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/induct.ML"
26 "~~/src/Tools/code/code_name.ML"
27 "~~/src/Tools/code/code_funcgr.ML"
28 "~~/src/Tools/code/code_thingol.ML"
29 "~~/src/Tools/code/code_target.ML"
30 "~~/src/Tools/code/code_package.ML"
34 subsection {* Primitive logic *}
36 subsubsection {* Core syntax *}
47 "fun" :: (type, type) type
52 Trueprop :: "bool => prop" ("(_)" 5)
55 Not :: "bool => bool" ("~ _" [40] 40)
60 The :: "('a => bool) => 'a"
61 All :: "('a => bool) => bool" (binder "ALL " 10)
62 Ex :: "('a => bool) => bool" (binder "EX " 10)
63 Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
64 Let :: "['a, 'a => 'b] => 'b"
66 "op =" :: "['a, 'a] => bool" (infixl "=" 50)
67 "op &" :: "[bool, bool] => bool" (infixr "&" 35)
68 "op |" :: "[bool, bool] => bool" (infixr "|" 30)
69 "op -->" :: "[bool, bool] => bool" (infixr "-->" 25)
74 If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
77 subsubsection {* Additional concrete syntax *}
83 not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
87 not_equal (infix "~=" 50)
90 Not ("\<not> _" [40] 40) and
91 "op &" (infixr "\<and>" 35) and
92 "op |" (infixr "\<or>" 30) and
93 "op -->" (infixr "\<longrightarrow>" 25) and
94 not_equal (infix "\<noteq>" 50)
96 notation (HTML output)
97 Not ("\<not> _" [40] 40) and
98 "op &" (infixr "\<and>" 35) and
99 "op |" (infixr "\<or>" 30) and
100 not_equal (infix "\<noteq>" 50)
103 iff :: "[bool, bool] => bool" (infixr "<->" 25) where
107 iff (infixr "\<longleftrightarrow>" 25)
115 "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
117 "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
118 "" :: "letbind => letbinds" ("_")
119 "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
120 "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
122 "_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
123 "_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
124 "" :: "case_syn => cases_syn" ("_")
125 "_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
128 "THE x. P" == "The (%x. P)"
129 "_Let (_binds b bs) e" == "_Let b (_Let bs e)"
130 "let x = a in e" == "Let a (%x. e)"
133 (* To avoid eta-contraction of body: *)
134 [("The", fn [Abs abs] =>
135 let val (x,t) = atomic_abs_tr' abs
136 in Syntax.const "_The" $ x $ t end)]
140 "_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
143 All (binder "\<forall>" 10) and
144 Ex (binder "\<exists>" 10) and
145 Ex1 (binder "\<exists>!" 10)
147 notation (HTML output)
148 All (binder "\<forall>" 10) and
149 Ex (binder "\<exists>" 10) and
150 Ex1 (binder "\<exists>!" 10)
153 All (binder "! " 10) and
154 Ex (binder "? " 10) and
155 Ex1 (binder "?! " 10)
158 subsubsection {* Axioms and basic definitions *}
161 eq_reflection: "(x=y) ==> (x==y)"
165 ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
166 -- {*Extensionality is built into the meta-logic, and this rule expresses
167 a related property. It is an eta-expanded version of the traditional
168 rule, and similar to the ABS rule of HOL*}
170 the_eq_trivial: "(THE x. x = a) = (a::'a)"
172 impI: "(P ==> Q) ==> P-->Q"
173 mp: "[| P-->Q; P |] ==> Q"
177 True_def: "True == ((%x::bool. x) = (%x. x))"
178 All_def: "All(P) == (P = (%x. True))"
179 Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
180 False_def: "False == (!P. P)"
181 not_def: "~ P == P-->False"
182 and_def: "P & Q == !R. (P-->Q-->R) --> R"
183 or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
184 Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
187 iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
188 True_or_False: "(P=True) | (P=False)"
191 Let_def: "Let s f == f(s)"
192 if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
204 undefined_fun: "undefined x = undefined"
207 subsubsection {* Generic classes and algebraic operations *}
209 class default = type +
213 fixes zero :: 'a ("0")
216 fixes one :: 'a ("1")
218 hide (open) const zero one
221 fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
224 fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
225 and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
228 fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
230 class inverse = type +
231 fixes inverse :: "'a \<Rightarrow> 'a"
232 and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
235 fixes abs :: "'a \<Rightarrow> 'a"
239 abs ("\<bar>_\<bar>")
241 notation (HTML output)
242 abs ("\<bar>_\<bar>")
247 fixes sgn :: "'a \<Rightarrow> 'a"
250 fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
251 and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
255 less_eq ("op <=") and
256 less_eq ("(_/ <= _)" [51, 51] 50) and
258 less ("(_/ < _)" [51, 51] 50)
261 less_eq ("op \<le>") and
262 less_eq ("(_/ \<le> _)" [51, 51] 50)
264 notation (HTML output)
265 less_eq ("op \<le>") and
266 less_eq ("(_/ \<le> _)" [51, 51] 50)
269 greater_eq (infix ">=" 50) where
270 "x >= y \<equiv> y <= x"
273 greater_eq (infix "\<ge>" 50)
276 greater (infix ">" 50) where
277 "x > y \<equiv> y < x"
280 Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
281 "Least P == (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y))"
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 text {* Thanks to Stephan Merz *}
305 assumes eq: "s = t" and p: "P s"
308 from eq have meta: "s \<equiv> t"
309 by (rule eq_reflection)
314 lemma sym: "s = t ==> t = s"
315 by (erule subst) (rule refl)
