
1300   #> (fn thy => (change_simpset_of thy (fn _ => HOL.simpset_simprocs); thy))
1301   #> Splitter.setup
1302   #> Clasimp.setup
1303   #> EqSubst.setup
1304 *}
1306 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1307 proof
1308   assume prem: "True \<Longrightarrow> PROP P"
1309   from prem [OF TrueI] show "PROP P" .
1310 next
1311   assume "PROP P"
1312   show "PROP P" .
1313 qed
1315 lemma ex_simps:
1316   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
1317   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
1318   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
1319   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
1320   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
1321   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
1322   -- {* Miniscoping: pushing in existential quantifiers. *}
1323   by (iprover | blast)+
1325 lemma all_simps:
1326   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
1327   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
1328   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
1329   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
1330   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
1331   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
1332   -- {* Miniscoping: pushing in universal quantifiers. *}
1333   by (iprover | blast)+
1335 declare triv_forall_equality [simp] (*prunes params*)
1336   and True_implies_equals [simp] (*prune asms `True'*)
1337   and if_True [simp]
1338   and if_False [simp]
1339   and if_cancel [simp]
1340   and if_eq_cancel [simp]
1341   and imp_disjL [simp]
1342   (*In general it seems wrong to add distributive laws by default: they
1343     might cause exponential blow-up.  But imp_disjL has been in for a while
1344     and cannot be removed without affecting existing proofs.  Moreover,
1345     rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
1346     grounds that it allows simplification of R in the two cases.*)
1347   and conj_assoc [simp]
1348   and disj_assoc [simp]
1349   and de_Morgan_conj [simp]
1350   and de_Morgan_disj [simp]
1351   and imp_disj1 [simp]
1352   and imp_disj2 [simp]
1353   and not_imp [simp]
1354   and disj_not1 [simp]
1355   and not_all [simp]
1356   and not_ex [simp]
1357   and cases_simp [simp]
1358   and the_eq_trivial [simp]
1359   and the_sym_eq_trivial [simp]
1360   and ex_simps [simp]
1361   and all_simps [simp]
1362   and simp_thms [simp]
1363   and imp_cong [cong]
1364   and simp_implies_cong [cong]
1365   and split_if [split]
1367 ML {*
1368 structure HOL =
1369 struct
1371 open HOL;
1373 val simpset = Simplifier.simpset_of (the_context ());
1375 end;
1376 *}
1378 text {* Simplifies x assuming c and y assuming ~c *}
1379 lemma if_cong:
1380   assumes "b = c"
1381       and "c \<Longrightarrow> x = u"
1382       and "\<not> c \<Longrightarrow> y = v"
1383   shows "(if b then x else y) = (if c then u else v)"
1384   unfolding if_def using prems by simp
1386 text {* Prevents simplification of x and y:
1387   faster and allows the execution of functional programs. *}
1388 lemma if_weak_cong [cong]:
1389   assumes "b = c"
1390   shows "(if b then x else y) = (if c then x else y)"
1391   using prems by (rule arg_cong)
1393 text {* Prevents simplification of t: much faster *}
1394 lemma let_weak_cong:
1395   assumes "a = b"
1396   shows "(let x = a in t x) = (let x = b in t x)"
1397   using prems by (rule arg_cong)
1399 text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
1400 lemma eq_cong2:
1401   assumes "u = u'"
1402   shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1403   using prems by simp
1405 lemma if_distrib:
1406   "f (if c then x else y) = (if c then f x else f y)"
1407   by simp
1409 text {* For @{text expand_case_tac} *}
1410 lemma expand_case:
1411   assumes "P \<Longrightarrow> Q True"
1412       and "~P \<Longrightarrow> Q False"
1413   shows "Q P"
1414 proof (tactic {* HOL.case_tac "P" 1 *})
1415   assume P
1416   then show "Q P" by simp
1417 next
1418   assume "\<not> P"
1419   then have "P = False" by simp
1420   with prems show "Q P" by simp
1421 qed
1423 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
1424   side of an equality.  Used in @{text "{Integ,Real}/simproc.ML"} *}
1425 lemma restrict_to_left:
1426   assumes "x = y"
1427   shows "(x = z) = (y = z)"
1428   using prems by simp
1431 subsubsection {* Generic cases and induction *}
1433 text {* Rule projections: *}
1435 ML {*
1436 structure ProjectRule = ProjectRuleFun
1437 (struct
1438   val conjunct1 = thm "conjunct1";
1439   val conjunct2 = thm "conjunct2";
1440   val mp = thm "mp";
1441 end)
1442 *}
1444 constdefs
1445   induct_forall where "induct_forall P == \<forall>x. P x"
1446   induct_implies where "induct_implies A B == A \<longrightarrow> B"
1447   induct_equal where "induct_equal x y == x = y"
1448   induct_conj where "induct_conj A B == A \<and> B"
1450 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
1451   by (unfold atomize_all induct_forall_def)
1453 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
1454   by (unfold atomize_imp induct_implies_def)
1456 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
1457   by (unfold atomize_eq induct_equal_def)
1459 lemma induct_conj_eq:
1460   includes meta_conjunction_syntax
1461   shows "(A && B) == Trueprop (induct_conj A B)"
1462   by (unfold atomize_conj induct_conj_def)
1464 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
1465 lemmas induct_rulify [symmetric, standard] = induct_atomize
1466 lemmas induct_rulify_fallback =
1467   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1470 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1471     induct_conj (induct_forall A) (induct_forall B)"
1472   by (unfold induct_forall_def induct_conj_def) iprover
1474 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1475     induct_conj (induct_implies C A) (induct_implies C B)"
1476   by (unfold induct_implies_def induct_conj_def) iprover
1478 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
1479 proof
1480   assume r: "induct_conj A B ==> PROP C" and A B
1481   show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
1482 next
1483   assume r: "A ==> B ==> PROP C" and "induct_conj A B"
1484   show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
1485 qed
1487 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1489 hide const induct_forall induct_implies induct_equal induct_conj
1491 text {* Method setup. *}
1493 ML {*
1494   structure InductMethod = InductMethodFun
1495   (struct
1496     val cases_default = thm "case_split"
1497     val atomize = thms "induct_atomize"
1498     val rulify = thms "induct_rulify"
1499     val rulify_fallback = thms "induct_rulify_fallback"
1500   end);
1501 *}
1503 setup InductMethod.setup
1507 subsection {* Other simple lemmas and lemma duplicates *}
1509 lemmas eq_sym_conv = eq_commute
1510 lemmas if_def2 = if_bool_eq_conj
1512 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
1513   by blast+
1515 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
1516   apply (rule iffI)
1517   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
1518   apply (fast dest!: theI')
1519   apply (fast intro: ext the1_equality [symmetric])
1520   apply (erule ex1E)
1521   apply (rule allI)
1522   apply (rule ex1I)
1523   apply (erule spec)
1524   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
1525   apply (erule impE)
1526   apply (rule allI)
1527   apply (rule_tac P = "xa = x" in case_split_thm)
1528   apply (drule_tac  x = x in fun_cong, simp_all)
1529   done
1531 text {* Needs only HOL-lemmas *}
1532 lemma mk_left_commute:
1533   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
1534           c: "\<And>x y. f x y = f y x"
1535   shows "f x (f y z) = f y (f x z)"
1536   by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
1538 end