src/HOL/HOL.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62151 dc4c9748a86e child 62390 842917225d56 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
1 (*  Title:      HOL/HOL.thy
2     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
3 *)
5 section \<open>The basis of Higher-Order Logic\<close>
7 theory HOL
8 imports Pure "~~/src/Tools/Code_Generator"
9 keywords
10   "try" "solve_direct" "quickcheck" "print_coercions" "print_claset"
11     "print_induct_rules" :: diag and
12   "quickcheck_params" :: thy_decl
13 begin
15 ML_file "~~/src/Tools/misc_legacy.ML"
16 ML_file "~~/src/Tools/try.ML"
17 ML_file "~~/src/Tools/quickcheck.ML"
18 ML_file "~~/src/Tools/solve_direct.ML"
19 ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
20 ML_file "~~/src/Tools/IsaPlanner/isand.ML"
21 ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
22 ML_file "~~/src/Provers/hypsubst.ML"
23 ML_file "~~/src/Provers/splitter.ML"
24 ML_file "~~/src/Provers/classical.ML"
25 ML_file "~~/src/Provers/blast.ML"
26 ML_file "~~/src/Provers/clasimp.ML"
27 ML_file "~~/src/Tools/eqsubst.ML"
28 ML_file "~~/src/Provers/quantifier1.ML"
29 ML_file "~~/src/Tools/atomize_elim.ML"
30 ML_file "~~/src/Tools/cong_tac.ML"
31 ML_file "~~/src/Tools/intuitionistic.ML" setup \<open>Intuitionistic.method_setup @{binding iprover}\<close>
32 ML_file "~~/src/Tools/project_rule.ML"
33 ML_file "~~/src/Tools/subtyping.ML"
34 ML_file "~~/src/Tools/case_product.ML"
37 ML \<open>Plugin_Name.declare_setup @{binding extraction}\<close>
39 ML \<open>
40   Plugin_Name.declare_setup @{binding quickcheck_random};
41   Plugin_Name.declare_setup @{binding quickcheck_exhaustive};
42   Plugin_Name.declare_setup @{binding quickcheck_bounded_forall};
43   Plugin_Name.declare_setup @{binding quickcheck_full_exhaustive};
44   Plugin_Name.declare_setup @{binding quickcheck_narrowing};
45 \<close>
46 ML \<open>
47   Plugin_Name.define_setup @{binding quickcheck}
48    [@{plugin quickcheck_exhaustive},
49     @{plugin quickcheck_random},
50     @{plugin quickcheck_bounded_forall},
51     @{plugin quickcheck_full_exhaustive},
52     @{plugin quickcheck_narrowing}]
53 \<close>
56 subsection \<open>Primitive logic\<close>
58 subsubsection \<open>Core syntax\<close>
60 setup \<open>Axclass.class_axiomatization (@{binding type}, [])\<close>
61 default_sort type
62 setup \<open>Object_Logic.add_base_sort @{sort type}\<close>
64 axiomatization where fun_arity: "OFCLASS('a \<Rightarrow> 'b, type_class)"
65 instance "fun" :: (type, type) type by (rule fun_arity)
67 axiomatization where itself_arity: "OFCLASS('a itself, type_class)"
68 instance itself :: (type) type by (rule itself_arity)
70 typedecl bool
72 judgment Trueprop :: "bool \<Rightarrow> prop"  ("(_)" 5)
74 axiomatization implies :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longrightarrow>" 25)
75   and eq :: "['a, 'a] \<Rightarrow> bool"  (infixl "=" 50)
76   and The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
79 subsubsection \<open>Defined connectives and quantifiers\<close>
81 definition True :: bool
82   where "True \<equiv> ((\<lambda>x::bool. x) = (\<lambda>x. x))"
84 definition All :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<forall>" 10)
85   where "All P \<equiv> (P = (\<lambda>x. True))"
87 definition Ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>" 10)
88   where "Ex P \<equiv> \<forall>Q. (\<forall>x. P x \<longrightarrow> Q) \<longrightarrow> Q"
90 definition False :: bool
91   where "False \<equiv> (\<forall>P. P)"
93 definition Not :: "bool \<Rightarrow> bool"  ("\<not> _"  40)
94   where not_def: "\<not> P \<equiv> P \<longrightarrow> False"
96 definition conj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<and>" 35)
97   where and_def: "P \<and> Q \<equiv> \<forall>R. (P \<longrightarrow> Q \<longrightarrow> R) \<longrightarrow> R"
99 definition disj :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<or>" 30)
100   where or_def: "P \<or> Q \<equiv> \<forall>R. (P \<longrightarrow> R) \<longrightarrow> (Q \<longrightarrow> R) \<longrightarrow> R"
102 definition Ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>!" 10)
103   where "Ex1 P \<equiv> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y = x)"
106 subsubsection \<open>Additional concrete syntax\<close>
108 abbreviation not_equal :: "['a, 'a] \<Rightarrow> bool"  (infixl "\<noteq>" 50)
109   where "x \<noteq> y \<equiv> \<not> (x = y)"
111 notation (output)
112   eq  (infix "=" 50) and
113   not_equal  (infix "\<noteq>" 50)
115 notation (ASCII output)
116   not_equal  (infix "~=" 50)
118 notation (ASCII)
119   Not  ("~ _"  40) and
120   conj  (infixr "&" 35) and
121   disj  (infixr "|" 30) and
122   implies  (infixr "-->" 25) and
123   not_equal  (infixl "~=" 50)
125 abbreviation (iff)
126   iff :: "[bool, bool] \<Rightarrow> bool"  (infixr "\<longleftrightarrow>" 25)
127   where "A \<longleftrightarrow> B \<equiv> A = B"
129 syntax "_The" :: "[pttrn, bool] \<Rightarrow> 'a"  ("(3THE _./ _)" [0, 10] 10)
130 translations "THE x. P" \<rightleftharpoons> "CONST The (\<lambda>x. P)"
131 print_translation \<open>
132   [(@{const_syntax The}, fn _ => fn [Abs abs] =>
133       let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
134       in Syntax.const @{syntax_const "_The"} \$ x \$ t end)]
135 \<close>  \<comment> \<open>To avoid eta-contraction of body\<close>
137 nonterminal letbinds and letbind
138 syntax
139   "_bind"       :: "[pttrn, 'a] \<Rightarrow> letbind"              ("(2_ =/ _)" 10)
140   ""            :: "letbind \<Rightarrow> letbinds"                 ("_")
141   "_binds"      :: "[letbind, letbinds] \<Rightarrow> letbinds"     ("_;/ _")
142   "_Let"        :: "[letbinds, 'a] \<Rightarrow> 'a"                ("(let (_)/ in (_))" [0, 10] 10)
144 nonterminal case_syn and cases_syn
145 syntax
146   "_case_syntax" :: "['a, cases_syn] \<Rightarrow> 'b"  ("(case _ of/ _)" 10)
147   "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ \<Rightarrow>/ _)" 10)
148   "" :: "case_syn \<Rightarrow> cases_syn"  ("_")
149   "_case2" :: "[case_syn, cases_syn] \<Rightarrow> cases_syn"  ("_/ | _")
150 syntax (ASCII)
151   "_case1" :: "['a, 'b] \<Rightarrow> case_syn"  ("(2_ =>/ _)" 10)
153 notation (ASCII)
154   All  (binder "ALL " 10) and
155   Ex  (binder "EX " 10) and
156   Ex1  (binder "EX! " 10)
158 notation (HOL)
159   All  (binder "! " 10) and
160   Ex  (binder "? " 10) and
161   Ex1  (binder "?! " 10)
164 subsubsection \<open>Axioms and basic definitions\<close>
166 axiomatization where
167   refl: "t = (t::'a)" and
168   subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t" and
169   ext: "(\<And>x::'a. (f x ::'b) = g x) \<Longrightarrow> (\<lambda>x. f x) = (\<lambda>x. g x)"
170     \<comment> \<open>Extensionality is built into the meta-logic, and this rule expresses
171          a related property.  It is an eta-expanded version of the traditional
172          rule, and similar to the ABS rule of HOL\<close> and
174   the_eq_trivial: "(THE x. x = a) = (a::'a)"
176 axiomatization where
177   impI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q" and
178   mp: "\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q" and
180   iff: "(P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P) \<longrightarrow> (P = Q)" and
181   True_or_False: "(P = True) \<or> (P = False)"
183 definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
184   where "If P x y \<equiv> (THE z::'a. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y))"
186 definition Let :: "'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"
187   where "Let s f \<equiv> f s"
189 translations
190   "_Let (_binds b bs) e"  \<rightleftharpoons> "_Let b (_Let bs e)"
191   "let x = a in e"        \<rightleftharpoons> "CONST Let a (\<lambda>x. e)"
193 axiomatization undefined :: 'a
195 class default = fixes default :: 'a
198 subsection \<open>Fundamental rules\<close>
200 subsubsection \<open>Equality\<close>
202 lemma sym: "s = t \<Longrightarrow> t = s"
203   by (erule subst) (rule refl)
205 lemma ssubst: "t = s \<Longrightarrow> P s \<Longrightarrow> P t"
206   by (drule sym) (erule subst)
208 lemma trans: "\<lbrakk>r = s; s = t\<rbrakk> \<Longrightarrow> r = t"
209   by (erule subst)
211 lemma trans_sym [Pure.elim?]: "r = s \<Longrightarrow> t = s \<Longrightarrow> r = t"
212   by (rule trans [OF _ sym])
214 lemma meta_eq_to_obj_eq:
215   assumes meq: "A \<equiv> B"
216   shows "A = B"
217   by (unfold meq) (rule refl)
219 text \<open>Useful with \<open>erule\<close> for proving equalities from known equalities.\<close>
220      (* a = b
221         |   |
222         c = d   *)
223 lemma box_equals: "\<lbrakk>a = b; a = c; b = d\<rbrakk> \<Longrightarrow> c = d"
224 apply (rule trans)
225 apply (rule trans)
226 apply (rule sym)
227 apply assumption+
228 done
230 text \<open>For calculational reasoning:\<close>
232 lemma forw_subst: "a = b \<Longrightarrow> P b \<Longrightarrow> P a"
233   by (rule ssubst)
235 lemma back_subst: "P a \<Longrightarrow> a = b \<Longrightarrow> P b"
236   by (rule subst)
