3 Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
6 header {* The basis of Higher-Order Logic *}
12 "~~/src/Tools/IsaPlanner/zipper.ML"
13 "~~/src/Tools/IsaPlanner/isand.ML"
14 "~~/src/Tools/IsaPlanner/rw_tools.ML"
15 "~~/src/Tools/IsaPlanner/rw_inst.ML"
16 "~~/src/Provers/project_rule.ML"
17 "~~/src/Provers/hypsubst.ML"
18 "~~/src/Provers/splitter.ML"
19 "~~/src/Provers/classical.ML"
20 "~~/src/Provers/blast.ML"
21 "~~/src/Provers/clasimp.ML"
22 "~~/src/Provers/eqsubst.ML"
23 "~~/src/Provers/quantifier1.ML"
25 "~~/src/Tools/random_word.ML"
26 "~~/src/Tools/atomize_elim.ML"
27 "~~/src/Tools/induct.ML"
28 ("~~/src/Tools/induct_tacs.ML")
29 "~~/src/Tools/code/code_name.ML"
30 "~~/src/Tools/code/code_funcgr.ML"
31 "~~/src/Tools/code/code_thingol.ML"
32 "~~/src/Tools/code/code_target.ML"
36 subsection {* Primitive logic *}
38 subsubsection {* Core syntax *}
42 setup {* ObjectLogic.add_base_sort @{sort type} *}
45 "fun" :: (type, type) type
53 Trueprop :: "bool => prop" ("(_)" 5)
56 Not :: "bool => bool" ("~ _" [40] 40)
61 The :: "('a => bool) => 'a"
62 All :: "('a => bool) => bool" (binder "ALL " 10)
63 Ex :: "('a => bool) => bool" (binder "EX " 10)
64 Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
65 Let :: "['a, 'a => 'b] => 'b"
67 "op =" :: "['a, 'a] => bool" (infixl "=" 50)
68 "op &" :: "[bool, bool] => bool" (infixr "&" 35)
69 "op |" :: "[bool, bool] => bool" (infixr "|" 30)
70 "op -->" :: "[bool, bool] => bool" (infixr "-->" 25)
75 If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
78 subsubsection {* Additional concrete syntax *}
84 not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
88 not_equal (infix "~=" 50)
91 Not ("\<not> _" [40] 40) and
92 "op &" (infixr "\<and>" 35) and
93 "op |" (infixr "\<or>" 30) and
94 "op -->" (infixr "\<longrightarrow>" 25) and
95 not_equal (infix "\<noteq>" 50)
97 notation (HTML output)
98 Not ("\<not> _" [40] 40) and
99 "op &" (infixr "\<and>" 35) and
100 "op |" (infixr "\<or>" 30) and
101 not_equal (infix "\<noteq>" 50)
104 iff :: "[bool, bool] => bool" (infixr "<->" 25) where
108 iff (infixr "\<longleftrightarrow>" 25)
116 "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
118 "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
119 "" :: "letbind => letbinds" ("_")
120 "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
121 "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
123 "_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
124 "_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
125 "" :: "case_syn => cases_syn" ("_")
126 "_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
129 "THE x. P" == "The (%x. P)"
130 "_Let (_binds b bs) e" == "_Let b (_Let bs e)"
131 "let x = a in e" == "Let a (%x. e)"
134 (* To avoid eta-contraction of body: *)
135 [("The", fn [Abs abs] =>
136 let val (x,t) = atomic_abs_tr' abs
137 in Syntax.const "_The" $ x $ t end)]
141 "_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
144 All (binder "\<forall>" 10) and
145 Ex (binder "\<exists>" 10) and
146 Ex1 (binder "\<exists>!" 10)
148 notation (HTML output)
149 All (binder "\<forall>" 10) and
150 Ex (binder "\<exists>" 10) and
151 Ex1 (binder "\<exists>!" 10)
154 All (binder "! " 10) and
155 Ex (binder "? " 10) and
156 Ex1 (binder "?! " 10)
159 subsubsection {* Axioms and basic definitions *}
162 eq_reflection: "(x=y) ==> (x==y)"
166 ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
167 -- {*Extensionality is built into the meta-logic, and this rule expresses
168 a related property. It is an eta-expanded version of the traditional
169 rule, and similar to the ABS rule of HOL*}
171 the_eq_trivial: "(THE x. x = a) = (a::'a)"
173 impI: "(P ==> Q) ==> P-->Q"
174 mp: "[| P-->Q; P |] ==> Q"
178 True_def: "True == ((%x::bool. x) = (%x. x))"
179 All_def: "All(P) == (P = (%x. True))"
180 Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
181 False_def: "False == (!P. P)"
182 not_def: "~ P == P-->False"
183 and_def: "P & Q == !R. (P-->Q-->R) --> R"
184 or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
185 Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
188 iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
189 True_or_False: "(P=True) | (P=False)"
192 Let_def: "Let s f == f(s)"
193 if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
