2 Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
5 header {* The basis of Higher-Order Logic *}
11 "~~/src/Tools/IsaPlanner/zipper.ML"
12 "~~/src/Tools/IsaPlanner/isand.ML"
13 "~~/src/Tools/IsaPlanner/rw_tools.ML"
14 "~~/src/Tools/IsaPlanner/rw_inst.ML"
15 "~~/src/Provers/project_rule.ML"
16 "~~/src/Provers/hypsubst.ML"
17 "~~/src/Provers/splitter.ML"
18 "~~/src/Provers/classical.ML"
19 "~~/src/Provers/blast.ML"
20 "~~/src/Provers/clasimp.ML"
21 "~~/src/Provers/coherent.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/value.ML"
30 "~~/src/Tools/code/code_name.ML"
31 "~~/src/Tools/code/code_funcgr.ML"
32 "~~/src/Tools/code/code_thingol.ML"
33 "~~/src/Tools/code/code_printer.ML"
34 "~~/src/Tools/code/code_target.ML"
35 "~~/src/Tools/code/code_ml.ML"
36 "~~/src/Tools/code/code_haskell.ML"
38 ("Tools/recfun_codegen.ML")
41 subsection {* Primitive logic *}
43 subsubsection {* Core syntax *}
47 setup {* ObjectLogic.add_base_sort @{sort type} *}
50 "fun" :: (type, type) type
58 Trueprop :: "bool => prop" ("(_)" 5)
61 Not :: "bool => bool" ("~ _" [40] 40)
65 The :: "('a => bool) => 'a"
66 All :: "('a => bool) => bool" (binder "ALL " 10)
67 Ex :: "('a => bool) => bool" (binder "EX " 10)
68 Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
69 Let :: "['a, 'a => 'b] => 'b"
71 "op =" :: "['a, 'a] => bool" (infixl "=" 50)
72 "op &" :: "[bool, bool] => bool" (infixr "&" 35)
73 "op |" :: "[bool, bool] => bool" (infixr "|" 30)
74 "op -->" :: "[bool, bool] => bool" (infixr "-->" 25)
79 If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
82 subsubsection {* Additional concrete syntax *}
88 not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
92 not_equal (infix "~=" 50)
95 Not ("\<not> _" [40] 40) and
96 "op &" (infixr "\<and>" 35) and
97 "op |" (infixr "\<or>" 30) and
98 "op -->" (infixr "\<longrightarrow>" 25) and
99 not_equal (infix "\<noteq>" 50)
101 notation (HTML output)
102 Not ("\<not> _" [40] 40) and
103 "op &" (infixr "\<and>" 35) and
104 "op |" (infixr "\<or>" 30) and
105 not_equal (infix "\<noteq>" 50)
108 iff :: "[bool, bool] => bool" (infixr "<->" 25) where
112 iff (infixr "\<longleftrightarrow>" 25)
120 "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
122 "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
123 "" :: "letbind => letbinds" ("_")
124 "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
125 "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
127 "_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
128 "_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
129 "" :: "case_syn => cases_syn" ("_")
130 "_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
133 "THE x. P" == "The (%x. P)"
134 "_Let (_binds b bs) e" == "_Let b (_Let bs e)"
135 "let x = a in e" == "Let a (%x. e)"
138 (* To avoid eta-contraction of body: *)
139 [("The", fn [Abs abs] =>
140 let val (x,t) = atomic_abs_tr' abs
141 in Syntax.const "_The" $ x $ t end)]
145 "_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
148 All (binder "\<forall>" 10) and
149 Ex (binder "\<exists>" 10) and
150 Ex1 (binder "\<exists>!" 10)
152 notation (HTML output)
153 All (binder "\<forall>" 10) and
154 Ex (binder "\<exists>" 10) and
155 Ex1 (binder "\<exists>!" 10)
158 All (binder "! " 10) and
159 Ex (binder "? " 10) and
160 Ex1 (binder "?! " 10)
163 subsubsection {* Axioms and basic definitions *}
167 subst: "s = t \<Longrightarrow> P s \<Longrightarrow> P t"
168 ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
169 -- {*Extensionality is built into the meta-logic, and this rule expresses
170 a related property. It is an eta-expanded version of the traditional
171 rule, and similar to the ABS rule of HOL*}
173 the_eq_trivial: "(THE x. x = a) = (a::'a)"
175 impI: "(P ==> Q) ==> P-->Q"
176 mp: "[| P-->Q; P |] ==> Q"
180 True_def: "True == ((%x::bool. x) = (%x. x))"
181 All_def: "All(P) == (P = (%x. True))"
182 Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
183 False_def: "False == (!P. P)"
184 not_def: "~ P == P-->False"
185 and_def: "P & Q == !R. (P-->Q-->R) --> R"
186 or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
187 Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
190 iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