239 subsubsection \<open>Congruence rules for application\<close>
241 text \<open>Similar to \<open>AP_THM\<close> in Gordon's HOL.\<close>
242 lemma fun_cong: "(f :: 'a \<Rightarrow> 'b) = g \<Longrightarrow> f x = g x"
243 apply (erule subst)
244 apply (rule refl)
245 done
247 text \<open>Similar to \<open>AP_TERM\<close> in Gordon's HOL and FOL's \<open>subst_context\<close>.\<close>
248 lemma arg_cong: "x = y \<Longrightarrow> f x = f y"
249 apply (erule subst)
250 apply (rule refl)
251 done
253 lemma arg_cong2: "\<lbrakk>a = b; c = d\<rbrakk> \<Longrightarrow> f a c = f b d"
254 apply (erule ssubst)+
255 apply (rule refl)
256 done
258 lemma cong: "\<lbrakk>f = g; (x::'a) = y\<rbrakk> \<Longrightarrow> f x = g y"
259 apply (erule subst)+
260 apply (rule refl)
261 done
263 ML \<open>fun cong_tac ctxt = Cong_Tac.cong_tac ctxt @{thm cong}\<close>
266 subsubsection \<open>Equality of booleans -- iff\<close>
268 lemma iffI: assumes "P \<Longrightarrow> Q" and "Q \<Longrightarrow> P" shows "P = Q"
269   by (iprover intro: iff [THEN mp, THEN mp] impI assms)
271 lemma iffD2: "\<lbrakk>P = Q; Q\<rbrakk> \<Longrightarrow> P"
272   by (erule ssubst)
274 lemma rev_iffD2: "\<lbrakk>Q; P = Q\<rbrakk> \<Longrightarrow> P"
275   by (erule iffD2)
277 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
278   by (drule sym) (rule iffD2)
280 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
281   by (drule sym) (rule rev_iffD2)
283 lemma iffE:
284   assumes major: "P = Q"
285     and minor: "\<lbrakk>P \<longrightarrow> Q; Q \<longrightarrow> P\<rbrakk> \<Longrightarrow> R"
286   shows R
287   by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
290 subsubsection \<open>True\<close>
292 lemma TrueI: "True"
293   unfolding True_def by (rule refl)
295 lemma eqTrueI: "P \<Longrightarrow> P = True"
296   by (iprover intro: iffI TrueI)
298 lemma eqTrueE: "P = True \<Longrightarrow> P"
299   by (erule iffD2) (rule TrueI)
302 subsubsection \<open>Universal quantifier\<close>
304 lemma allI: assumes "\<And>x::'a. P x" shows "\<forall>x. P x"
305   unfolding All_def by (iprover intro: ext eqTrueI assms)
307 lemma spec: "\<forall>x::'a. P x \<Longrightarrow> P x"
308 apply (unfold All_def)
309 apply (rule eqTrueE)
310 apply (erule fun_cong)
311 done
313 lemma allE:
314   assumes major: "\<forall>x. P x"
315     and minor: "P x \<Longrightarrow> R"
316   shows R
317   by (iprover intro: minor major [THEN spec])
319 lemma all_dupE:
320   assumes major: "\<forall>x. P x"
321     and minor: "\<lbrakk>P x; \<forall>x. P x\<rbrakk> \<Longrightarrow> R"
322   shows R
323   by (iprover intro: minor major major [THEN spec])
326 subsubsection \<open>False\<close>
328 text \<open>
329   Depends upon \<open>spec\<close>; it is impossible to do propositional
330   logic before quantifiers!
331 \<close>
333 lemma FalseE: "False \<Longrightarrow> P"
334   apply (unfold False_def)
335   apply (erule spec)
336   done
338 lemma False_neq_True: "False = True \<Longrightarrow> P"
339   by (erule eqTrueE [THEN FalseE])
342 subsubsection \<open>Negation\<close>
344 lemma notI:
345   assumes "P \<Longrightarrow> False"
346   shows "\<not> P"
347   apply (unfold not_def)
348   apply (iprover intro: impI assms)
349   done
351 lemma False_not_True: "False \<noteq> True"
352   apply (rule notI)
353   apply (erule False_neq_True)
354   done
356 lemma True_not_False: "True \<noteq> False"
357   apply (rule notI)
358   apply (drule sym)
359   apply (erule False_neq_True)
360   done
362 lemma notE: "\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R"
363   apply (unfold not_def)
364   apply (erule mp [THEN FalseE])
365   apply assumption
366   done
368 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
369   by (erule notE [THEN notI]) (erule meta_mp)
372 subsubsection \<open>Implication\<close>
374 lemma impE:
375   assumes "P \<longrightarrow> Q" P "Q \<Longrightarrow> R"
376   shows R
377 by (iprover intro: assms mp)
379 (* Reduces Q to P \<longrightarrow> Q, allowing substitution in P. *)
380 lemma rev_mp: "\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
381 by (iprover intro: mp)
383 lemma contrapos_nn:
384   assumes major: "\<not> Q"
385       and minor: "P \<Longrightarrow> Q"
386   shows "\<not> P"
387 by (iprover intro: notI minor major [THEN notE])
389 (*not used at all, but we already have the other 3 combinations *)
390 lemma contrapos_pn:
391   assumes major: "Q"
392       and minor: "P \<Longrightarrow> \<not> Q"
393   shows "\<not> P"
394 by (iprover intro: notI minor major notE)
396 lemma not_sym: "t \<noteq> s \<Longrightarrow> s \<noteq> t"
397   by (erule contrapos_nn) (erule sym)
399 lemma eq_neq_eq_imp_neq: "\<lbrakk>x = a; a \<noteq> b; b = y\<rbrakk> \<Longrightarrow> x \<noteq> y"
400   by (erule subst, erule ssubst, assumption)
403 subsubsection \<open>Existential quantifier\<close>
405 lemma exI: "P x \<Longrightarrow> \<exists>x::'a. P x"
406 apply (unfold Ex_def)
407 apply (iprover intro: allI allE impI mp)
408 done
410 lemma exE:
411   assumes major: "\<exists>x::'a. P x"
412       and minor: "\<And>x. P x \<Longrightarrow> Q"
413   shows "Q"
414 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
415 apply (iprover intro: impI [THEN allI] minor)
416 done
419 subsubsection \<open>Conjunction\<close>
421 lemma conjI: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> P \<and> Q"
422 apply (unfold and_def)
423 apply (iprover intro: impI [THEN allI] mp)
424 done
426 lemma conjunct1: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> P"
427 apply (unfold and_def)
428 apply (iprover intro: impI dest: spec mp)
429 done
431 lemma conjunct2: "\<lbrakk>P \<and> Q\<rbrakk> \<Longrightarrow> Q"
432 apply (unfold and_def)
433 apply (iprover intro: impI dest: spec mp)
434 done
436 lemma conjE:
437   assumes major: "P \<and> Q"
438       and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R"
439   shows R
440 apply (rule minor)
441 apply (rule major [THEN conjunct1])
442 apply (rule major [THEN conjunct2])
443 done
445 lemma context_conjI:
446   assumes P "P \<Longrightarrow> Q" shows "P \<and> Q"
447 by (iprover intro: conjI assms)
450 subsubsection \<open>Disjunction\<close>
452 lemma disjI1: "P \<Longrightarrow> P \<or> Q"
453 apply (unfold or_def)
454 apply (iprover intro: allI impI mp)
455 done
457 lemma disjI2: "Q \<Longrightarrow> P \<or> Q"
458 apply (unfold or_def)
459 apply (iprover intro: allI impI mp)
460 done
462 lemma disjE:
463   assumes major: "P \<or> Q"
464       and minorP: "P \<Longrightarrow> R"
465       and minorQ: "Q \<Longrightarrow> R"
466   shows R
467 by (iprover intro: minorP minorQ impI
468                  major [unfolded or_def, THEN spec, THEN mp, THEN mp])
471 subsubsection \<open>Classical logic\<close>
473 lemma classical:
474   assumes prem: "\<not> P \<Longrightarrow> P"
475   shows P
476 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
477 apply assumption
478 apply (rule notI [THEN prem, THEN eqTrueI])
479 apply (erule subst)
480 apply assumption
481 done
483 lemmas ccontr = FalseE [THEN classical]
485 (*notE with premises exchanged; it discharges \<not> R so that it can be used to
486   make elimination rules*)
487 lemma rev_notE:
488   assumes premp: P
489       and premnot: "\<not> R \<Longrightarrow> \<not> P"
490   shows R
491 apply (rule ccontr)
492 apply (erule notE [OF premnot premp])
493 done
495 (*Double negation law*)
496 lemma notnotD: "\<not>\<not> P \<Longrightarrow> P"
497 apply (rule classical)
498 apply (erule notE)
499 apply assumption
500 done
502 lemma contrapos_pp:
503   assumes p1: Q
504       and p2: "\<not> P \<Longrightarrow> \<not> Q"
505   shows P
506 by (iprover intro: classical p1 p2 notE)
509 subsubsection \<open>Unique existence\<close>
511 lemma ex1I:
512   assumes "P a" "\<And>x. P x \<Longrightarrow> x = a"
513   shows "\<exists>!x. P x"
514 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
516 text\<open>Sometimes easier to use: the premises have no shared variables.  Safe!\<close>
517 lemma ex_ex1I:
518   assumes ex_prem: "\<exists>x. P x"
519       and eq: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y"
520   shows "\<exists>!x. P x"
521 by (iprover intro: ex_prem [THEN exE] ex1I eq)
523 lemma ex1E:
524   assumes major: "\<exists>!x. P x"
525       and minor: "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R"
526   shows R
527 apply (rule major [unfolded Ex1_def, THEN exE])
528 apply (erule conjE)
529 apply (iprover intro: minor)
530 done
532 lemma ex1_implies_ex: "\<exists>!x. P x \<Longrightarrow> \<exists>x. P x"
533 apply (erule ex1E)
534 apply (rule exI)
535 apply assumption
536 done
539 subsubsection \<open>Classical intro rules for disjunction and existential quantifiers\<close>
541 lemma disjCI:
542   assumes "\<not> Q \<Longrightarrow> P" shows "P \<or> Q"
543 apply (rule classical)
544 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
545 done
547 lemma excluded_middle: "\<not> P \<or> P"
548 by (iprover intro: disjCI)
550 text \<open>
551   case distinction as a natural deduction rule.