205 undefined_fun: "undefined x = undefined"
208 subsubsection {* Generic classes and algebraic operations *}
210 class default = type +
214 fixes zero :: 'a ("0")
217 fixes one :: 'a ("1")
219 hide (open) const zero one
222 fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
225 fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
227 class uminus = type +
228 fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
231 fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
233 class inverse = type +
234 fixes inverse :: "'a \<Rightarrow> 'a"
235 and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
238 fixes abs :: "'a \<Rightarrow> 'a"
242 abs ("\<bar>_\<bar>")
244 notation (HTML output)
245 abs ("\<bar>_\<bar>")
250 fixes sgn :: "'a \<Rightarrow> 'a"
253 fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
254 and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
258 less_eq ("op <=") and
259 less_eq ("(_/ <= _)" [51, 51] 50) and
261 less ("(_/ < _)" [51, 51] 50)
264 less_eq ("op \<le>") and
265 less_eq ("(_/ \<le> _)" [51, 51] 50)
267 notation (HTML output)
268 less_eq ("op \<le>") and
269 less_eq ("(_/ \<le> _)" [51, 51] 50)
272 greater_eq (infix ">=" 50) where
273 "x >= y \<equiv> y <= x"
276 greater_eq (infix "\<ge>" 50)
279 greater (infix ">" 50) where
280 "x > y \<equiv> y < x"
285 "_index1" :: index ("\<^sub>1")
287 (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
289 typed_print_translation {*
291 fun tr' c = (c, fn show_sorts => fn T => fn ts =>
292 if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
293 else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
294 in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
295 *} -- {* show types that are presumably too general *}
298 subsection {* Fundamental rules *}
300 subsubsection {* Equality *}
302 text {* Thanks to Stephan Merz *}
304 assumes eq: "s = t" and p: "P s"
307 from eq have meta: "s \<equiv> t"
308 by (rule eq_reflection)
313 lemma sym: "s = t ==> t = s"
314 by (erule subst) (rule refl)
316 lemma ssubst: "t = s ==> P s ==> P t"
317 by (drule sym) (erule subst)
319 lemma trans: "[| r=s; s=t |] ==> r=t"
322 lemma meta_eq_to_obj_eq:
323 assumes meq: "A == B"
325 by (unfold meq) (rule refl)
327 text {* Useful with @{text erule} for proving equalities from known equalities. *}
331 lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
338 text {* For calculational reasoning: *}
340 lemma forw_subst: "a = b ==> P b ==> P a"
343 lemma back_subst: "P a ==> a = b ==> P b"
347 subsubsection {*Congruence rules for application*}
349 (*similar to AP_THM in Gordon's HOL*)
350 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
355 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
356 lemma arg_cong: "x=y ==> f(x)=f(y)"
361 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
362 apply (erule ssubst)+
366 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
372 subsubsection {*Equality of booleans -- iff*}
374 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
375 by (iprover intro: iff [THEN mp, THEN mp] impI assms)
377 lemma iffD2: "[| P=Q; Q |] ==> P"
380 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
383 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
384 by (drule sym) (rule iffD2)
386 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
387 by (drule sym) (rule rev_iffD2)
391 and minor: "[| P --> Q; Q --> P |] ==> R"
393 by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
396 subsubsection {*True*}
399 unfolding True_def by (rule refl)
401 lemma eqTrueI: "P ==> P = True"
402 by (iprover intro: iffI TrueI)
404 lemma eqTrueE: "P = True ==> P"
405 by (erule iffD2) (rule TrueI)
408 subsubsection {*Universal quantifier*}
410 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
411 unfolding All_def by (iprover intro: ext eqTrueI assms)
413 lemma spec: "ALL x::'a. P(x) ==> P(x)"
414 apply (unfold All_def)
416 apply (erule fun_cong)
420 assumes major: "ALL x. P(x)"
421 and minor: "P(x) ==> R"
423 by (iprover intro: minor major [THEN spec])
426 assumes major: "ALL x. P(x)"
427 and minor: "[| P(x); ALL x. P(x) |] ==> R"
429 by (iprover intro: minor major major [THEN spec])
432 subsubsection {* False *}
435 Depends upon @{text spec}; it is impossible to do propositional
436 logic before quantifiers!