191 True_or_False: "(P=True) | (P=False)"
194 Let_def: "Let s f == f(s)"
195 if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
206 "arbitrary \<equiv> undefined"
209 subsubsection {* Generic classes and algebraic operations *}
215 fixes zero :: 'a ("0")
218 fixes one :: 'a ("1")
220 hide (open) const zero one
223 fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
226 fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
229 fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
232 fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
235 fixes inverse :: "'a \<Rightarrow> 'a"
236 and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
239 fixes abs :: "'a \<Rightarrow> 'a"
243 abs ("\<bar>_\<bar>")
245 notation (HTML output)
246 abs ("\<bar>_\<bar>")
251 fixes sgn :: "'a \<Rightarrow> 'a"
254 fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
255 and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
259 less_eq ("op <=") and
260 less_eq ("(_/ <= _)" [51, 51] 50) and
262 less ("(_/ < _)" [51, 51] 50)
265 less_eq ("op \<le>") and
266 less_eq ("(_/ \<le> _)" [51, 51] 50)
268 notation (HTML output)
269 less_eq ("op \<le>") and
270 less_eq ("(_/ \<le> _)" [51, 51] 50)
273 greater_eq (infix ">=" 50) where
274 "x >= y \<equiv> y <= x"
277 greater_eq (infix "\<ge>" 50)
280 greater (infix ">" 50) where
281 "x > y \<equiv> y < x"
286 "_index1" :: index ("\<^sub>1")
288 (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
290 typed_print_translation {*
292 fun tr' c = (c, fn show_sorts => fn T => fn ts =>
293 if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
294 else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
295 in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
296 *} -- {* show types that are presumably too general *}
299 subsection {* Fundamental rules *}
301 subsubsection {* Equality *}
303 lemma sym: "s = t ==> t = s"
304 by (erule subst) (rule refl)
306 lemma ssubst: "t = s ==> P s ==> P t"
307 by (drule sym) (erule subst)
309 lemma trans: "[| r=s; s=t |] ==> r=t"
312 lemma meta_eq_to_obj_eq:
313 assumes meq: "A == B"
315 by (unfold meq) (rule refl)
317 text {* Useful with @{text erule} for proving equalities from known equalities. *}
321 lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
328 text {* For calculational reasoning: *}
330 lemma forw_subst: "a = b ==> P b ==> P a"
333 lemma back_subst: "P a ==> a = b ==> P b"
337 subsubsection {*Congruence rules for application*}
339 (*similar to AP_THM in Gordon's HOL*)
340 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
345 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
346 lemma arg_cong: "x=y ==> f(x)=f(y)"
351 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
352 apply (erule ssubst)+
356 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
362 subsubsection {*Equality of booleans -- iff*}
364 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
365 by (iprover intro: iff [THEN mp, THEN mp] impI assms)
367 lemma iffD2: "[| P=Q; Q |] ==> P"
370 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
373 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
374 by (drule sym) (rule iffD2)
376 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
377 by (drule sym) (rule rev_iffD2)
381 and minor: "[| P --> Q; Q --> P |] ==> R"
383 by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
386 subsubsection {*True*}
389 unfolding True_def by (rule refl)
391 lemma eqTrueI: "P ==> P = True"
392 by (iprover intro: iffI TrueI)
394 lemma eqTrueE: "P = True ==> P"
395 by (erule iffD2) (rule TrueI)
398 subsubsection {*Universal quantifier*}
400 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
401 unfolding All_def by (iprover intro: ext eqTrueI assms)
403 lemma spec: "ALL x::'a. P(x) ==> P(x)"
404 apply (unfold All_def)
406 apply (erule fun_cong)
410 assumes major: "ALL x. P(x)"
411 and minor: "P(x) ==> R"
413 by (iprover intro: minor major [THEN spec])
416 assumes major: "ALL x. P(x)"
417 and minor: "[| P(x); ALL x. P(x) |] ==> R"
419 by (iprover intro: minor major major [THEN spec])
422 subsubsection {* False *}
425 Depends upon @{text spec}; it is impossible to do propositional
426 logic before quantifiers!