552   Note that @{term "\<not> P"} is the second case, not the first
553 \<close>
554 lemma case_split [case_names True False]:
555   assumes prem1: "P \<Longrightarrow> Q"
556       and prem2: "\<not> P \<Longrightarrow> Q"
557   shows Q
558 apply (rule excluded_middle [THEN disjE])
559 apply (erule prem2)
560 apply (erule prem1)
561 done
563 (*Classical implies (\<longrightarrow>) elimination. *)
564 lemma impCE:
565   assumes major: "P \<longrightarrow> Q"
566       and minor: "\<not> P \<Longrightarrow> R" "Q \<Longrightarrow> R"
567   shows R
568 apply (rule excluded_middle [of P, THEN disjE])
569 apply (iprover intro: minor major [THEN mp])+
570 done
572 (*This version of \<longrightarrow> elimination works on Q before P.  It works best for
573   those cases in which P holds "almost everywhere".  Can't install as
574   default: would break old proofs.*)
575 lemma impCE':
576   assumes major: "P \<longrightarrow> Q"
577       and minor: "Q \<Longrightarrow> R" "\<not> P \<Longrightarrow> R"
578   shows R
579 apply (rule excluded_middle [of P, THEN disjE])
580 apply (iprover intro: minor major [THEN mp])+
581 done
583 (*Classical <-> elimination. *)
584 lemma iffCE:
585   assumes major: "P = Q"
586       and minor: "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R" "\<lbrakk>\<not> P; \<not> Q\<rbrakk> \<Longrightarrow> R"
587   shows R
588 apply (rule major [THEN iffE])
589 apply (iprover intro: minor elim: impCE notE)
590 done
592 lemma exCI:
593   assumes "\<forall>x. \<not> P x \<Longrightarrow> P a"
594   shows "\<exists>x. P x"
595 apply (rule ccontr)
596 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
597 done
600 subsubsection \<open>Intuitionistic Reasoning\<close>
602 lemma impE':
603   assumes 1: "P \<longrightarrow> Q"
604     and 2: "Q \<Longrightarrow> R"
605     and 3: "P \<longrightarrow> Q \<Longrightarrow> P"
606   shows R
607 proof -
608   from 3 and 1 have P .
609   with 1 have Q by (rule impE)
610   with 2 show R .
611 qed
613 lemma allE':
614   assumes 1: "\<forall>x. P x"
615     and 2: "P x \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q"
616   shows Q
617 proof -
618   from 1 have "P x" by (rule spec)
619   from this and 1 show Q by (rule 2)
620 qed
622 lemma notE':
623   assumes 1: "\<not> P"
624     and 2: "\<not> P \<Longrightarrow> P"
625   shows R
626 proof -
627   from 2 and 1 have P .
628   with 1 show R by (rule notE)
629 qed
631 lemma TrueE: "True \<Longrightarrow> P \<Longrightarrow> P" .
632 lemma notFalseE: "\<not> False \<Longrightarrow> P \<Longrightarrow> P" .
634 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
635   and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
636   and [Pure.elim 2] = allE notE' impE'
637   and [Pure.intro] = exI disjI2 disjI1
639 lemmas [trans] = trans
640   and [sym] = sym not_sym
641   and [Pure.elim?] = iffD1 iffD2 impE
644 subsubsection \<open>Atomizing meta-level connectives\<close>
646 axiomatization where
647   eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
649 lemma atomize_all [atomize]: "(\<And>x. P x) \<equiv> Trueprop (\<forall>x. P x)"
650 proof
651   assume "\<And>x. P x"
652   then show "\<forall>x. P x" ..
653 next
654   assume "\<forall>x. P x"
655   then show "\<And>x. P x" by (rule allE)
656 qed
658 lemma atomize_imp [atomize]: "(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)"
659 proof
660   assume r: "A \<Longrightarrow> B"
661   show "A \<longrightarrow> B" by (rule impI) (rule r)
662 next
663   assume "A \<longrightarrow> B" and A
664   then show B by (rule mp)
665 qed
667 lemma atomize_not: "(A \<Longrightarrow> False) \<equiv> Trueprop (\<not> A)"
668 proof
669   assume r: "A \<Longrightarrow> False"
670   show "\<not> A" by (rule notI) (rule r)
671 next
672   assume "\<not> A" and A
673   then show False by (rule notE)
674 qed
676 lemma atomize_eq [atomize, code]: "(x \<equiv> y) \<equiv> Trueprop (x = y)"
677 proof
678   assume "x \<equiv> y"
679   show "x = y" by (unfold \<open>x \<equiv> y\<close>) (rule refl)
680 next
681   assume "x = y"
682   then show "x \<equiv> y" by (rule eq_reflection)
683 qed
685 lemma atomize_conj [atomize]: "(A &&& B) \<equiv> Trueprop (A \<and> B)"
686 proof
687   assume conj: "A &&& B"
688   show "A \<and> B"
689   proof (rule conjI)
690     from conj show A by (rule conjunctionD1)
691     from conj show B by (rule conjunctionD2)
692   qed
693 next
694   assume conj: "A \<and> B"
695   show "A &&& B"
696   proof -
697     from conj show A ..
698     from conj show B ..
699   qed
700 qed
702 lemmas [symmetric, rulify] = atomize_all atomize_imp
703   and [symmetric, defn] = atomize_all atomize_imp atomize_eq
706 subsubsection \<open>Atomizing elimination rules\<close>
708 lemma atomize_exL[atomize_elim]: "(\<And>x. P x \<Longrightarrow> Q) \<equiv> ((\<exists>x. P x) \<Longrightarrow> Q)"
709   by rule iprover+
711 lemma atomize_conjL[atomize_elim]: "(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)"
712   by rule iprover+
714 lemma atomize_disjL[atomize_elim]: "((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)"
715   by rule iprover+
717 lemma atomize_elimL[atomize_elim]: "(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop A" ..
720 subsection \<open>Package setup\<close>
722 ML_file "Tools/hologic.ML"
725 subsubsection \<open>Sledgehammer setup\<close>
727 text \<open>
728 Theorems blacklisted to Sledgehammer. These theorems typically produce clauses
729 that are prolific (match too many equality or membership literals) and relate to
730 seldom-used facts. Some duplicate other rules.
731 \<close>
733 named_theorems no_atp "theorems that should be filtered out by Sledgehammer"
736 subsubsection \<open>Classical Reasoner setup\<close>
738 lemma imp_elim: "P \<longrightarrow> Q \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
739   by (rule classical) iprover
741 lemma swap: "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"
742   by (rule classical) iprover
744 lemma thin_refl: "\<And>X. \<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W" .
746 ML \<open>
747 structure Hypsubst = Hypsubst
748 (
749   val dest_eq = HOLogic.dest_eq
750   val dest_Trueprop = HOLogic.dest_Trueprop
751   val dest_imp = HOLogic.dest_imp
752   val eq_reflection = @{thm eq_reflection}
753   val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
754   val imp_intr = @{thm impI}
755   val rev_mp = @{thm rev_mp}
756   val subst = @{thm subst}
757   val sym = @{thm sym}
758   val thin_refl = @{thm thin_refl};
759 );
760 open Hypsubst;
762 structure Classical = Classical
763 (
764   val imp_elim = @{thm imp_elim}
765   val not_elim = @{thm notE}
766   val swap = @{thm swap}
767   val classical = @{thm classical}
768   val sizef = Drule.size_of_thm
769   val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
770 );
772 structure Basic_Classical: BASIC_CLASSICAL = Classical;
773 open Basic_Classical;
774 \<close>
776 setup \<open>
777   (*prevent substitution on bool*)
778   let
779     fun non_bool_eq (@{const_name HOL.eq}, Type (_, [T, _])) = T <> @{typ bool}
780       | non_bool_eq _ = false;
781     fun hyp_subst_tac' ctxt =
782       SUBGOAL (fn (goal, i) =>
783         if Term.exists_Const non_bool_eq goal
784         then Hypsubst.hyp_subst_tac ctxt i
785         else no_tac);
786   in
787     Context_Rules.addSWrapper (fn ctxt => fn tac => hyp_subst_tac' ctxt ORELSE' tac)
788   end
789 \<close>
791 declare iffI [intro!]
792   and notI [intro!]
793   and impI [intro!]
794   and disjCI [intro!]
795   and conjI [intro!]
796   and TrueI [intro!]
797   and refl [intro!]
799 declare iffCE [elim!]
800   and FalseE [elim!]
801   and impCE [elim!]
802   and disjE [elim!]
803   and conjE [elim!]
805 declare ex_ex1I [intro!]
806   and allI [intro!]
807   and exI [intro]
809 declare exE [elim!]