439 lemma FalseE: "False ==> P"
440 apply (unfold False_def)
444 lemma False_neq_True: "False = True ==> P"
445 by (erule eqTrueE [THEN FalseE])
448 subsubsection {* Negation *}
451 assumes "P ==> False"
453 apply (unfold not_def)
454 apply (iprover intro: impI assms)
457 lemma False_not_True: "False ~= True"
459 apply (erule False_neq_True)
462 lemma True_not_False: "True ~= False"
465 apply (erule False_neq_True)
468 lemma notE: "[| ~P; P |] ==> R"
469 apply (unfold not_def)
470 apply (erule mp [THEN FalseE])
474 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
475 by (erule notE [THEN notI]) (erule meta_mp)
478 subsubsection {*Implication*}
481 assumes "P-->Q" "P" "Q ==> R"
483 by (iprover intro: assms mp)
485 (* Reduces Q to P-->Q, allowing substitution in P. *)
486 lemma rev_mp: "[| P; P --> Q |] ==> Q"
487 by (iprover intro: mp)
493 by (iprover intro: notI minor major [THEN notE])
495 (*not used at all, but we already have the other 3 combinations *)
498 and minor: "P ==> ~Q"
500 by (iprover intro: notI minor major notE)
502 lemma not_sym: "t ~= s ==> s ~= t"
503 by (erule contrapos_nn) (erule sym)
505 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
506 by (erule subst, erule ssubst, assumption)
508 (*still used in HOLCF*)
510 assumes pq: "P ==> Q"
513 apply (rule nq [THEN contrapos_nn])
517 subsubsection {*Existential quantifier*}
519 lemma exI: "P x ==> EX x::'a. P x"
520 apply (unfold Ex_def)
521 apply (iprover intro: allI allE impI mp)
525 assumes major: "EX x::'a. P(x)"
526 and minor: "!!x. P(x) ==> Q"
528 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
529 apply (iprover intro: impI [THEN allI] minor)
533 subsubsection {*Conjunction*}
535 lemma conjI: "[| P; Q |] ==> P&Q"
536 apply (unfold and_def)
537 apply (iprover intro: impI [THEN allI] mp)
540 lemma conjunct1: "[| P & Q |] ==> P"
541 apply (unfold and_def)
542 apply (iprover intro: impI dest: spec mp)
545 lemma conjunct2: "[| P & Q |] ==> Q"
546 apply (unfold and_def)
547 apply (iprover intro: impI dest: spec mp)
552 and minor: "[| P; Q |] ==> R"
555 apply (rule major [THEN conjunct1])
556 apply (rule major [THEN conjunct2])
560 assumes "P" "P ==> Q" shows "P & Q"
561 by (iprover intro: conjI assms)
564 subsubsection {*Disjunction*}
566 lemma disjI1: "P ==> P|Q"
567 apply (unfold or_def)
568 apply (iprover intro: allI impI mp)
571 lemma disjI2: "Q ==> P|Q"
572 apply (unfold or_def)
573 apply (iprover intro: allI impI mp)
578 and minorP: "P ==> R"
579 and minorQ: "Q ==> R"
581 by (iprover intro: minorP minorQ impI
582 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
585 subsubsection {*Classical logic*}
588 assumes prem: "~P ==> P"
590 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
592 apply (rule notI [THEN prem, THEN eqTrueI])
597 lemmas ccontr = FalseE [THEN classical, standard]
599 (*notE with premises exchanged; it discharges ~R so that it can be used to
600 make elimination rules*)
603 and premnot: "~R ==> ~P"
606 apply (erule notE [OF premnot premp])
609 (*Double negation law*)
610 lemma notnotD: "~~P ==> P"
611 apply (rule classical)
620 by (iprover intro: classical p1 p2 notE)
623 subsubsection {*Unique existence*}
626 assumes "P a" "!!x. P(x) ==> x=a"
628 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
630 text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
632 assumes ex_prem: "EX x. P(x)"
633 and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
635 by (iprover intro: ex_prem [THEN exE] ex1I eq)
638 assumes major: "EX! x. P(x)"
639 and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
641 apply (rule major [unfolded Ex1_def, THEN exE])
643 apply (iprover intro: minor)
646 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
653 subsubsection {*THE: definite description operator*}
657 and premx: "!!x. P x ==> x=a"
658 shows "(THE x. P x) = a"
659 apply (rule trans [OF _ the_eq_trivial])
660 apply (rule_tac f = "The" in arg_cong)
664 apply (erule ssubst, rule prema)
668 assumes "P a" and "!!x. P x ==> x=a"
669 shows "P (THE x. P x)"
670 by (iprover intro: assms the_equality [THEN ssubst])
672 lemma theI': "EX! x. P x ==> P (THE x. P x)"
680 (*Easier to apply than theI: only one occurrence of P*)
682 assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
683 shows "Q (THE x. P x)"
684 by (iprover intro: assms theI)
686 lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
687 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
690 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
691 apply (rule the_equality)
694 apply (erule all_dupE)
703 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
704 apply (rule the_equality)
710 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
713 assumes "~Q ==> P" shows "P|Q"
714 apply (rule classical)
715 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
718 lemma excluded_middle: "~P | P"
719 by (iprover intro: disjCI)
722 case distinction as a natural deduction rule.