429 lemma FalseE: "False ==> P"
430 apply (unfold False_def)
434 lemma False_neq_True: "False = True ==> P"
435 by (erule eqTrueE [THEN FalseE])
438 subsubsection {* Negation *}
441 assumes "P ==> False"
443 apply (unfold not_def)
444 apply (iprover intro: impI assms)
447 lemma False_not_True: "False ~= True"
449 apply (erule False_neq_True)
452 lemma True_not_False: "True ~= False"
455 apply (erule False_neq_True)
458 lemma notE: "[| ~P; P |] ==> R"
459 apply (unfold not_def)
460 apply (erule mp [THEN FalseE])
464 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
465 by (erule notE [THEN notI]) (erule meta_mp)
468 subsubsection {*Implication*}
471 assumes "P-->Q" "P" "Q ==> R"
473 by (iprover intro: assms mp)
475 (* Reduces Q to P-->Q, allowing substitution in P. *)
476 lemma rev_mp: "[| P; P --> Q |] ==> Q"
477 by (iprover intro: mp)
483 by (iprover intro: notI minor major [THEN notE])
485 (*not used at all, but we already have the other 3 combinations *)
488 and minor: "P ==> ~Q"
490 by (iprover intro: notI minor major notE)
492 lemma not_sym: "t ~= s ==> s ~= t"
493 by (erule contrapos_nn) (erule sym)
495 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
496 by (erule subst, erule ssubst, assumption)
498 (*still used in HOLCF*)
500 assumes pq: "P ==> Q"
503 apply (rule nq [THEN contrapos_nn])
507 subsubsection {*Existential quantifier*}
509 lemma exI: "P x ==> EX x::'a. P x"
510 apply (unfold Ex_def)
511 apply (iprover intro: allI allE impI mp)
515 assumes major: "EX x::'a. P(x)"
516 and minor: "!!x. P(x) ==> Q"
518 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
519 apply (iprover intro: impI [THEN allI] minor)
523 subsubsection {*Conjunction*}
525 lemma conjI: "[| P; Q |] ==> P&Q"
526 apply (unfold and_def)
527 apply (iprover intro: impI [THEN allI] mp)
530 lemma conjunct1: "[| P & Q |] ==> P"
531 apply (unfold and_def)
532 apply (iprover intro: impI dest: spec mp)
535 lemma conjunct2: "[| P & Q |] ==> Q"
536 apply (unfold and_def)
537 apply (iprover intro: impI dest: spec mp)
542 and minor: "[| P; Q |] ==> R"
545 apply (rule major [THEN conjunct1])
546 apply (rule major [THEN conjunct2])
550 assumes "P" "P ==> Q" shows "P & Q"
551 by (iprover intro: conjI assms)
554 subsubsection {*Disjunction*}
556 lemma disjI1: "P ==> P|Q"
557 apply (unfold or_def)
558 apply (iprover intro: allI impI mp)
561 lemma disjI2: "Q ==> P|Q"
562 apply (unfold or_def)
563 apply (iprover intro: allI impI mp)
568 and minorP: "P ==> R"
569 and minorQ: "Q ==> R"
571 by (iprover intro: minorP minorQ impI
572 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
575 subsubsection {*Classical logic*}
578 assumes prem: "~P ==> P"
580 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
582 apply (rule notI [THEN prem, THEN eqTrueI])
587 lemmas ccontr = FalseE [THEN classical, standard]
589 (*notE with premises exchanged; it discharges ~R so that it can be used to
590 make elimination rules*)
593 and premnot: "~R ==> ~P"
596 apply (erule notE [OF premnot premp])
599 (*Double negation law*)
600 lemma notnotD: "~~P ==> P"
601 apply (rule classical)
610 by (iprover intro: classical p1 p2 notE)
613 subsubsection {*Unique existence*}
616 assumes "P a" "!!x. P(x) ==> x=a"
618 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
620 text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
622 assumes ex_prem: "EX x. P(x)"
623 and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
625 by (iprover intro: ex_prem [THEN exE] ex1I eq)
628 assumes major: "EX! x. P(x)"
629 and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
631 apply (rule major [unfolded Ex1_def, THEN exE])
633 apply (iprover intro: minor)
636 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
643 subsubsection {*THE: definite description operator*}
647 and premx: "!!x. P x ==> x=a"
648 shows "(THE x. P x) = a"
649 apply (rule trans [OF _ the_eq_trivial])
650 apply (rule_tac f = "The" in arg_cong)
654 apply (erule ssubst, rule prema)
658 assumes "P a" and "!!x. P x ==> x=a"
659 shows "P (THE x. P x)"
660 by (iprover intro: assms the_equality [THEN ssubst])
662 lemma theI': "EX! x. P x ==> P (THE x. P x)"
670 (*Easier to apply than theI: only one occurrence of P*)
672 assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
673 shows "Q (THE x. P x)"
674 by (iprover intro: assms theI)
676 lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
677 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
680 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
681 apply (rule the_equality)
684 apply (erule all_dupE)
693 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
694 apply (rule the_equality)
700 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
703 assumes "~Q ==> P" shows "P|Q"
704 apply (rule classical)
705 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
708 lemma excluded_middle: "~P | P"
709 by (iprover intro: disjCI)
712 case distinction as a natural deduction rule.
713 Note that @{term "~P"} is the second case, not the first
715 lemma case_split [case_names True False]:
716 assumes prem1: "P ==> Q"
717 and prem2: "~P ==> Q"
719 apply (rule excluded_middle [THEN disjE])
724 (*Classical implies (-->) elimination. *)
726 assumes major: "P-->Q"
727 and minor: "~P ==> R" "Q ==> R"
729 apply (rule excluded_middle [of P, THEN disjE])
730 apply (iprover intro: minor major [THEN mp])+
733 (*This version of --> elimination works on Q before P. It works best for
734 those cases in which P holds "almost everywhere". Can't install as
735 default: would break old proofs.*)
737 assumes major: "P-->Q"
738 and minor: "Q ==> R" "~P ==> R"
740 apply (rule excluded_middle [of P, THEN disjE])
741 apply (iprover intro: minor major [THEN mp])+
744 (*Classical <-> elimination. *)
747 and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
749 apply (rule major [THEN iffE])
750 apply (iprover intro: minor elim: impCE notE)
754 assumes "ALL x. ~P(x) ==> P(a)"
757 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
761 subsubsection {* Intuitionistic Reasoning *}
766 and 3: "P --> Q ==> P"
769 from 3 and 1 have P .