810   allE [elim]
812 ML \<open>val HOL_cs = claset_of @{context}\<close>
814 lemma contrapos_np: "\<not> Q \<Longrightarrow> (\<not> P \<Longrightarrow> Q) \<Longrightarrow> P"
815   apply (erule swap)
816   apply (erule (1) meta_mp)
817   done
819 declare ex_ex1I [rule del, intro! 2]
820   and ex1I [intro]
822 declare ext [intro]
824 lemmas [intro?] = ext
825   and [elim?] = ex1_implies_ex
827 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
828 lemma alt_ex1E [elim!]:
829   assumes major: "\<exists>!x. P x"
830       and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
831   shows R
832 apply (rule ex1E [OF major])
833 apply (rule prem)
834 apply assumption
835 apply (rule allI)+
836 apply (tactic \<open>eresolve_tac @{context} [Classical.dup_elim @{context} @{thm allE}] 1\<close>)
837 apply iprover
838 done
840 ML \<open>
841   structure Blast = Blast
842   (
843     structure Classical = Classical
844     val Trueprop_const = dest_Const @{const Trueprop}
845     val equality_name = @{const_name HOL.eq}
846     val not_name = @{const_name Not}
847     val notE = @{thm notE}
848     val ccontr = @{thm ccontr}
849     val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
850   );
851   val blast_tac = Blast.blast_tac;
852 \<close>
855 subsubsection \<open>THE: definite description operator\<close>
857 lemma the_equality [intro]:
858   assumes "P a"
859       and "\<And>x. P x \<Longrightarrow> x = a"
860   shows "(THE x. P x) = a"
861   by (blast intro: assms trans [OF arg_cong [where f=The] the_eq_trivial])
863 lemma theI:
864   assumes "P a" and "\<And>x. P x \<Longrightarrow> x = a"
865   shows "P (THE x. P x)"
866 by (iprover intro: assms the_equality [THEN ssubst])
868 lemma theI': "\<exists>!x. P x \<Longrightarrow> P (THE x. P x)"
869   by (blast intro: theI)
871 (*Easier to apply than theI: only one occurrence of P*)
872 lemma theI2:
873   assumes "P a" "\<And>x. P x \<Longrightarrow> x = a" "\<And>x. P x \<Longrightarrow> Q x"
874   shows "Q (THE x. P x)"
875 by (iprover intro: assms theI)
877 lemma the1I2: assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
878 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
879            elim:allE impE)
881 lemma the1_equality [elim?]: "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> (THE x. P x) = a"
882   by blast
884 lemma the_sym_eq_trivial: "(THE y. x = y) = x"
885   by blast
888 subsubsection \<open>Simplifier\<close>
890 lemma eta_contract_eq: "(\<lambda>s. f s) = f" ..
892 lemma simp_thms:
893   shows not_not: "(\<not> \<not> P) = P"
894   and Not_eq_iff: "((\<not> P) = (\<not> Q)) = (P = Q)"
895   and
896     "(P \<noteq> Q) = (P = (\<not> Q))"
897     "(P \<or> \<not>P) = True"    "(\<not> P \<or> P) = True"
898     "(x = x) = True"
899   and not_True_eq_False [code]: "(\<not> True) = False"
900   and not_False_eq_True [code]: "(\<not> False) = True"
901   and
902     "(\<not> P) \<noteq> P"  "P \<noteq> (\<not> P)"
903     "(True = P) = P"
904   and eq_True: "(P = True) = P"
905   and "(False = P) = (\<not> P)"
906   and eq_False: "(P = False) = (\<not> P)"
907   and
908     "(True \<longrightarrow> P) = P"  "(False \<longrightarrow> P) = True"
909     "(P \<longrightarrow> True) = True"  "(P \<longrightarrow> P) = True"
910     "(P \<longrightarrow> False) = (\<not> P)"  "(P \<longrightarrow> \<not> P) = (\<not> P)"
911     "(P \<and> True) = P"  "(True \<and> P) = P"
912     "(P \<and> False) = False"  "(False \<and> P) = False"
913     "(P \<and> P) = P"  "(P \<and> (P \<and> Q)) = (P \<and> Q)"
914     "(P \<and> \<not> P) = False"    "(\<not> P \<and> P) = False"
915     "(P \<or> True) = True"  "(True \<or> P) = True"
916     "(P \<or> False) = P"  "(False \<or> P) = P"
917     "(P \<or> P) = P"  "(P \<or> (P \<or> Q)) = (P \<or> Q)" and
918     "(\<forall>x. P) = P"  "(\<exists>x. P) = P"  "\<exists>x. x = t"  "\<exists>x. t = x"
919   and
920     "\<And>P. (\<exists>x. x = t \<and> P x) = P t"
921     "\<And>P. (\<exists>x. t = x \<and> P x) = P t"
922     "\<And>P. (\<forall>x. x = t \<longrightarrow> P x) = P t"
923     "\<And>P. (\<forall>x. t = x \<longrightarrow> P x) = P t"
924   by (blast, blast, blast, blast, blast, iprover+)
926 lemma disj_absorb: "(A \<or> A) = A"
927   by blast
929 lemma disj_left_absorb: "(A \<or> (A \<or> B)) = (A \<or> B)"
930   by blast
932 lemma conj_absorb: "(A \<and> A) = A"
933   by blast
935 lemma conj_left_absorb: "(A \<and> (A \<and> B)) = (A \<and> B)"
936   by blast
938 lemma eq_ac:
939   shows eq_commute: "a = b \<longleftrightarrow> b = a"
940     and iff_left_commute: "(P \<longleftrightarrow> (Q \<longleftrightarrow> R)) \<longleftrightarrow> (Q \<longleftrightarrow> (P \<longleftrightarrow> R))"
941     and iff_assoc: "((P \<longleftrightarrow> Q) \<longleftrightarrow> R) \<longleftrightarrow> (P \<longleftrightarrow> (Q \<longleftrightarrow> R))" by (iprover, blast+)
942 lemma neq_commute: "a \<noteq> b \<longleftrightarrow> b \<noteq> a" by iprover
944 lemma conj_comms:
945   shows conj_commute: "(P \<and> Q) = (Q \<and> P)"
946     and conj_left_commute: "(P \<and> (Q \<and> R)) = (Q \<and> (P \<and> R))" by iprover+
947 lemma conj_assoc: "((P \<and> Q) \<and> R) = (P \<and> (Q \<and> R))" by iprover
949 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
951 lemma disj_comms:
952   shows disj_commute: "(P \<or> Q) = (Q \<or> P)"
953     and disj_left_commute: "(P \<or> (Q \<or> R)) = (Q \<or> (P \<or> R))" by iprover+
954 lemma disj_assoc: "((P \<or> Q) \<or> R) = (P \<or> (Q \<or> R))" by iprover
956 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
958 lemma conj_disj_distribL: "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)" by iprover
959 lemma conj_disj_distribR: "((P \<or> Q) \<and> R) = (P \<and> R \<or> Q \<and> R)" by iprover
961 lemma disj_conj_distribL: "(P \<or> (Q \<and> R)) = ((P \<or> Q) \<and> (P \<or> R))" by iprover
962 lemma disj_conj_distribR: "((P \<and> Q) \<or> R) = ((P \<or> R) \<and> (Q \<or> R))" by iprover
964 lemma imp_conjR: "(P \<longrightarrow> (Q \<and> R)) = ((P \<longrightarrow> Q) \<and> (P \<longrightarrow> R))" by iprover
965 lemma imp_conjL: "((P \<and> Q) \<longrightarrow> R) = (P \<longrightarrow> (Q \<longrightarrow> R))" by iprover
966 lemma imp_disjL: "((P \<or> Q) \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" by iprover
968 text \<open>These two are specialized, but \<open>imp_disj_not1\<close> is useful in \<open>Auth/Yahalom\<close>.\<close>
969 lemma imp_disj_not1: "(P \<longrightarrow> Q \<or> R) = (\<not> Q \<longrightarrow> P \<longrightarrow> R)" by blast
970 lemma imp_disj_not2: "(P \<longrightarrow> Q \<or> R) = (\<not> R \<longrightarrow> P \<longrightarrow> Q)" by blast
972 lemma imp_disj1: "((P \<longrightarrow> Q) \<or> R) = (P \<longrightarrow> Q \<or> R)" by blast
973 lemma imp_disj2: "(Q \<or> (P \<longrightarrow> R)) = (P \<longrightarrow> Q \<or> R)" by blast
975 lemma imp_cong: "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<longrightarrow> Q) = (P' \<longrightarrow> Q'))"
976   by iprover
978 lemma de_Morgan_disj: "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)" by iprover
979 lemma de_Morgan_conj: "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)" by blast
980 lemma not_imp: "(\<not> (P \<longrightarrow> Q)) = (P \<and> \<not> Q)" by blast
981 lemma not_iff: "(P \<noteq> Q) = (P = (\<not> Q))" by blast
982 lemma disj_not1: "(\<not> P \<or> Q) = (P \<longrightarrow> Q)" by blast
983 lemma disj_not2: "(P \<or> \<not> Q) = (Q \<longrightarrow> P)"  \<comment> \<open>changes orientation :-(\<close>
984   by blast
985 lemma imp_conv_disj: "(P \<longrightarrow> Q) = ((\<not> P) \<or> Q)" by blast
987 lemma iff_conv_conj_imp: "(P = Q) = ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))" by iprover
990 lemma cases_simp: "((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> Q)) = Q"
991   \<comment> \<open>Avoids duplication of subgoals after \<open>split_if\<close>, when the true and false\<close>
992   \<comment> \<open>cases boil down to the same thing.\<close>
993   by blast
995 lemma not_all: "(\<not> (\<forall>x. P x)) = (\<exists>x. \<not> P x)" by blast
996 lemma imp_all: "((\<forall>x. P x) \<longrightarrow> Q) = (\<exists>x. P x \<longrightarrow> Q)" by blast
997 lemma not_ex: "(\<not> (\<exists>x. P x)) = (\<forall>x. \<not> P x)" by iprover
998 lemma imp_ex: "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)" by iprover
999 lemma all_not_ex: "(\<forall>x. P x) = (\<not> (\<exists>x. \<not> P x ))" by blast
1001 declare All_def [no_atp]
1003 lemma ex_disj_distrib: "(\<exists>x. P x \<or> Q x) = ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by iprover
1004 lemma all_conj_distrib: "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))" by iprover
1006 text \<open>
1007   \medskip The \<open>\<and>\<close> congruence rule: not included by default!