723 Note that @{term "~P"} is the second case, not the first
725 lemma case_split [case_names True False]:
726 assumes prem1: "P ==> Q"
727 and prem2: "~P ==> Q"
729 apply (rule excluded_middle [THEN disjE])
734 (*Classical implies (-->) elimination. *)
736 assumes major: "P-->Q"
737 and minor: "~P ==> R" "Q ==> R"
739 apply (rule excluded_middle [of P, THEN disjE])
740 apply (iprover intro: minor major [THEN mp])+
743 (*This version of --> elimination works on Q before P. It works best for
744 those cases in which P holds "almost everywhere". Can't install as
745 default: would break old proofs.*)
747 assumes major: "P-->Q"
748 and minor: "Q ==> R" "~P ==> R"
750 apply (rule excluded_middle [of P, THEN disjE])
751 apply (iprover intro: minor major [THEN mp])+
754 (*Classical <-> elimination. *)
757 and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
759 apply (rule major [THEN iffE])
760 apply (iprover intro: minor elim: impCE notE)
764 assumes "ALL x. ~P(x) ==> P(a)"
767 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
771 subsubsection {* Intuitionistic Reasoning *}
776 and 3: "P --> Q ==> P"
779 from 3 and 1 have P .
780 with 1 have Q by (rule impE)
785 assumes 1: "ALL x. P x"
786 and 2: "P x ==> ALL x. P x ==> Q"
789 from 1 have "P x" by (rule spec)
790 from this and 1 show Q by (rule 2)
798 from 2 and 1 have P .
799 with 1 show R by (rule notE)
802 lemma TrueE: "True ==> P ==> P" .
803 lemma notFalseE: "~ False ==> P ==> P" .
805 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
806 and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
807 and [Pure.elim 2] = allE notE' impE'
808 and [Pure.intro] = exI disjI2 disjI1
810 lemmas [trans] = trans
811 and [sym] = sym not_sym
812 and [Pure.elim?] = iffD1 iffD2 impE
817 subsubsection {* Atomizing meta-level connectives *}
819 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
822 then show "ALL x. P x" ..
825 then show "!!x. P x" by (rule allE)
828 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
831 show "A --> B" by (rule impI) (rule r)
833 assume "A --> B" and A
834 then show B by (rule mp)
837 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
839 assume r: "A ==> False"
840 show "~A" by (rule notI) (rule r)
843 then show False by (rule notE)
846 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
849 show "x = y" by (unfold `x == y`) (rule refl)
852 then show "x == y" by (rule eq_reflection)
855 lemma atomize_conj [atomize]:
856 includes meta_conjunction_syntax
857 shows "(A && B) == Trueprop (A & B)"
859 assume conj: "A && B"
862 from conj show A by (rule conjunctionD1)
863 from conj show B by (rule conjunctionD2)
874 lemmas [symmetric, rulify] = atomize_all atomize_imp
875 and [symmetric, defn] = atomize_all atomize_imp atomize_eq
878 subsubsection {* Atomizing elimination rules *}
880 setup AtomizeElim.setup
882 lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
885 lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
888 lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
891 lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
894 subsection {* Package setup *}
896 subsubsection {* Classical Reasoner setup *}
898 lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
899 by (rule classical) iprover
901 lemma swap: "~ P ==> (~ R ==> P) ==> R"
902 by (rule classical) iprover
905 "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
908 structure Hypsubst = HypsubstFun(
910 structure Simplifier = Simplifier
911 val dest_eq = HOLogic.dest_eq
912 val dest_Trueprop = HOLogic.dest_Trueprop
913 val dest_imp = HOLogic.dest_imp
914 val eq_reflection = @{thm eq_reflection}
915 val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
916 val imp_intr = @{thm impI}
917 val rev_mp = @{thm rev_mp}
918 val subst = @{thm subst}
920 val thin_refl = @{thm thin_refl};
924 structure Classical = ClassicalFun(
926 val imp_elim = @{thm imp_elim}
927 val not_elim = @{thm notE}
928 val swap = @{thm swap}
929 val classical = @{thm classical}
930 val sizef = Drule.size_of_thm
931 val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
934 structure BasicClassical: BASIC_CLASSICAL = Classical;
937 ML_Antiquote.value "claset"
938 (Scan.succeed "Classical.local_claset_of (ML_Context.the_local_context ())");
940 structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
942 structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
945 text {*ResBlacklist holds theorems blacklisted to sledgehammer.
946 These theorems typically produce clauses that are prolific (match too many equality or
947 membership literals) and relate to seldom-used facts. Some duplicate other rules.*}
951 (*prevent substitution on bool*)
952 fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
953 Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
954 (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
956 Hypsubst.hypsubst_setup
957 #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
960 #> ResBlacklist.setup
964 declare iffI [intro!]
972 declare iffCE [elim!]
979 declare ex_ex1I [intro!]
981 and the_equality [intro]
987 ML {* val HOL_cs = @{claset} *}
989 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
991 apply (erule (1) meta_mp)
994 declare ex_ex1I [rule del, intro! 2]
997 lemmas [intro?] = ext
998 and [elim?] = ex1_implies_ex
1000 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
1001 lemma alt_ex1E [elim!]:
1002 assumes major: "\<exists>!x. P x"
1003 and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
1005 apply (rule ex1E [OF major])
1007 apply (tactic {* ares_tac @{thms allI} 1 *})+
1008 apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
1013 structure Blast = BlastFun
1015 type claset = Classical.claset
1016 val equality_name = @{const_name "op ="}
1017 val not_name = @{const_name Not}
1018 val notE = @{thm notE}
1019 val ccontr = @{thm ccontr}
1020 val contr_tac = Classical.contr_tac
1021 val dup_intr = Classical.dup_intr
1022 val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
1023 val claset = Classical.claset
1024 val rep_cs = Classical.rep_cs
1025 val cla_modifiers = Classical.cla_modifiers
1026 val cla_meth' = Classical.cla_meth'
1028 val Blast_tac = Blast.Blast_tac;
1029 val blast_tac = Blast.blast_tac;
1035 subsubsection {* Simplifier *}
1037 lemma eta_contract_eq: "(%s. f s) = f" ..