770 with 1 have Q by (rule impE)
775 assumes 1: "ALL x. P x"
776 and 2: "P x ==> ALL x. P x ==> Q"
779 from 1 have "P x" by (rule spec)
780 from this and 1 show Q by (rule 2)
788 from 2 and 1 have P .
789 with 1 show R by (rule notE)
792 lemma TrueE: "True ==> P ==> P" .
793 lemma notFalseE: "~ False ==> P ==> P" .
795 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
796 and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
797 and [Pure.elim 2] = allE notE' impE'
798 and [Pure.intro] = exI disjI2 disjI1
800 lemmas [trans] = trans
801 and [sym] = sym not_sym
802 and [Pure.elim?] = iffD1 iffD2 impE
804 use "Tools/hologic.ML"
807 subsubsection {* Atomizing meta-level connectives *}
810 eq_reflection: "x = y \<Longrightarrow> x \<equiv> y" (*admissible axiom*)
812 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
815 then show "ALL x. P x" ..
818 then show "!!x. P x" by (rule allE)
821 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
824 show "A --> B" by (rule impI) (rule r)
826 assume "A --> B" and A
827 then show B by (rule mp)
830 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
832 assume r: "A ==> False"
833 show "~A" by (rule notI) (rule r)
836 then show False by (rule notE)
839 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
842 show "x = y" by (unfold `x == y`) (rule refl)
845 then show "x == y" by (rule eq_reflection)
848 lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
850 assume conj: "A &&& B"
853 from conj show A by (rule conjunctionD1)
854 from conj show B by (rule conjunctionD2)
865 lemmas [symmetric, rulify] = atomize_all atomize_imp
866 and [symmetric, defn] = atomize_all atomize_imp atomize_eq
869 subsubsection {* Atomizing elimination rules *}
871 setup AtomizeElim.setup
873 lemma atomize_exL[atomize_elim]: "(!!x. P x ==> Q) == ((EX x. P x) ==> Q)"
876 lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
879 lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
882 lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop A" ..
885 subsection {* Package setup *}
887 subsubsection {* Classical Reasoner setup *}
889 lemma imp_elim: "P --> Q ==> (~ R ==> P) ==> (Q ==> R) ==> R"
890 by (rule classical) iprover
892 lemma swap: "~ P ==> (~ R ==> P) ==> R"
893 by (rule classical) iprover
896 "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
899 structure Hypsubst = HypsubstFun(
901 structure Simplifier = Simplifier
902 val dest_eq = HOLogic.dest_eq
903 val dest_Trueprop = HOLogic.dest_Trueprop
904 val dest_imp = HOLogic.dest_imp
905 val eq_reflection = @{thm eq_reflection}
906 val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
907 val imp_intr = @{thm impI}
908 val rev_mp = @{thm rev_mp}
909 val subst = @{thm subst}
911 val thin_refl = @{thm thin_refl};
912 val prop_subst = @{lemma "PROP P t ==> PROP prop (x = t ==> PROP P x)"
913 by (unfold prop_def) (drule eq_reflection, unfold)}
917 structure Classical = ClassicalFun(
919 val imp_elim = @{thm imp_elim}
920 val not_elim = @{thm notE}
921 val swap = @{thm swap}
922 val classical = @{thm classical}
923 val sizef = Drule.size_of_thm
924 val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
927 structure BasicClassical: BASIC_CLASSICAL = Classical;
930 ML_Antiquote.value "claset"
931 (Scan.succeed "Classical.local_claset_of (ML_Context.the_local_context ())");
933 structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
935 structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
938 text {*ResBlacklist holds theorems blacklisted to sledgehammer.
939 These theorems typically produce clauses that are prolific (match too many equality or
940 membership literals) and relate to seldom-used facts. Some duplicate other rules.*}
944 (*prevent substitution on bool*)
945 fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
946 Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
947 (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
949 Hypsubst.hypsubst_setup
950 #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
953 #> ResBlacklist.setup
957 declare iffI [intro!]
965 declare iffCE [elim!]
972 declare ex_ex1I [intro!]
974 and the_equality [intro]
980 ML {* val HOL_cs = @{claset} *}
982 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
984 apply (erule (1) meta_mp)
987 declare ex_ex1I [rule del, intro! 2]
990 lemmas [intro?] = ext
991 and [elim?] = ex1_implies_ex
993 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
994 lemma alt_ex1E [elim!]:
995 assumes major: "\<exists>!x. P x"
996 and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
998 apply (rule ex1E [OF major])
1000 apply (tactic {* ares_tac @{thms allI} 1 *})+
1001 apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
1006 structure Blast = BlastFun
1008 type claset = Classical.claset
1009 val equality_name = @{const_name "op ="}
1010 val not_name = @{const_name Not}
1011 val notE = @{thm notE}
1012 val ccontr = @{thm ccontr}
1013 val contr_tac = Classical.contr_tac
1014 val dup_intr = Classical.dup_intr
1015 val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
1016 val claset = Classical.claset
1017 val rep_cs = Classical.rep_cs
1018 val cla_modifiers = Classical.cla_modifiers
1019 val cla_meth' = Classical.cla_meth'
1021 val Blast_tac = Blast.Blast_tac;
1022 val blast_tac = Blast.blast_tac;
1028 subsubsection {* Simplifier *}
1030 lemma eta_contract_eq: "(%s. f s) = f" ..