1008   May slow rewrite proofs down by as much as 50\%\<close>
1010 lemma conj_cong:
1011     "(P = P') \<Longrightarrow> (P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<and> Q) = (P' \<and> Q'))"
1012   by iprover
1014 lemma rev_conj_cong:
1015     "(Q = Q') \<Longrightarrow> (Q' \<Longrightarrow> (P = P')) \<Longrightarrow> ((P \<and> Q) = (P' \<and> Q'))"
1016   by iprover
1018 text \<open>The \<open>|\<close> congruence rule: not included by default!\<close>
1020 lemma disj_cong:
1021     "(P = P') \<Longrightarrow> (\<not> P' \<Longrightarrow> (Q = Q')) \<Longrightarrow> ((P \<or> Q) = (P' \<or> Q'))"
1022   by blast
1025 text \<open>\medskip if-then-else rules\<close>
1027 lemma if_True [code]: "(if True then x else y) = x"
1028   by (unfold If_def) blast
1030 lemma if_False [code]: "(if False then x else y) = y"
1031   by (unfold If_def) blast
1033 lemma if_P: "P \<Longrightarrow> (if P then x else y) = x"
1034   by (unfold If_def) blast
1036 lemma if_not_P: "\<not> P \<Longrightarrow> (if P then x else y) = y"
1037   by (unfold If_def) blast
1039 lemma split_if: "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
1040   apply (rule case_split [of Q])
1041    apply (simplesubst if_P)
1042     prefer 3 apply (simplesubst if_not_P, blast+)
1043   done
1045 lemma split_if_asm: "P (if Q then x else y) = (\<not> ((Q \<and> \<not> P x) \<or> (\<not> Q \<and> \<not> P y)))"
1046 by (simplesubst split_if, blast)
1048 lemmas if_splits [no_atp] = split_if split_if_asm
1050 lemma if_cancel: "(if c then x else x) = x"
1051 by (simplesubst split_if, blast)
1053 lemma if_eq_cancel: "(if x = y then y else x) = x"
1054 by (simplesubst split_if, blast)
1056 lemma if_bool_eq_conj: "(if P then Q else R) = ((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R))"
1057   \<comment> \<open>This form is useful for expanding \<open>if\<close>s on the RIGHT of the \<open>\<Longrightarrow>\<close> symbol.\<close>
1058   by (rule split_if)
1060 lemma if_bool_eq_disj: "(if P then Q else R) = ((P \<and> Q) \<or> (\<not> P \<and> R))"
1061   \<comment> \<open>And this form is useful for expanding \<open>if\<close>s on the LEFT.\<close>
1062   by (simplesubst split_if) blast
1064 lemma Eq_TrueI: "P \<Longrightarrow> P \<equiv> True" by (unfold atomize_eq) iprover
1065 lemma Eq_FalseI: "\<not> P \<Longrightarrow> P \<equiv> False" by (unfold atomize_eq) iprover
1067 text \<open>\medskip let rules for simproc\<close>
1069 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
1070   by (unfold Let_def)
1072 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
1073   by (unfold Let_def)
1075 text \<open>
1076   The following copy of the implication operator is useful for
1077   fine-tuning congruence rules.  It instructs the simplifier to simplify
1078   its premise.
1079 \<close>
1081 definition simp_implies :: "[prop, prop] \<Rightarrow> prop"  (infixr "=simp=>" 1) where
1082   "simp_implies \<equiv> op \<Longrightarrow>"
1084 lemma simp_impliesI:
1085   assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1086   shows "PROP P =simp=> PROP Q"
1087   apply (unfold simp_implies_def)
1088   apply (rule PQ)
1089   apply assumption
1090   done
1092 lemma simp_impliesE:
1093   assumes PQ: "PROP P =simp=> PROP Q"
1094   and P: "PROP P"
1095   and QR: "PROP Q \<Longrightarrow> PROP R"
1096   shows "PROP R"
1097   apply (rule QR)
1098   apply (rule PQ [unfolded simp_implies_def])
1099   apply (rule P)
1100   done
1102 lemma simp_implies_cong:
1103   assumes PP' :"PROP P \<equiv> PROP P'"
1104   and P'QQ': "PROP P' \<Longrightarrow> (PROP Q \<equiv> PROP Q')"
1105   shows "(PROP P =simp=> PROP Q) \<equiv> (PROP P' =simp=> PROP Q')"
1106 proof (unfold simp_implies_def, rule equal_intr_rule)
1107   assume PQ: "PROP P \<Longrightarrow> PROP Q"
1108   and P': "PROP P'"
1109   from PP' [symmetric] and P' have "PROP P"
1110     by (rule equal_elim_rule1)
1111   then have "PROP Q" by (rule PQ)
1112   with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1113 next
1114   assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1115   and P: "PROP P"
1116   from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1117   then have "PROP Q'" by (rule P'Q')
1118   with P'QQ' [OF P', symmetric] show "PROP Q"
1119     by (rule equal_elim_rule1)
1120 qed
1122 lemma uncurry:
1123   assumes "P \<longrightarrow> Q \<longrightarrow> R"
1124   shows "P \<and> Q \<longrightarrow> R"
1125   using assms by blast
1127 lemma iff_allI:
1128   assumes "\<And>x. P x = Q x"
1129   shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1130   using assms by blast
1132 lemma iff_exI:
1133   assumes "\<And>x. P x = Q x"
1134   shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1135   using assms by blast
1137 lemma all_comm:
1138   "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1139   by blast
1141 lemma ex_comm:
1142   "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1143   by blast
1145 ML_file "Tools/simpdata.ML"
1146 ML \<open>open Simpdata\<close>
1148 setup \<open>
1149   map_theory_simpset (put_simpset HOL_basic_ss) #>
1150   Simplifier.method_setup Splitter.split_modifiers
1151 \<close>
1153 simproc_setup defined_Ex ("\<exists>x. P x") = \<open>fn _ => Quantifier1.rearrange_ex\<close>
1154 simproc_setup defined_All ("\<forall>x. P x") = \<open>fn _ => Quantifier1.rearrange_all\<close>
1156 text \<open>Simproc for proving \<open>(y = x) \<equiv> False\<close> from premise \<open>\<not> (x = y)\<close>:\<close>
1158 simproc_setup neq ("x = y") = \<open>fn _ =>
1159 let
1160   val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
1161   fun is_neq eq lhs rhs thm =
1162     (case Thm.prop_of thm of
1163       _ \$ (Not \$ (eq' \$ l' \$ r')) =>
1164         Not = HOLogic.Not andalso eq' = eq andalso
1165         r' aconv lhs andalso l' aconv rhs
1166     | _ => false);
1167   fun proc ss ct =
1168     (case Thm.term_of ct of
1169       eq \$ lhs \$ rhs =>
1170         (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of ss) of
1171           SOME thm => SOME (thm RS neq_to_EQ_False)
1172         | NONE => NONE)
1173      | _ => NONE);
1174 in proc end;
1175 \<close>
1177 simproc_setup let_simp ("Let x f") = \<open>
1178 let
1179   fun count_loose (Bound i) k = if i >= k then 1 else 0
1180     | count_loose (s \$ t) k = count_loose s k + count_loose t k
1181     | count_loose (Abs (_, _, t)) k = count_loose  t (k + 1)
1182     | count_loose _ _ = 0;
1183   fun is_trivial_let (Const (@{const_name Let}, _) \$ x \$ t) =
1184     (case t of
1185       Abs (_, _, t') => count_loose t' 0 <= 1
1186     | _ => true);
1187 in
1188   fn _ => fn ctxt => fn ct =>
1189     if is_trivial_let (Thm.term_of ct)
1190     then SOME @{thm Let_def} (*no or one ocurrence of bound variable*)
1191     else
1192       let (*Norbert Schirmer's case*)
1193         val t = Thm.term_of ct;
1194         val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
1195       in
1196         Option.map (hd o Variable.export ctxt' ctxt o single)
1197           (case t' of Const (@{const_name Let},_) \$ x \$ f => (* x and f are already in normal form *)
1198             if is_Free x orelse is_Bound x orelse is_Const x
1199             then SOME @{thm Let_def}
1200             else
1201               let
1202                 val n = case f of (Abs (x, _, _)) => x | _ => "x";
1203                 val cx = Thm.cterm_of ctxt x;
1204                 val xT = Thm.typ_of_cterm cx;
1205                 val cf = Thm.cterm_of ctxt f;
1206                 val fx_g = Simplifier.rewrite ctxt (Thm.apply cf cx);
1207                 val (_ \$ _ \$ g) = Thm.prop_of fx_g;
1208                 val g' = abstract_over (x, g);
1209                 val abs_g'= Abs (n, xT, g');
1210               in
1211                 if g aconv g' then
1212                   let
1213                     val rl =
1214                       infer_instantiate ctxt [(("f", 0), cf), (("x", 0), cx)] @{thm Let_unfold};
1215                   in SOME (rl OF [fx_g]) end
1216                 else if (Envir.beta_eta_contract f) aconv (Envir.beta_eta_contract abs_g')
1217                 then NONE (*avoid identity conversion*)
1218                 else
1219                   let
1220                     val g'x = abs_g' \$ x;
1221                     val g_g'x = Thm.symmetric (Thm.beta_conversion false (Thm.cterm_of ctxt g'x));
1222                     val rl =
1223                       @{thm Let_folded} |> infer_instantiate ctxt
1224                         [(("f", 0), Thm.cterm_of ctxt f),
1225                          (("x", 0), cx),
1226                          (("g", 0), Thm.cterm_of ctxt abs_g')];
1227                   in SOME (rl OF [Thm.transitive fx_g g_g'x]) end
1228               end
1229           | _ => NONE)
1230       end
1231 end\<close>
1233 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1234 proof
1235   assume "True \<Longrightarrow> PROP P"
1236   from this [OF TrueI] show "PROP P" .
1237 next
1238   assume "PROP P"
1239   then show "PROP P" .