1040 shows not_not: "(~ ~ P) = P"
1041 and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
1043 "(P ~= Q) = (P = (~Q))"
1044 "(P | ~P) = True" "(~P | P) = True"
1046 and not_True_eq_False: "(\<not> True) = False"
1047 and not_False_eq_True: "(\<not> False) = True"
1049 "(~P) ~= P" "P ~= (~P)"
1051 and eq_True: "(P = True) = P"
1052 and "(False=P) = (~P)"
1053 and eq_False: "(P = False) = (\<not> P)"
1055 "(True --> P) = P" "(False --> P) = True"
1056 "(P --> True) = True" "(P --> P) = True"
1057 "(P --> False) = (~P)" "(P --> ~P) = (~P)"
1058 "(P & True) = P" "(True & P) = P"
1059 "(P & False) = False" "(False & P) = False"
1060 "(P & P) = P" "(P & (P & Q)) = (P & Q)"
1061 "(P & ~P) = False" "(~P & P) = False"
1062 "(P | True) = True" "(True | P) = True"
1063 "(P | False) = P" "(False | P) = P"
1064 "(P | P) = P" "(P | (P | Q)) = (P | Q)" and
1065 "(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
1066 -- {* needed for the one-point-rule quantifier simplification procs *}
1067 -- {* essential for termination!! *} and
1068 "!!P. (EX x. x=t & P(x)) = P(t)"
1069 "!!P. (EX x. t=x & P(x)) = P(t)"
1070 "!!P. (ALL x. x=t --> P(x)) = P(t)"
1071 "!!P. (ALL x. t=x --> P(x)) = P(t)"
1072 by (blast, blast, blast, blast, blast, iprover+)
1074 lemma disj_absorb: "(A | A) = A"
1077 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
1080 lemma conj_absorb: "(A & A) = A"
1083 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
1087 shows eq_commute: "(a=b) = (b=a)"
1088 and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
1089 and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
1090 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
1093 shows conj_commute: "(P&Q) = (Q&P)"
1094 and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
1095 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
1097 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
1100 shows disj_commute: "(P|Q) = (Q|P)"
1101 and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
1102 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
1104 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
1106 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
1107 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
1109 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
1110 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
1112 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
1113 lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
1114 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
1116 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
1117 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
1118 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
1120 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
1121 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
1123 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
1126 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
1127 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
1128 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
1129 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
1130 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
1131 lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
1133 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
1135 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
1138 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
1139 -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
1140 -- {* cases boil down to the same thing. *}
1143 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
1144 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
1145 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
1146 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
1147 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
1149 declare All_def [noatp]
1151 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
1152 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
1155 \medskip The @{text "&"} congruence rule: not included by default!
1156 May slow rewrite proofs down by as much as 50\% *}
1159 "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
1162 lemma rev_conj_cong:
1163 "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
1166 text {* The @{text "|"} congruence rule: not included by default! *}
1169 "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
1173 text {* \medskip if-then-else rules *}
1175 lemma if_True: "(if True then x else y) = x"
1176 by (unfold if_def) blast
1178 lemma if_False: "(if False then x else y) = y"
1179 by (unfold if_def) blast
1181 lemma if_P: "P ==> (if P then x else y) = x"
1182 by (unfold if_def) blast
1184 lemma if_not_P: "~P ==> (if P then x else y) = y"
1185 by (unfold if_def) blast
1187 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
1188 apply (rule case_split [of Q])
1189 apply (simplesubst if_P)
1190 prefer 3 apply (simplesubst if_not_P, blast+)
1193 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