1033 shows not_not: "(~ ~ P) = P"
1034 and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
1036 "(P ~= Q) = (P = (~Q))"
1037 "(P | ~P) = True" "(~P | P) = True"
1039 and not_True_eq_False: "(\<not> True) = False"
1040 and not_False_eq_True: "(\<not> False) = True"
1042 "(~P) ~= P" "P ~= (~P)"
1044 and eq_True: "(P = True) = P"
1045 and "(False=P) = (~P)"
1046 and eq_False: "(P = False) = (\<not> P)"
1048 "(True --> P) = P" "(False --> P) = True"
1049 "(P --> True) = True" "(P --> P) = True"
1050 "(P --> False) = (~P)" "(P --> ~P) = (~P)"
1051 "(P & True) = P" "(True & P) = P"
1052 "(P & False) = False" "(False & P) = False"
1053 "(P & P) = P" "(P & (P & Q)) = (P & Q)"
1054 "(P & ~P) = False" "(~P & P) = False"
1055 "(P | True) = True" "(True | P) = True"
1056 "(P | False) = P" "(False | P) = P"
1057 "(P | P) = P" "(P | (P | Q)) = (P | Q)" and
1058 "(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
1059 -- {* needed for the one-point-rule quantifier simplification procs *}
1060 -- {* essential for termination!! *} and
1061 "!!P. (EX x. x=t & P(x)) = P(t)"
1062 "!!P. (EX x. t=x & P(x)) = P(t)"
1063 "!!P. (ALL x. x=t --> P(x)) = P(t)"
1064 "!!P. (ALL x. t=x --> P(x)) = P(t)"
1065 by (blast, blast, blast, blast, blast, iprover+)
1067 lemma disj_absorb: "(A | A) = A"
1070 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
1073 lemma conj_absorb: "(A & A) = A"
1076 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
1080 shows eq_commute: "(a=b) = (b=a)"
1081 and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
1082 and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
1083 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
1086 shows conj_commute: "(P&Q) = (Q&P)"
1087 and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
1088 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
1090 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
1093 shows disj_commute: "(P|Q) = (Q|P)"
1094 and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
1095 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
1097 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
1099 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
1100 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
1102 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
1103 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
1105 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
1106 lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
1107 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
1109 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
1110 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
1111 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
1113 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
1114 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
1116 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
1119 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
1120 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
1121 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
1122 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
1123 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
1124 lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
1126 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
1128 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
1131 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
1132 -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
1133 -- {* cases boil down to the same thing. *}
1136 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
1137 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
1138 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
1139 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
1140 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
1142 declare All_def [noatp]
1144 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
1145 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
1148 \medskip The @{text "&"} congruence rule: not included by default!
1149 May slow rewrite proofs down by as much as 50\% *}
1152 "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
1155 lemma rev_conj_cong:
1156 "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
1159 text {* The @{text "|"} congruence rule: not included by default! *}
1162 "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
1166 text {* \medskip if-then-else rules *}
1168 lemma if_True: "(if True then x else y) = x"
1169 by (unfold if_def) blast
1171 lemma if_False: "(if False then x else y) = y"
1172 by (unfold if_def) blast
1174 lemma if_P: "P ==> (if P then x else y) = x"
1175 by (unfold if_def) blast
1177 lemma if_not_P: "~P ==> (if P then x else y) = y"
1178 by (unfold if_def) blast
1180 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
1181 apply (rule case_split [of Q])
1182 apply (simplesubst if_P)
1183 prefer 3 apply (simplesubst if_not_P, blast+)
1186 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
1187 by (simplesubst split_if, blast)
1189 lemmas if_splits [noatp] = split_if split_if_asm
1191 lemma if_cancel: "(if c then x else x) = x"
1192 by (simplesubst split_if, blast)
1194 lemma if_eq_cancel: "(if x = y then y else x) = x"
1195 by (simplesubst split_if, blast)
1197 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