1240 qed
1242 lemma implies_True_equals: "(PROP P \<Longrightarrow> True) \<equiv> Trueprop True"
1243   by standard (intro TrueI)
1245 lemma False_implies_equals: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
1246   by standard simp_all
1248 (* This is not made a simp rule because it does not improve any proofs
1249    but slows some AFP entries down by 5% (cpu time). May 2015 *)
1250 lemma implies_False_swap: "NO_MATCH (Trueprop False) P \<Longrightarrow>
1251   (False \<Longrightarrow> PROP P \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> False \<Longrightarrow> PROP Q)"
1252 by(rule swap_prems_eq)
1254 lemma ex_simps:
1255   "\<And>P Q. (\<exists>x. P x \<and> Q)   = ((\<exists>x. P x) \<and> Q)"
1256   "\<And>P Q. (\<exists>x. P \<and> Q x)   = (P \<and> (\<exists>x. Q x))"
1257   "\<And>P Q. (\<exists>x. P x \<or> Q)   = ((\<exists>x. P x) \<or> Q)"
1258   "\<And>P Q. (\<exists>x. P \<or> Q x)   = (P \<or> (\<exists>x. Q x))"
1259   "\<And>P Q. (\<exists>x. P x \<longrightarrow> Q) = ((\<forall>x. P x) \<longrightarrow> Q)"
1260   "\<And>P Q. (\<exists>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<exists>x. Q x))"
1261   \<comment> \<open>Miniscoping: pushing in existential quantifiers.\<close>
1262   by (iprover | blast)+
1264 lemma all_simps:
1265   "\<And>P Q. (\<forall>x. P x \<and> Q)   = ((\<forall>x. P x) \<and> Q)"
1266   "\<And>P Q. (\<forall>x. P \<and> Q x)   = (P \<and> (\<forall>x. Q x))"
1267   "\<And>P Q. (\<forall>x. P x \<or> Q)   = ((\<forall>x. P x) \<or> Q)"
1268   "\<And>P Q. (\<forall>x. P \<or> Q x)   = (P \<or> (\<forall>x. Q x))"
1269   "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q) = ((\<exists>x. P x) \<longrightarrow> Q)"
1270   "\<And>P Q. (\<forall>x. P \<longrightarrow> Q x) = (P \<longrightarrow> (\<forall>x. Q x))"
1271   \<comment> \<open>Miniscoping: pushing in universal quantifiers.\<close>
1272   by (iprover | blast)+
1274 lemmas [simp] =
1275   triv_forall_equality (*prunes params*)
1276   True_implies_equals implies_True_equals (*prune True in asms*)
1277   False_implies_equals (*prune False in asms*)
1278   if_True
1279   if_False
1280   if_cancel
1281   if_eq_cancel
1282   imp_disjL
1283   (*In general it seems wrong to add distributive laws by default: they
1284     might cause exponential blow-up.  But imp_disjL has been in for a while
1285     and cannot be removed without affecting existing proofs.  Moreover,
1286     rewriting by "(P \<or> Q \<longrightarrow> R) = ((P \<longrightarrow> R) \<and> (Q \<longrightarrow> R))" might be justified on the
1287     grounds that it allows simplification of R in the two cases.*)
1288   conj_assoc
1289   disj_assoc
1290   de_Morgan_conj
1291   de_Morgan_disj
1292   imp_disj1
1293   imp_disj2
1294   not_imp
1295   disj_not1
1296   not_all
1297   not_ex
1298   cases_simp
1299   the_eq_trivial
1300   the_sym_eq_trivial
1301   ex_simps
1302   all_simps
1303   simp_thms
1305 lemmas [cong] = imp_cong simp_implies_cong
1306 lemmas [split] = split_if
1308 ML \<open>val HOL_ss = simpset_of @{context}\<close>
1310 text \<open>Simplifies @{term x} assuming @{prop c} and @{term y} assuming @{prop "\<not> c"}\<close>
1311 lemma if_cong:
1312   assumes "b = c"
1313       and "c \<Longrightarrow> x = u"
1314       and "\<not> c \<Longrightarrow> y = v"
1315   shows "(if b then x else y) = (if c then u else v)"
1316   using assms by simp
1318 text \<open>Prevents simplification of x and y:
1319   faster and allows the execution of functional programs.\<close>
1320 lemma if_weak_cong [cong]:
1321   assumes "b = c"
1322   shows "(if b then x else y) = (if c then x else y)"
1323   using assms by (rule arg_cong)
1325 text \<open>Prevents simplification of t: much faster\<close>
1326 lemma let_weak_cong:
1327   assumes "a = b"
1328   shows "(let x = a in t x) = (let x = b in t x)"
1329   using assms by (rule arg_cong)
1331 text \<open>To tidy up the result of a simproc.  Only the RHS will be simplified.\<close>
1332 lemma eq_cong2:
1333   assumes "u = u'"
1334   shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1335   using assms by simp
1337 lemma if_distrib:
1338   "f (if c then x else y) = (if c then f x else f y)"
1339   by simp
1341 text\<open>As a simplification rule, it replaces all function equalities by
1342   first-order equalities.\<close>
1343 lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
1344   by auto
1347 subsubsection \<open>Generic cases and induction\<close>
1349 text \<open>Rule projections:\<close>
1350 ML \<open>
1351 structure Project_Rule = Project_Rule
1352 (
1353   val conjunct1 = @{thm conjunct1}
1354   val conjunct2 = @{thm conjunct2}
1355   val mp = @{thm mp}
1356 );
1357 \<close>
1359 context
1360 begin
1362 qualified definition "induct_forall P \<equiv> \<forall>x. P x"
1363 qualified definition "induct_implies A B \<equiv> A \<longrightarrow> B"
1364 qualified definition "induct_equal x y \<equiv> x = y"
1365 qualified definition "induct_conj A B \<equiv> A \<and> B"
1366 qualified definition "induct_true \<equiv> True"
1367 qualified definition "induct_false \<equiv> False"
1369 lemma induct_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (induct_forall (\<lambda>x. P x))"
1370   by (unfold atomize_all induct_forall_def)
1372 lemma induct_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (induct_implies A B)"
1373   by (unfold atomize_imp induct_implies_def)
1375 lemma induct_equal_eq: "(x \<equiv> y) \<equiv> Trueprop (induct_equal x y)"
1376   by (unfold atomize_eq induct_equal_def)
1378 lemma induct_conj_eq: "(A &&& B) \<equiv> Trueprop (induct_conj A B)"
1379   by (unfold atomize_conj induct_conj_def)
1381 lemmas induct_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
1382 lemmas induct_atomize = induct_atomize' induct_equal_eq
1383 lemmas induct_rulify' [symmetric] = induct_atomize'
1384 lemmas induct_rulify [symmetric] = induct_atomize
1385 lemmas induct_rulify_fallback =
1386   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1387   induct_true_def induct_false_def
1389 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1390     induct_conj (induct_forall A) (induct_forall B)"
1391   by (unfold induct_forall_def induct_conj_def) iprover
1393 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1394     induct_conj (induct_implies C A) (induct_implies C B)"
1395   by (unfold induct_implies_def induct_conj_def) iprover
1397 lemma induct_conj_curry: "(induct_conj A B \<Longrightarrow> PROP C) \<equiv> (A \<Longrightarrow> B \<Longrightarrow> PROP C)"
1398 proof
1399   assume r: "induct_conj A B \<Longrightarrow> PROP C"
1400   assume ab: A B
1401   show "PROP C" by (rule r) (simp add: induct_conj_def ab)
1402 next
1403   assume r: "A \<Longrightarrow> B \<Longrightarrow> PROP C"
1404   assume ab: "induct_conj A B"
1405   show "PROP C" by (rule r) (simp_all add: ab [unfolded induct_conj_def])
1406 qed
1408 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1410 lemma induct_trueI: "induct_true"
1411   by (simp add: induct_true_def)
1413 text \<open>Method setup.\<close>
1415 ML_file "~~/src/Tools/induct.ML"
1416 ML \<open>
1417 structure Induct = Induct
1418 (
1419   val cases_default = @{thm case_split}
1420   val atomize = @{thms induct_atomize}
1421   val rulify = @{thms induct_rulify'}
1422   val rulify_fallback = @{thms induct_rulify_fallback}
1423   val equal_def = @{thm induct_equal_def}
1424   fun dest_def (Const (@{const_name induct_equal}, _) \$ t \$ u) = SOME (t, u)
1425     | dest_def _ = NONE
1426   fun trivial_tac ctxt = match_tac ctxt @{thms induct_trueI}
1427 )
1428 \<close>
1430 ML_file "~~/src/Tools/induction.ML"
1432 declaration \<open>
1433   fn _ => Induct.map_simpset (fn ss => ss
1435       [Simplifier.make_simproc @{context} "swap_induct_false"
1436         {lhss = [@{term "induct_false \<Longrightarrow> PROP P \<Longrightarrow> PROP Q"}],
1437          proc = fn _ => fn _ => fn ct =>
1438           (case Thm.term_of ct of
1439             _ \$ (P as _ \$ @{const induct_false}) \$ (_ \$ Q \$ _) =>
1440               if P <> Q then SOME Drule.swap_prems_eq else NONE
1441           | _ => NONE),
1442          identifier = []},
1443        Simplifier.make_simproc @{context} "induct_equal_conj_curry"
1444         {lhss = [@{term "induct_conj P Q \<Longrightarrow> PROP R"}],
1445          proc = fn _ => fn _ => fn ct =>
1446           (case Thm.term_of ct of
1447             _ \$ (_ \$ P) \$ _ =>
1448               let
1449                 fun is_conj (@{const induct_conj} \$ P \$ Q) =
1450                       is_conj P andalso is_conj Q
1451                   | is_conj (Const (@{const_name induct_equal}, _) \$ _ \$ _) = true
1452                   | is_conj @{const induct_true} = true
1453                   | is_conj @{const induct_false} = true
1454                   | is_conj _ = false
1455               in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
1456             | _ => NONE),
1457           identifier = []}]
1458     |> Simplifier.set_mksimps (fn ctxt =>
1459         Simpdata.mksimps Simpdata.mksimps_pairs ctxt #>
1460         map (rewrite_rule ctxt (map Thm.symmetric @{thms induct_rulify_fallback}))))
1461 \<close>
1463 text \<open>Pre-simplification of induction and cases rules\<close>
1465 lemma [induct_simp]: "(\<And>x. induct_equal x t \<Longrightarrow> PROP P x) \<equiv> PROP P t"
1466   unfolding induct_equal_def
1467 proof
1468   assume r: "\<And>x. x = t \<Longrightarrow> PROP P x"
1469   show "PROP P t" by (rule r [OF refl])
1470 next
1471   fix x
1472   assume "PROP P t" "x = t"
1473   then show "PROP P x" by simp
1474 qed
1476 lemma [induct_simp]: "(\<And>x. induct_equal t x \<Longrightarrow> PROP P x) \<equiv> PROP P t"
1477   unfolding induct_equal_def
1478 proof
1479   assume r: "\<And>x. t = x \<Longrightarrow> PROP P x"
1480   show "PROP P t" by (rule r [OF refl])
1481 next
1482   fix x
1483   assume "PROP P t" "t = x"
1484   then show "PROP P x" by simp
1485 qed
1487 lemma [induct_simp]: "(induct_false \<Longrightarrow> P) \<equiv> Trueprop induct_true"
1488   unfolding induct_false_def induct_true_def
1489   by (iprover intro: equal_intr_rule)
1491 lemma [induct_simp]: "(induct_true \<Longrightarrow> PROP P) \<equiv> PROP P"
1492   unfolding induct_true_def
1493 proof
1494   assume "True \<Longrightarrow> PROP P"
1495   then show "PROP P" using TrueI .