1194 by (simplesubst split_if, blast)
1196 lemmas if_splits [noatp] = split_if split_if_asm
1198 lemma if_cancel: "(if c then x else x) = x"
1199 by (simplesubst split_if, blast)
1201 lemma if_eq_cancel: "(if x = y then y else x) = x"
1202 by (simplesubst split_if, blast)
1204 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
1205 -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
1208 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
1209 -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
1210 apply (simplesubst split_if, blast)
1213 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
1214 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
1216 text {* \medskip let rules for simproc *}
1218 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
1221 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
1225 The following copy of the implication operator is useful for
1226 fine-tuning congruence rules. It instructs the simplifier to simplify
1231 simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
1232 [code func del]: "simp_implies \<equiv> op ==>"
1234 lemma simp_impliesI:
1235 assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1236 shows "PROP P =simp=> PROP Q"
1237 apply (unfold simp_implies_def)
1242 lemma simp_impliesE:
1243 assumes PQ: "PROP P =simp=> PROP Q"
1245 and QR: "PROP Q \<Longrightarrow> PROP R"
1248 apply (rule PQ [unfolded simp_implies_def])
1252 lemma simp_implies_cong:
1253 assumes PP' :"PROP P == PROP P'"
1254 and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
1255 shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
1256 proof (unfold simp_implies_def, rule equal_intr_rule)
1257 assume PQ: "PROP P \<Longrightarrow> PROP Q"
1259 from PP' [symmetric] and P' have "PROP P"
1260 by (rule equal_elim_rule1)
1261 then have "PROP Q" by (rule PQ)
1262 with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1264 assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1266 from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1267 then have "PROP Q'" by (rule P'Q')
1268 with P'QQ' [OF P', symmetric] show "PROP Q"
1269 by (rule equal_elim_rule1)
1273 assumes "P \<longrightarrow> Q \<longrightarrow> R"
1274 shows "P \<and> Q \<longrightarrow> R"
1275 using assms by blast
1278 assumes "\<And>x. P x = Q x"
1279 shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1280 using assms by blast
1283 assumes "\<And>x. P x = Q x"
1284 shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1285 using assms by blast
1288 "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1292 "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1296 ML {* open Simpdata *}
1299 Simplifier.method_setup Splitter.split_modifiers
1300 #> Simplifier.map_simpset (K Simpdata.simpset_simprocs)
1306 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
1308 simproc_setup neq ("x = y") = {* fn _ =>
1310 val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
1311 fun is_neq eq lhs rhs thm =
1312 (case Thm.prop_of thm of
1313 _ $ (Not $ (eq' $ l' $ r')) =>
1314 Not = HOLogic.Not andalso eq' = eq andalso
1315 r' aconv lhs andalso l' aconv rhs
1318 (case Thm.term_of ct of
1320 (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
1321 SOME thm => SOME (thm RS neq_to_EQ_False)
1327 simproc_setup let_simp ("Let x f") = {*
1329 val (f_Let_unfold, x_Let_unfold) =
1330 let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
1331 in (cterm_of @{theory} f, cterm_of @{theory} x) end
1332 val (f_Let_folded, x_Let_folded) =
1333 let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
1334 in (cterm_of @{theory} f, cterm_of @{theory} x) end;
1336 let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
1340 val ctxt = Simplifier.the_context ss;
1341 val thy = ProofContext.theory_of ctxt;
1342 val t = Thm.term_of ct;
1343 val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
1344 in Option.map (hd o Variable.export ctxt' ctxt o single)
1345 (case t' of Const ("Let",_) $ x $ f => (* x and f are already in normal form *)
1346 if is_Free x orelse is_Bound x orelse is_Const x
1347 then SOME @{thm Let_def}
1350 val n = case f of (Abs (x,_,_)) => x | _ => "x";
1351 val cx = cterm_of thy x;
1352 val {T=xT,...} = rep_cterm cx;
1353 val cf = cterm_of thy f;
1354 val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
1355 val (_$_$g) = prop_of fx_g;
1356 val g' = abstract_over (x,g);
1361 cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
1362 in SOME (rl OF [fx_g]) end
1363 else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
1365 val abs_g'= Abs (n,xT,g');
1367 val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
1368 val rl = cterm_instantiate
1369 [(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
1370 (g_Let_folded,cterm_of thy abs_g')]
1372 in SOME (rl OF [transitive fx_g g_g'x])
1380 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1382 assume "True \<Longrightarrow> PROP P"
1383 from this [OF TrueI] show "PROP P" .
1386 then show "PROP P" .