1198 -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
1201 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
1202 -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
1203 apply (simplesubst split_if, blast)
1206 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
1207 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
1209 text {* \medskip let rules for simproc *}
1211 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
1214 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
1218 The following copy of the implication operator is useful for
1219 fine-tuning congruence rules. It instructs the simplifier to simplify
1224 simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
1225 [code del]: "simp_implies \<equiv> op ==>"
1227 lemma simp_impliesI:
1228 assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1229 shows "PROP P =simp=> PROP Q"
1230 apply (unfold simp_implies_def)
1235 lemma simp_impliesE:
1236 assumes PQ: "PROP P =simp=> PROP Q"
1238 and QR: "PROP Q \<Longrightarrow> PROP R"
1241 apply (rule PQ [unfolded simp_implies_def])
1245 lemma simp_implies_cong:
1246 assumes PP' :"PROP P == PROP P'"
1247 and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
1248 shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
1249 proof (unfold simp_implies_def, rule equal_intr_rule)
1250 assume PQ: "PROP P \<Longrightarrow> PROP Q"
1252 from PP' [symmetric] and P' have "PROP P"
1253 by (rule equal_elim_rule1)
1254 then have "PROP Q" by (rule PQ)
1255 with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1257 assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1259 from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1260 then have "PROP Q'" by (rule P'Q')
1261 with P'QQ' [OF P', symmetric] show "PROP Q"
1262 by (rule equal_elim_rule1)
1266 assumes "P \<longrightarrow> Q \<longrightarrow> R"
1267 shows "P \<and> Q \<longrightarrow> R"
1268 using assms by blast
1271 assumes "\<And>x. P x = Q x"
1272 shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1273 using assms by blast
1276 assumes "\<And>x. P x = Q x"
1277 shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1278 using assms by blast
1281 "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1285 "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1288 use "Tools/simpdata.ML"
1289 ML {* open Simpdata *}
1292 Simplifier.method_setup Splitter.split_modifiers
1293 #> Simplifier.map_simpset (K Simpdata.simpset_simprocs)
1299 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
1301 simproc_setup neq ("x = y") = {* fn _ =>
1303 val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
1304 fun is_neq eq lhs rhs thm =
1305 (case Thm.prop_of thm of
1306 _ $ (Not $ (eq' $ l' $ r')) =>
1307 Not = HOLogic.Not andalso eq' = eq andalso
1308 r' aconv lhs andalso l' aconv rhs
1311 (case Thm.term_of ct of
1313 (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
1314 SOME thm => SOME (thm RS neq_to_EQ_False)
1320 simproc_setup let_simp ("Let x f") = {*
1322 val (f_Let_unfold, x_Let_unfold) =
1323 let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_unfold}
1324 in (cterm_of @{theory} f, cterm_of @{theory} x) end
1325 val (f_Let_folded, x_Let_folded) =
1326 let val [(_ $ (f $ x) $ _)] = prems_of @{thm Let_folded}
1327 in (cterm_of @{theory} f, cterm_of @{theory} x) end;
1329 let val [(_ $ _ $ (g $ _))] = prems_of @{thm Let_folded}
1330 in cterm_of @{theory} g end;
1331 fun count_loose (Bound i) k = if i >= k then 1 else 0
1332 | count_loose (s $ t) k = count_loose s k + count_loose t k
1333 | count_loose (Abs (_, _, t)) k = count_loose t (k + 1)
1334 | count_loose _ _ = 0;
1335 fun is_trivial_let (Const (@{const_name Let}, _) $ x $ t) =
1337 of Abs (_, _, t') => count_loose t' 0 <= 1
1339 in fn _ => fn ss => fn ct => if is_trivial_let (Thm.term_of ct)
1340 then SOME @{thm Let_def} (*no or one ocurrenc of bound variable*)
1341 else let (*Norbert Schirmer's case*)
1342 val ctxt = Simplifier.the_context ss;
1343 val thy = ProofContext.theory_of ctxt;
1344 val t = Thm.term_of ct;
1345 val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
1346 in Option.map (hd o Variable.export ctxt' ctxt o single)
1347 (case t' of Const (@{const_name Let},_) $ x $ f => (* x and f are already in normal form *)
1348 if is_Free x orelse is_Bound x orelse is_Const x
1349 then SOME @{thm Let_def}
1352 val n = case f of (Abs (x, _, _)) => x | _ => "x";
1353 val cx = cterm_of thy x;
1354 val {T = xT, ...} = rep_cterm cx;
1355 val cf = cterm_of thy f;
1356 val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
1357 val (_ $ _ $ g) = prop_of fx_g;
1358 val g' = abstract_over (x,g);
1363 cterm_instantiate [(f_Let_unfold, cf), (x_Let_unfold, cx)] @{thm Let_unfold};
1364 in SOME (rl OF [fx_g]) end
1365 else if Term.betapply (f, x) aconv g then NONE (*avoid identity conversion*)
1367 val abs_g'= Abs (n,xT,g');
1369 val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
1370 val rl = cterm_instantiate
1371 [(f_Let_folded, cterm_of thy f), (x_Let_folded, cx),
1372 (g_Let_folded, cterm_of thy abs_g')]
1374 in SOME (rl OF [transitive fx_g g_g'x])
1381 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1383 assume "True \<Longrightarrow> PROP P"
1384 from this [OF TrueI] show "PROP P" .
1387 then show "PROP P" .