1496 next
1497   assume "PROP P"
1498   then show "PROP P" .
1499 qed
1501 lemma [induct_simp]: "(PROP P \<Longrightarrow> induct_true) \<equiv> Trueprop induct_true"
1502   unfolding induct_true_def
1503   by (iprover intro: equal_intr_rule)
1505 lemma [induct_simp]: "(\<And>x. induct_true) \<equiv> Trueprop induct_true"
1506   unfolding induct_true_def
1507   by (iprover intro: equal_intr_rule)
1509 lemma [induct_simp]: "induct_implies induct_true P \<equiv> P"
1510   by (simp add: induct_implies_def induct_true_def)
1512 lemma [induct_simp]: "x = x \<longleftrightarrow> True"
1513   by (rule simp_thms)
1515 end
1517 ML_file "~~/src/Tools/induct_tacs.ML"
1520 subsubsection \<open>Coherent logic\<close>
1522 ML_file "~~/src/Tools/coherent.ML"
1523 ML \<open>
1524 structure Coherent = Coherent
1525 (
1526   val atomize_elimL = @{thm atomize_elimL};
1527   val atomize_exL = @{thm atomize_exL};
1528   val atomize_conjL = @{thm atomize_conjL};
1529   val atomize_disjL = @{thm atomize_disjL};
1530   val operator_names = [@{const_name HOL.disj}, @{const_name HOL.conj}, @{const_name Ex}];
1531 );
1532 \<close>
1535 subsubsection \<open>Reorienting equalities\<close>
1537 ML \<open>
1538 signature REORIENT_PROC =
1539 sig
1540   val add : (term -> bool) -> theory -> theory
1541   val proc : morphism -> Proof.context -> cterm -> thm option
1542 end;
1544 structure Reorient_Proc : REORIENT_PROC =
1545 struct
1546   structure Data = Theory_Data
1547   (
1548     type T = ((term -> bool) * stamp) list;
1549     val empty = [];
1550     val extend = I;
1551     fun merge data : T = Library.merge (eq_snd op =) data;
1552   );
1553   fun add m = Data.map (cons (m, stamp ()));
1554   fun matches thy t = exists (fn (m, _) => m t) (Data.get thy);
1556   val meta_reorient = @{thm eq_commute [THEN eq_reflection]};
1557   fun proc phi ctxt ct =
1558     let
1559       val thy = Proof_Context.theory_of ctxt;
1560     in
1561       case Thm.term_of ct of
1562         (_ \$ t \$ u) => if matches thy u then NONE else SOME meta_reorient
1563       | _ => NONE
1564     end;
1565 end;
1566 \<close>
1569 subsection \<open>Other simple lemmas and lemma duplicates\<close>
1571 lemma ex1_eq [iff]: "\<exists>!x. x = t" "\<exists>!x. t = x"
1572   by blast+
1574 lemma choice_eq: "(\<forall>x. \<exists>!y. P x y) = (\<exists>!f. \<forall>x. P x (f x))"
1575   apply (rule iffI)
1576   apply (rule_tac a = "\<lambda>x. THE y. P x y" in ex1I)
1577   apply (fast dest!: theI')
1578   apply (fast intro: the1_equality [symmetric])
1579   apply (erule ex1E)
1580   apply (rule allI)
1581   apply (rule ex1I)
1582   apply (erule spec)
1583   apply (erule_tac x = "\<lambda>z. if z = x then y else f z" in allE)
1584   apply (erule impE)
1585   apply (rule allI)
1586   apply (case_tac "xa = x")
1587   apply (drule_tac  x = x in fun_cong, simp_all)
1588   done
1590 lemmas eq_sym_conv = eq_commute
1592 lemma nnf_simps:
1593   "(\<not>(P \<and> Q)) = (\<not> P \<or> \<not> Q)" "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
1594   "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
1595   "(\<not> \<not>(P)) = P"
1596 by blast+
1598 subsection \<open>Basic ML bindings\<close>
1600 ML \<open>
1601 val FalseE = @{thm FalseE}
1602 val Let_def = @{thm Let_def}
1603 val TrueI = @{thm TrueI}
1604 val allE = @{thm allE}
1605 val allI = @{thm allI}
1606 val all_dupE = @{thm all_dupE}
1607 val arg_cong = @{thm arg_cong}
1608 val box_equals = @{thm box_equals}
1609 val ccontr = @{thm ccontr}
1610 val classical = @{thm classical}
1611 val conjE = @{thm conjE}
1612 val conjI = @{thm conjI}
1613 val conjunct1 = @{thm conjunct1}
1614 val conjunct2 = @{thm conjunct2}
1615 val disjCI = @{thm disjCI}
1616 val disjE = @{thm disjE}
1617 val disjI1 = @{thm disjI1}
1618 val disjI2 = @{thm disjI2}
1619 val eq_reflection = @{thm eq_reflection}
1620 val ex1E = @{thm ex1E}
1621 val ex1I = @{thm ex1I}
1622 val ex1_implies_ex = @{thm ex1_implies_ex}
1623 val exE = @{thm exE}
1624 val exI = @{thm exI}
1625 val excluded_middle = @{thm excluded_middle}
1626 val ext = @{thm ext}
1627 val fun_cong = @{thm fun_cong}
1628 val iffD1 = @{thm iffD1}
1629 val iffD2 = @{thm iffD2}
1630 val iffI = @{thm iffI}
1631 val impE = @{thm impE}
1632 val impI = @{thm impI}
1633 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
1634 val mp = @{thm mp}
1635 val notE = @{thm notE}
1636 val notI = @{thm notI}
1637 val not_all = @{thm not_all}
1638 val not_ex = @{thm not_ex}
1639 val not_iff = @{thm not_iff}
1640 val not_not = @{thm not_not}
1641 val not_sym = @{thm not_sym}
1642 val refl = @{thm refl}
1643 val rev_mp = @{thm rev_mp}
1644 val spec = @{thm spec}
1645 val ssubst = @{thm ssubst}
1646 val subst = @{thm subst}
1647 val sym = @{thm sym}
1648 val trans = @{thm trans}
1649 \<close>
1651 ML_file "Tools/cnf.ML"
1654 section \<open>\<open>NO_MATCH\<close> simproc\<close>
1656 text \<open>
1657  The simplification procedure can be used to avoid simplification of terms of a certain form
1658 \<close>
1660 definition NO_MATCH :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where "NO_MATCH pat val \<equiv> True"
1662 lemma NO_MATCH_cong[cong]: "NO_MATCH pat val = NO_MATCH pat val" by (rule refl)
1664 declare [[coercion_args NO_MATCH - -]]
1666 simproc_setup NO_MATCH ("NO_MATCH pat val") = \<open>fn _ => fn ctxt => fn ct =>
1667   let
1668     val thy = Proof_Context.theory_of ctxt
1669     val dest_binop = Term.dest_comb #> apfst (Term.dest_comb #> snd)
1670     val m = Pattern.matches thy (dest_binop (Thm.term_of ct))
1671   in if m then NONE else SOME @{thm NO_MATCH_def} end
1672 \<close>
1674 text \<open>
1675   This setup ensures that a rewrite rule of the form @{term "NO_MATCH pat val \<Longrightarrow> t"}
1676   is only applied, if the pattern @{term pat} does not match the value @{term val}.