1390 "!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
1391 "!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
1392 "!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
1393 "!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
1394 "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
1395 "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
1396 -- {* Miniscoping: pushing in existential quantifiers. *}
1397 by (iprover | blast)+
1400 "!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
1401 "!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
1402 "!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
1403 "!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
1404 "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
1405 "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
1406 -- {* Miniscoping: pushing in universal quantifiers. *}
1407 by (iprover | blast)+
1410 triv_forall_equality (*prunes params*)
1411 True_implies_equals (*prune asms `True'*)
1417 (*In general it seems wrong to add distributive laws by default: they
1418 might cause exponential blow-up. But imp_disjL has been in for a while
1419 and cannot be removed without affecting existing proofs. Moreover,
1420 rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
1421 grounds that it allows simplification of R in the two cases.*)
1439 lemmas [cong] = imp_cong simp_implies_cong
1440 lemmas [split] = split_if
1442 ML {* val HOL_ss = @{simpset} *}
1444 text {* Simplifies x assuming c and y assuming ~c *}
1447 and "c \<Longrightarrow> x = u"
1448 and "\<not> c \<Longrightarrow> y = v"
1449 shows "(if b then x else y) = (if c then u else v)"
1450 unfolding if_def using assms by simp
1452 text {* Prevents simplification of x and y:
1453 faster and allows the execution of functional programs. *}
1454 lemma if_weak_cong [cong]:
1456 shows "(if b then x else y) = (if c then x else y)"
1457 using assms by (rule arg_cong)
1459 text {* Prevents simplification of t: much faster *}
1460 lemma let_weak_cong:
1462 shows "(let x = a in t x) = (let x = b in t x)"
1463 using assms by (rule arg_cong)
1465 text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
1468 shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1472 "f (if c then x else y) = (if c then f x else f y)"
1475 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
1476 side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *}
1477 lemma restrict_to_left:
1479 shows "(x = z) = (y = z)"
1483 subsubsection {* Generic cases and induction *}
1485 text {* Rule projections: *}
1488 structure ProjectRule = ProjectRuleFun
1490 val conjunct1 = @{thm conjunct1}
1491 val conjunct2 = @{thm conjunct2}
1497 induct_forall where "induct_forall P == \<forall>x. P x"
1498 induct_implies where "induct_implies A B == A \<longrightarrow> B"
1499 induct_equal where "induct_equal x y == x = y"
1500 induct_conj where "induct_conj A B == A \<and> B"
1502 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
1503 by (unfold atomize_all induct_forall_def)
1505 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
1506 by (unfold atomize_imp induct_implies_def)
1508 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
1509 by (unfold atomize_eq induct_equal_def)
1511 lemma induct_conj_eq:
1512 includes meta_conjunction_syntax
1513 shows "(A && B) == Trueprop (induct_conj A B)"
1514 by (unfold atomize_conj induct_conj_def)
1516 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
1517 lemmas induct_rulify [symmetric, standard] = induct_atomize
1518 lemmas induct_rulify_fallback =
1519 induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1522 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1523 induct_conj (induct_forall A) (induct_forall B)"
1524 by (unfold induct_forall_def induct_conj_def) iprover
1526 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1527 induct_conj (induct_implies C A) (induct_implies C B)"
1528 by (unfold induct_implies_def induct_conj_def) iprover
1530 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
1532 assume r: "induct_conj A B ==> PROP C" and A B
1533 show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
1535 assume r: "A ==> B ==> PROP C" and "induct_conj A B"
1536 show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
1539 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1541 hide const induct_forall induct_implies induct_equal induct_conj
1543 text {* Method setup. *}
1546 structure Induct = InductFun
1548 val cases_default = @{thm case_split}
1549 val atomize = @{thms induct_atomize}
1550 val rulify = @{thms induct_rulify}
1551 val rulify_fallback = @{thms induct_rulify_fallback}
1557 use "~~/src/Tools/induct_tacs.ML"
1558 setup InductTacs.setup
1561 subsection {* Other simple lemmas and lemma duplicates *}
1563 lemma Let_0 [simp]: "Let 0 f = f 0"
1564 unfolding Let_def ..
1566 lemma Let_1 [simp]: "Let 1 f = f 1"
1567 unfolding Let_def ..
1569 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
1572 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
1574 apply (rule_tac a = "%x. THE y. P x y" in ex1I)
1575 apply (fast dest!: theI')
1576 apply (fast intro: ext the1_equality [symmetric])
1581 apply (erule_tac x = "%z. if z = x then y else f z" in allE)
1584 apply (case_tac "xa = x")
1585 apply (drule_tac [3] x = x in fun_cong, simp_all)
1588 lemma mk_left_commute:
1589 fixes f (infix "\<otimes>" 60)
1590 assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
1591 c: "\<And>x y. x \<otimes> y = y \<otimes> x"
1592 shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