1391 "!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
1392 "!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
1393 "!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
1394 "!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
1395 "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
1396 "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
1397 -- {* Miniscoping: pushing in existential quantifiers. *}
1398 by (iprover | blast)+
1401 "!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
1402 "!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
1403 "!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
1404 "!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
1405 "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
1406 "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
1407 -- {* Miniscoping: pushing in universal quantifiers. *}
1408 by (iprover | blast)+
1411 triv_forall_equality (*prunes params*)
1412 True_implies_equals (*prune asms `True'*)
1418 (*In general it seems wrong to add distributive laws by default: they
1419 might cause exponential blow-up. But imp_disjL has been in for a while
1420 and cannot be removed without affecting existing proofs. Moreover,
1421 rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
1422 grounds that it allows simplification of R in the two cases.*)
1440 lemmas [cong] = imp_cong simp_implies_cong
1441 lemmas [split] = split_if
1443 ML {* val HOL_ss = @{simpset} *}
1445 text {* Simplifies x assuming c and y assuming ~c *}
1448 and "c \<Longrightarrow> x = u"
1449 and "\<not> c \<Longrightarrow> y = v"
1450 shows "(if b then x else y) = (if c then u else v)"
1451 unfolding if_def using assms by simp
1453 text {* Prevents simplification of x and y:
1454 faster and allows the execution of functional programs. *}
1455 lemma if_weak_cong [cong]:
1457 shows "(if b then x else y) = (if c then x else y)"
1458 using assms by (rule arg_cong)
1460 text {* Prevents simplification of t: much faster *}
1461 lemma let_weak_cong:
1463 shows "(let x = a in t x) = (let x = b in t x)"
1464 using assms by (rule arg_cong)
1466 text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
1469 shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1473 "f (if c then x else y) = (if c then f x else f y)"
1476 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
1477 side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *}
1478 lemma restrict_to_left:
1480 shows "(x = z) = (y = z)"
1484 subsubsection {* Generic cases and induction *}
1486 text {* Rule projections: *}
1489 structure ProjectRule = ProjectRuleFun
1491 val conjunct1 = @{thm conjunct1}
1492 val conjunct2 = @{thm conjunct2}
1498 induct_forall where "induct_forall P == \<forall>x. P x"
1499 induct_implies where "induct_implies A B == A \<longrightarrow> B"
1500 induct_equal where "induct_equal x y == x = y"
1501 induct_conj where "induct_conj A B == A \<and> B"
1503 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
1504 by (unfold atomize_all induct_forall_def)
1506 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
1507 by (unfold atomize_imp induct_implies_def)
1509 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
1510 by (unfold atomize_eq induct_equal_def)
1512 lemma induct_conj_eq: "(A &&& B) == Trueprop (induct_conj A B)"
1513 by (unfold atomize_conj induct_conj_def)
1515 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
1516 lemmas induct_rulify [symmetric, standard] = induct_atomize
1517 lemmas induct_rulify_fallback =
1518 induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1521 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1522 induct_conj (induct_forall A) (induct_forall B)"
1523 by (unfold induct_forall_def induct_conj_def) iprover
1525 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1526 induct_conj (induct_implies C A) (induct_implies C B)"
1527 by (unfold induct_implies_def induct_conj_def) iprover
1529 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
1531 assume r: "induct_conj A B ==> PROP C" and A B
1532 show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
1534 assume r: "A ==> B ==> PROP C" and "induct_conj A B"
1535 show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
1538 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1540 hide const induct_forall induct_implies induct_equal induct_conj
1542 text {* Method setup. *}
1545 structure Induct = InductFun
1547 val cases_default = @{thm case_split}
1548 val atomize = @{thms induct_atomize}
1549 val rulify = @{thms induct_rulify}
1550 val rulify_fallback = @{thms induct_rulify_fallback}
1556 use "~~/src/Tools/induct_tacs.ML"
1557 setup InductTacs.setup
1560 subsubsection {* Coherent logic *}
1563 structure Coherent = CoherentFun
1565 val atomize_elimL = @{thm atomize_elimL}
1566 val atomize_exL = @{thm atomize_exL}
1567 val atomize_conjL = @{thm atomize_conjL}
1568 val atomize_disjL = @{thm atomize_disjL}
1569 val operator_names =
1570 [@{const_name "op |"}, @{const_name "op &"}, @{const_name "Ex"}]
1574 setup Coherent.setup
1577 subsection {* Other simple lemmas and lemma duplicates *}
1579 lemma Let_0 [simp]: "Let 0 f = f 0"
1580 unfolding Let_def ..
1582 lemma Let_1 [simp]: "Let 1 f = f 1"
1583 unfolding Let_def ..