1677 \<close>
1680 text\<open>Tagging a premise of a simp rule with ASSUMPTION forces the simplifier
1681 not to simplify the argument and to solve it by an assumption.\<close>
1683 definition ASSUMPTION :: "bool \<Rightarrow> bool" where
1684 "ASSUMPTION A \<equiv> A"
1686 lemma ASSUMPTION_cong[cong]: "ASSUMPTION A = ASSUMPTION A"
1687 by (rule refl)
1689 lemma ASSUMPTION_I: "A \<Longrightarrow> ASSUMPTION A"
1690 by(simp add: ASSUMPTION_def)
1692 lemma ASSUMPTION_D: "ASSUMPTION A \<Longrightarrow> A"
1693 by(simp add: ASSUMPTION_def)
1695 setup \<open>
1696 let
1697   val asm_sol = mk_solver "ASSUMPTION" (fn ctxt =>
1698     resolve_tac ctxt [@{thm ASSUMPTION_I}] THEN'
1699     resolve_tac ctxt (Simplifier.prems_of ctxt))
1700 in
1701   map_theory_simpset (fn ctxt => Simplifier.addSolver (ctxt,asm_sol))
1702 end
1703 \<close>
1706 subsection \<open>Code generator setup\<close>
1708 subsubsection \<open>Generic code generator preprocessor setup\<close>
1710 lemma conj_left_cong:
1711   "P \<longleftrightarrow> Q \<Longrightarrow> P \<and> R \<longleftrightarrow> Q \<and> R"
1712   by (fact arg_cong)
1714 lemma disj_left_cong:
1715   "P \<longleftrightarrow> Q \<Longrightarrow> P \<or> R \<longleftrightarrow> Q \<or> R"
1716   by (fact arg_cong)
1718 setup \<open>
1719   Code_Preproc.map_pre (put_simpset HOL_basic_ss) #>
1720   Code_Preproc.map_post (put_simpset HOL_basic_ss) #>
1721   Code_Simp.map_ss (put_simpset HOL_basic_ss #>
1722   Simplifier.add_cong @{thm conj_left_cong} #>
1723   Simplifier.add_cong @{thm disj_left_cong})
1724 \<close>
1727 subsubsection \<open>Equality\<close>
1729 class equal =
1730   fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1731   assumes equal_eq: "equal x y \<longleftrightarrow> x = y"
1732 begin
1734 lemma equal: "equal = (op =)"
1735   by (rule ext equal_eq)+
1737 lemma equal_refl: "equal x x \<longleftrightarrow> True"
1738   unfolding equal by rule+
1740 lemma eq_equal: "(op =) \<equiv> equal"
1741   by (rule eq_reflection) (rule ext, rule ext, rule sym, rule equal_eq)
1743 end
1745 declare eq_equal [symmetric, code_post]
1746 declare eq_equal [code]
1748 setup \<open>
1749   Code_Preproc.map_pre (fn ctxt =>
1751       [Simplifier.make_simproc @{context} "equal"
1752         {lhss = [@{term HOL.eq}],
1753          proc = fn _ => fn _ => fn ct =>
1754           (case Thm.term_of ct of
1755             Const (_, Type (@{type_name fun}, [Type _, _])) => SOME @{thm eq_equal}
1756           | _ => NONE),
1757          identifier = []}])
1758 \<close>
1761 subsubsection \<open>Generic code generator foundation\<close>
1763 text \<open>Datatype @{typ bool}\<close>
1765 code_datatype True False
1767 lemma [code]:
1768   shows "False \<and> P \<longleftrightarrow> False"
1769     and "True \<and> P \<longleftrightarrow> P"
1770     and "P \<and> False \<longleftrightarrow> False"
1771     and "P \<and> True \<longleftrightarrow> P" by simp_all
1773 lemma [code]:
1774   shows "False \<or> P \<longleftrightarrow> P"
1775     and "True \<or> P \<longleftrightarrow> True"
1776     and "P \<or> False \<longleftrightarrow> P"
1777     and "P \<or> True \<longleftrightarrow> True" by simp_all
1779 lemma [code]:
1780   shows "(False \<longrightarrow> P) \<longleftrightarrow> True"
1781     and "(True \<longrightarrow> P) \<longleftrightarrow> P"
1782     and "(P \<longrightarrow> False) \<longleftrightarrow> \<not> P"
1783     and "(P \<longrightarrow> True) \<longleftrightarrow> True" by simp_all
1785 text \<open>More about @{typ prop}\<close>
1787 lemma [code nbe]:
1788   shows "(True \<Longrightarrow> PROP Q) \<equiv> PROP Q"
1789     and "(PROP Q \<Longrightarrow> True) \<equiv> Trueprop True"
1790     and "(P \<Longrightarrow> R) \<equiv> Trueprop (P \<longrightarrow> R)" by (auto intro!: equal_intr_rule)
1792 lemma Trueprop_code [code]:
1793   "Trueprop True \<equiv> Code_Generator.holds"
1794   by (auto intro!: equal_intr_rule holds)
1796 declare Trueprop_code [symmetric, code_post]
1798 text \<open>Equality\<close>
1800 declare simp_thms(6) [code nbe]
1802 instantiation itself :: (type) equal
1803 begin
1805 definition equal_itself :: "'a itself \<Rightarrow> 'a itself \<Rightarrow> bool" where
1806   "equal_itself x y \<longleftrightarrow> x = y"
1808 instance proof
1809 qed (fact equal_itself_def)
1811 end
1813 lemma equal_itself_code [code]:
1814   "equal TYPE('a) TYPE('a) \<longleftrightarrow> True"
1815   by (simp add: equal)
1817 setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::type \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
1819 lemma equal_alias_cert: "OFCLASS('a, equal_class) \<equiv> ((op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool) \<equiv> equal)" (is "?ofclass \<equiv> ?equal")
1820 proof
1821   assume "PROP ?ofclass"
1822   show "PROP ?equal"
1823     by (tactic \<open>ALLGOALS (resolve_tac @{context} [Thm.unconstrainT @{thm eq_equal}])\<close>)
1824       (fact \<open>PROP ?ofclass\<close>)
1825 next
1826   assume "PROP ?equal"
1827   show "PROP ?ofclass" proof
1828   qed (simp add: \<open>PROP ?equal\<close>)
1829 qed
1831 setup \<open>Sign.add_const_constraint (@{const_name equal}, SOME @{typ "'a::equal \<Rightarrow> 'a \<Rightarrow> bool"})\<close>
1833 setup \<open>Nbe.add_const_alias @{thm equal_alias_cert}\<close>
1835 text \<open>Cases\<close>
1837 lemma Let_case_cert:
1838   assumes "CASE \<equiv> (\<lambda>x. Let x f)"
1839   shows "CASE x \<equiv> f x"
1840   using assms by simp_all
1842 setup \<open>
1843   Code.add_case @{thm Let_case_cert} #>
1844   Code.add_undefined @{const_name undefined}
1845 \<close>
1847 declare [[code abort: undefined]]
1850 subsubsection \<open>Generic code generator target languages\<close>
1852 text \<open>type @{typ bool}\<close>
1854 code_printing
1855   type_constructor bool \<rightharpoonup>
1856     (SML) "bool" and (OCaml) "bool" and (Haskell) "Bool" and (Scala) "Boolean"
1857 | constant True \<rightharpoonup>
1858     (SML) "true" and (OCaml) "true" and (Haskell) "True" and (Scala) "true"
1859 | constant False \<rightharpoonup>
1860     (SML) "false" and (OCaml) "false" and (Haskell) "False" and (Scala) "false"
1862 code_reserved SML
1863   bool true false
1865 code_reserved OCaml
1866   bool
1868 code_reserved Scala
1869   Boolean
1871 code_printing
1872   constant Not \<rightharpoonup>
1873     (SML) "not" and (OCaml) "not" and (Haskell) "not" and (Scala) "'! _"
1874 | constant HOL.conj \<rightharpoonup>
1875     (SML) infixl 1 "andalso" and (OCaml) infixl 3 "&&" and (Haskell) infixr 3 "&&" and (Scala) infixl 3 "&&"
1876 | constant HOL.disj \<rightharpoonup>
1877     (SML) infixl 0 "orelse" and (OCaml) infixl 2 "||" and (Haskell) infixl 2 "||" and (Scala) infixl 1 "||"
1878 | constant HOL.implies \<rightharpoonup>
1879     (SML) "!(if (_)/ then (_)/ else true)"
1880     and (OCaml) "!(if (_)/ then (_)/ else true)"
1881     and (Haskell) "!(if (_)/ then (_)/ else True)"
1882     and (Scala) "!(if ((_))/ (_)/ else true)"
1883 | constant If \<rightharpoonup>
1884     (SML) "!(if (_)/ then (_)/ else (_))"
1885     and (OCaml) "!(if (_)/ then (_)/ else (_))"
1886     and (Haskell) "!(if (_)/ then (_)/ else (_))"
1887     and (Scala) "!(if ((_))/ (_)/ else (_))"
1889 code_reserved SML
1890   not
1892 code_reserved OCaml
1893   not
1895 code_identifier
1896   code_module Pure \<rightharpoonup>
1897     (SML) HOL and (OCaml) HOL and (Haskell) HOL and (Scala) HOL
1899 text \<open>using built-in Haskell equality\<close>
1901 code_printing
1902   type_class equal \<rightharpoonup> (Haskell) "Eq"
1903 | constant HOL.equal \<rightharpoonup> (Haskell) infix 4 "=="
1904 | constant HOL.eq \<rightharpoonup> (Haskell) infix 4 "=="
1906 text \<open>undefined\<close>
1908 code_printing
1909   constant undefined \<rightharpoonup>
1910     (SML) "!(raise/ Fail/ \"undefined\")"
1911     and (OCaml) "failwith/ \"undefined\""
1912     and (Haskell) "error/ \"undefined\""
1913     and (Scala) "!sys.error(\"undefined\")"
1916 subsubsection \<open>Evaluation and normalization by evaluation\<close>
1918 method_setup eval = \<open>
1919   let
1920     fun eval_tac ctxt =
1921       let val conv = Code_Runtime.dynamic_holds_conv ctxt
1922       in
1923         CONVERSION (Conv.params_conv ~1 (K (Conv.concl_conv ~1 conv)) ctxt) THEN'
1924         resolve_tac ctxt [TrueI]
1925       end
1926   in
1927     Scan.succeed (SIMPLE_METHOD' o eval_tac)
1928   end
1929 \<close> "solve goal by evaluation"
1931 method_setup normalization = \<open>
1932   Scan.succeed (fn ctxt =>
1933     SIMPLE_METHOD'
1934       (CHANGED_PROP o
1935         (CONVERSION (Nbe.dynamic_conv ctxt)
1936           THEN_ALL_NEW (TRY o resolve_tac ctxt [TrueI]))))
1937 \<close> "solve goal by normalization"
1940 subsection \<open>Counterexample Search Units\<close>
1942 subsubsection \<open>Quickcheck\<close>
1944 quickcheck_params [size = 5, iterations = 50]
1947 subsubsection \<open>Nitpick setup\<close>
1949 named_theorems nitpick_unfold "alternative definitions of constants as needed by Nitpick"
1950   and nitpick_simp "equational specification of constants as needed by Nitpick"
1951   and nitpick_psimp "partial equational specification of constants as needed by Nitpick"
1952   and nitpick_choice_spec "choice specification of constants as needed by Nitpick"
1954 declare if_bool_eq_conj [nitpick_unfold, no_atp]
1955         if_bool_eq_disj [no_atp]
1958 subsection \<open>Preprocessing for the predicate compiler\<close>
1960 named_theorems code_pred_def "alternative definitions of constants for the Predicate Compiler"
1961   and code_pred_inline "inlining definitions for the Predicate Compiler"
1962   and code_pred_simp "simplification rules for the optimisations in the Predicate Compiler"
1965 subsection \<open>Legacy tactics and ML bindings\<close>
1967 ML \<open>
1968   (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1969   local
1970     fun wrong_prem (Const (@{const_name All}, _) \$ Abs (_, _, t)) = wrong_prem t
1971       | wrong_prem (Bound _) = true
1972       | wrong_prem _ = false;
1973     val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
1974     fun smp i = funpow i (fn m => filter_right ([spec] RL m)) [mp];
1975   in
1976     fun smp_tac ctxt j = EVERY' [dresolve_tac ctxt (smp j), assume_tac ctxt];
1977   end;
1979   local
1980     val nnf_ss =
1981       simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms simp_thms nnf_simps});
1982   in
1983     fun nnf_conv ctxt = Simplifier.rewrite (put_simpset nnf_ss ctxt);
1984   end
1985 \<close>
1987 hide_const (open) eq equal
1989 end