1593 by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
1595 lemmas eq_sym_conv = eq_commute
1598 "(\<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)"
1599 "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
1600 "(\<not> \<not>(P)) = P"
1604 subsection {* Basic ML bindings *}
1607 val FalseE = @{thm FalseE}
1608 val Let_def = @{thm Let_def}
1609 val TrueI = @{thm TrueI}
1610 val allE = @{thm allE}
1611 val allI = @{thm allI}
1612 val all_dupE = @{thm all_dupE}
1613 val arg_cong = @{thm arg_cong}
1614 val box_equals = @{thm box_equals}
1615 val ccontr = @{thm ccontr}
1616 val classical = @{thm classical}
1617 val conjE = @{thm conjE}
1618 val conjI = @{thm conjI}
1619 val conjunct1 = @{thm conjunct1}
1620 val conjunct2 = @{thm conjunct2}
1621 val disjCI = @{thm disjCI}
1622 val disjE = @{thm disjE}
1623 val disjI1 = @{thm disjI1}
1624 val disjI2 = @{thm disjI2}
1625 val eq_reflection = @{thm eq_reflection}
1626 val ex1E = @{thm ex1E}
1627 val ex1I = @{thm ex1I}
1628 val ex1_implies_ex = @{thm ex1_implies_ex}
1629 val exE = @{thm exE}
1630 val exI = @{thm exI}
1631 val excluded_middle = @{thm excluded_middle}
1632 val ext = @{thm ext}
1633 val fun_cong = @{thm fun_cong}
1634 val iffD1 = @{thm iffD1}
1635 val iffD2 = @{thm iffD2}
1636 val iffI = @{thm iffI}
1637 val impE = @{thm impE}
1638 val impI = @{thm impI}
1639 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
1641 val notE = @{thm notE}
1642 val notI = @{thm notI}
1643 val not_all = @{thm not_all}
1644 val not_ex = @{thm not_ex}
1645 val not_iff = @{thm not_iff}
1646 val not_not = @{thm not_not}
1647 val not_sym = @{thm not_sym}
1648 val refl = @{thm refl}
1649 val rev_mp = @{thm rev_mp}
1650 val spec = @{thm spec}
1651 val ssubst = @{thm ssubst}
1652 val subst = @{thm subst}
1653 val sym = @{thm sym}
1654 val trans = @{thm trans}
1658 subsection {* Code generator basic setup -- see further @{text Code_Setup.thy} *}
1660 code_datatype Trueprop "prop"
1662 code_datatype "TYPE('a\<Colon>{})"
1664 lemma Let_case_cert:
1665 assumes "CASE \<equiv> (\<lambda>x. Let x f)"
1666 shows "CASE x \<equiv> f x"
1667 using assms by simp_all
1670 includes meta_conjunction_syntax
1671 assumes "CASE \<equiv> (\<lambda>b. If b f g)"
1672 shows "(CASE True \<equiv> f) && (CASE False \<equiv> g)"
1673 using assms by simp_all
1676 Code.add_case @{thm Let_case_cert}
1677 #> Code.add_case @{thm If_case_cert}
1678 #> Code.add_undefined @{const_name undefined}
1682 fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1683 assumes eq: "eq x y \<longleftrightarrow> x = y "
1686 lemma equals_eq [code inline, code func]: "op = \<equiv> eq"
1687 by (rule eq_reflection) (rule ext, rule ext, rule sym, rule eq)
1689 declare equals_eq [symmetric, code post]
1693 hide (open) const eq
1697 CodeUnit.add_const_alias @{thm equals_eq}
1704 shows "False \<and> x \<longleftrightarrow> False"
1705 and "True \<and> x \<longleftrightarrow> x"
1706 and "x \<and> False \<longleftrightarrow> False"
1707 and "x \<and> True \<longleftrightarrow> x" by simp_all
1710 shows "False \<or> x \<longleftrightarrow> x"
1711 and "True \<or> x \<longleftrightarrow> True"
1712 and "x \<or> False \<longleftrightarrow> x"
1713 and "x \<or> True \<longleftrightarrow> True" by simp_all
1716 shows "\<not> True \<longleftrightarrow> False"
1717 and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
1721 subsection {* Legacy tactics and ML bindings *}
1724 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
1726 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1728 fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1729 | wrong_prem (Bound _) = true
1730 | wrong_prem _ = false;
1731 val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
1733 fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
1734 fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
1737 val all_conj_distrib = thm "all_conj_distrib";
1738 val all_simps = thms "all_simps";
1739 val atomize_not = thm "atomize_not";
1740 val case_split = thm "case_split";
1741 val cases_simp = thm "cases_simp";
1742 val choice_eq = thm "choice_eq"
1743 val cong = thm "cong"
1744 val conj_comms = thms "conj_comms";
1745 val conj_cong = thm "conj_cong";
1746 val de_Morgan_conj = thm "de_Morgan_conj";
1747 val de_Morgan_disj = thm "de_Morgan_disj";
1748 val disj_assoc = thm "disj_assoc";
1749 val disj_comms = thms "disj_comms";
1750 val disj_cong = thm "disj_cong";
1751 val eq_ac = thms "eq_ac";
1752 val eq_cong2 = thm "eq_cong2"
1753 val Eq_FalseI = thm "Eq_FalseI";
1754 val Eq_TrueI = thm "Eq_TrueI";
1755 val Ex1_def = thm "Ex1_def"
1756 val ex_disj_distrib = thm "ex_disj_distrib";
1757 val ex_simps = thms "ex_simps";
1758 val if_cancel = thm "if_cancel";
1759 val if_eq_cancel = thm "if_eq_cancel";
1760 val if_False = thm "if_False";
1761 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
1763 val if_splits = thms "if_splits";
1764 val if_True = thm "if_True";
1765 val if_weak_cong = thm "if_weak_cong"
1766 val imp_all = thm "imp_all";
1767 val imp_cong = thm "imp_cong";
1768 val imp_conjL = thm "imp_conjL";
1769 val imp_conjR = thm "imp_conjR";
1770 val imp_conv_disj = thm "imp_conv_disj";
1771 val simp_implies_def = thm "simp_implies_def";
1772 val simp_thms = thms "simp_thms";
1773 val split_if = thm "split_if";
1774 val the1_equality = thm "the1_equality"
1775 val theI = thm "theI"
1776 val theI' = thm "theI'"
1777 val True_implies_equals = thm "True_implies_equals";
1778 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})