1585 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
1588 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
1590 apply (rule_tac a = "%x. THE y. P x y" in ex1I)
1591 apply (fast dest!: theI')
1592 apply (fast intro: ext the1_equality [symmetric])
1597 apply (erule_tac x = "%z. if z = x then y else f z" in allE)
1600 apply (case_tac "xa = x")
1601 apply (drule_tac [3] x = x in fun_cong, simp_all)
1604 lemma mk_left_commute:
1605 fixes f (infix "\<otimes>" 60)
1606 assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
1607 c: "\<And>x y. x \<otimes> y = y \<otimes> x"
1608 shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
1609 by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
1611 lemmas eq_sym_conv = eq_commute
1614 "(\<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)"
1615 "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
1616 "(\<not> \<not>(P)) = P"
1620 subsection {* Basic ML bindings *}
1623 val FalseE = @{thm FalseE}
1624 val Let_def = @{thm Let_def}
1625 val TrueI = @{thm TrueI}
1626 val allE = @{thm allE}
1627 val allI = @{thm allI}
1628 val all_dupE = @{thm all_dupE}
1629 val arg_cong = @{thm arg_cong}
1630 val box_equals = @{thm box_equals}
1631 val ccontr = @{thm ccontr}
1632 val classical = @{thm classical}
1633 val conjE = @{thm conjE}
1634 val conjI = @{thm conjI}
1635 val conjunct1 = @{thm conjunct1}
1636 val conjunct2 = @{thm conjunct2}
1637 val disjCI = @{thm disjCI}
1638 val disjE = @{thm disjE}
1639 val disjI1 = @{thm disjI1}
1640 val disjI2 = @{thm disjI2}
1641 val eq_reflection = @{thm eq_reflection}
1642 val ex1E = @{thm ex1E}
1643 val ex1I = @{thm ex1I}
1644 val ex1_implies_ex = @{thm ex1_implies_ex}
1645 val exE = @{thm exE}
1646 val exI = @{thm exI}
1647 val excluded_middle = @{thm excluded_middle}
1648 val ext = @{thm ext}
1649 val fun_cong = @{thm fun_cong}
1650 val iffD1 = @{thm iffD1}
1651 val iffD2 = @{thm iffD2}
1652 val iffI = @{thm iffI}
1653 val impE = @{thm impE}
1654 val impI = @{thm impI}
1655 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
1657 val notE = @{thm notE}
1658 val notI = @{thm notI}
1659 val not_all = @{thm not_all}
1660 val not_ex = @{thm not_ex}
1661 val not_iff = @{thm not_iff}
1662 val not_not = @{thm not_not}
1663 val not_sym = @{thm not_sym}
1664 val refl = @{thm refl}
1665 val rev_mp = @{thm rev_mp}
1666 val spec = @{thm spec}
1667 val ssubst = @{thm ssubst}
1668 val subst = @{thm subst}
1669 val sym = @{thm sym}
1670 val trans = @{thm trans}
1674 subsection {* Code generator basics -- see further theory @{text "Code_Setup"} *}
1679 fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1680 assumes eq_equals: "eq x y \<longleftrightarrow> x = y"
1683 lemma eq: "eq = (op =)"
1684 by (rule ext eq_equals)+
1686 lemma eq_refl: "eq x x \<longleftrightarrow> True"
1687 unfolding eq by rule+
1691 text {* Module setup *}
1693 use "Tools/recfun_codegen.ML"
1697 #> Code_Haskell.setup
1700 #> RecfunCodegen.setup
1704 subsection {* Legacy tactics and ML bindings *}
1707 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
1709 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1711 fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1712 | wrong_prem (Bound _) = true
1713 | wrong_prem _ = false;
1714 val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
1716 fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
1717 fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
1720 val all_conj_distrib = thm "all_conj_distrib";
1721 val all_simps = thms "all_simps";
1722 val atomize_not = thm "atomize_not";
1723 val case_split = thm "case_split";
1724 val cases_simp = thm "cases_simp";
1725 val choice_eq = thm "choice_eq"
1726 val cong = thm "cong"
1727 val conj_comms = thms "conj_comms";
1728 val conj_cong = thm "conj_cong";
1729 val de_Morgan_conj = thm "de_Morgan_conj";
1730 val de_Morgan_disj = thm "de_Morgan_disj";
1731 val disj_assoc = thm "disj_assoc";
1732 val disj_comms = thms "disj_comms";
1733 val disj_cong = thm "disj_cong";
1734 val eq_ac = thms "eq_ac";
1735 val eq_cong2 = thm "eq_cong2"
1736 val Eq_FalseI = thm "Eq_FalseI";
1737 val Eq_TrueI = thm "Eq_TrueI";
1738 val Ex1_def = thm "Ex1_def"
1739 val ex_disj_distrib = thm "ex_disj_distrib";
1740 val ex_simps = thms "ex_simps";
1741 val if_cancel = thm "if_cancel";
1742 val if_eq_cancel = thm "if_eq_cancel";
1743 val if_False = thm "if_False";
1744 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
1746 val if_splits = thms "if_splits";
1747 val if_True = thm "if_True";
1748 val if_weak_cong = thm "if_weak_cong"
1749 val imp_all = thm "imp_all";
1750 val imp_cong = thm "imp_cong";
1751 val imp_conjL = thm "imp_conjL";
1752 val imp_conjR = thm "imp_conjR";
1753 val imp_conv_disj = thm "imp_conv_disj";
1754 val simp_implies_def = thm "simp_implies_def";
1755 val simp_thms = thms "simp_thms";
1756 val split_if = thm "split_if";
1757 val the1_equality = thm "the1_equality"
1758 val theI = thm "theI"
1759 val theI' = thm "theI'"
1760 val True_implies_equals = thm "True_implies_equals";
1761 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})