317 lemma ssubst: "t = s ==> P s ==> P t"
318 by (drule sym) (erule subst)
320 lemma trans: "[| r=s; s=t |] ==> r=t"
323 lemma meta_eq_to_obj_eq:
324 assumes meq: "A == B"
326 by (unfold meq) (rule refl)
328 text {* Useful with @{text erule} for proving equalities from known equalities. *}
332 lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
339 text {* For calculational reasoning: *}
341 lemma forw_subst: "a = b ==> P b ==> P a"
344 lemma back_subst: "P a ==> a = b ==> P b"
348 subsubsection {*Congruence rules for application*}
350 (*similar to AP_THM in Gordon's HOL*)
351 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
356 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
357 lemma arg_cong: "x=y ==> f(x)=f(y)"
362 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
363 apply (erule ssubst)+
367 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
373 subsubsection {*Equality of booleans -- iff*}
375 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
376 by (iprover intro: iff [THEN mp, THEN mp] impI assms)
378 lemma iffD2: "[| P=Q; Q |] ==> P"
381 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
384 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
385 by (drule sym) (rule iffD2)
387 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
388 by (drule sym) (rule rev_iffD2)
392 and minor: "[| P --> Q; Q --> P |] ==> R"
394 by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
397 subsubsection {*True*}
400 unfolding True_def by (rule refl)
402 lemma eqTrueI: "P ==> P = True"
403 by (iprover intro: iffI TrueI)
405 lemma eqTrueE: "P = True ==> P"
406 by (erule iffD2) (rule TrueI)
409 subsubsection {*Universal quantifier*}
411 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
412 unfolding All_def by (iprover intro: ext eqTrueI assms)
414 lemma spec: "ALL x::'a. P(x) ==> P(x)"
415 apply (unfold All_def)
417 apply (erule fun_cong)
421 assumes major: "ALL x. P(x)"
422 and minor: "P(x) ==> R"
424 by (iprover intro: minor major [THEN spec])
427 assumes major: "ALL x. P(x)"
428 and minor: "[| P(x); ALL x. P(x) |] ==> R"
430 by (iprover intro: minor major major [THEN spec])
433 subsubsection {* False *}
436 Depends upon @{text spec}; it is impossible to do propositional
437 logic before quantifiers!
440 lemma FalseE: "False ==> P"
441 apply (unfold False_def)
445 lemma False_neq_True: "False = True ==> P"
446 by (erule eqTrueE [THEN FalseE])
449 subsubsection {* Negation *}
452 assumes "P ==> False"
454 apply (unfold not_def)
455 apply (iprover intro: impI assms)
458 lemma False_not_True: "False ~= True"
460 apply (erule False_neq_True)
463 lemma True_not_False: "True ~= False"
466 apply (erule False_neq_True)
469 lemma notE: "[| ~P; P |] ==> R"
470 apply (unfold not_def)
471 apply (erule mp [THEN FalseE])
475 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
476 by (erule notE [THEN notI]) (erule meta_mp)
479 subsubsection {*Implication*}
482 assumes "P-->Q" "P" "Q ==> R"
484 by (iprover intro: assms mp)
486 (* Reduces Q to P-->Q, allowing substitution in P. *)
487 lemma rev_mp: "[| P; P --> Q |] ==> Q"
488 by (iprover intro: mp)
494 by (iprover intro: notI minor major [THEN notE])
496 (*not used at all, but we already have the other 3 combinations *)
499 and minor: "P ==> ~Q"
501 by (iprover intro: notI minor major notE)
503 lemma not_sym: "t ~= s ==> s ~= t"
504 by (erule contrapos_nn) (erule sym)
506 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
507 by (erule subst, erule ssubst, assumption)
509 (*still used in HOLCF*)
511 assumes pq: "P ==> Q"
514 apply (rule nq [THEN contrapos_nn])
518 subsubsection {*Existential quantifier*}
520 lemma exI: "P x ==> EX x::'a. P x"
521 apply (unfold Ex_def)
522 apply (iprover intro: allI allE impI mp)
526 assumes major: "EX x::'a. P(x)"
527 and minor: "!!x. P(x) ==> Q"
529 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
530 apply (iprover intro: impI [THEN allI] minor)
534 subsubsection {*Conjunction*}
536 lemma conjI: "[| P; Q |] ==> P&Q"
537 apply (unfold and_def)
538 apply (iprover intro: impI [THEN allI] mp)
541 lemma conjunct1: "[| P & Q |] ==> P"
542 apply (unfold and_def)
543 apply (iprover intro: impI dest: spec mp)
546 lemma conjunct2: "[| P & Q |] ==> Q"
547 apply (unfold and_def)
548 apply (iprover intro: impI dest: spec mp)
553 and minor: "[| P; Q |] ==> R"
556 apply (rule major [THEN conjunct1])
557 apply (rule major [THEN conjunct2])
561 assumes "P" "P ==> Q" shows "P & Q"
562 by (iprover intro: conjI assms)
565 subsubsection {*Disjunction*}
567 lemma disjI1: "P ==> P|Q"
568 apply (unfold or_def)
569 apply (iprover intro: allI impI mp)
572 lemma disjI2: "Q ==> P|Q"
573 apply (unfold or_def)
574 apply (iprover intro: allI impI mp)
579 and minorP: "P ==> R"
580 and minorQ: "Q ==> R"
582 by (iprover intro: minorP minorQ impI
583 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
586 subsubsection {*Classical logic*}
589 assumes prem: "~P ==> P"
591 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
593 apply (rule notI [THEN prem, THEN eqTrueI])
598 lemmas ccontr = FalseE [THEN classical, standard]
600 (*notE with premises exchanged; it discharges ~R so that it can be used to
601 make elimination rules*)
604 and premnot: "~R ==> ~P"
607 apply (erule notE [OF premnot premp])
610 (*Double negation law*)
611 lemma notnotD: "~~P ==> P"
612 apply (rule classical)
621 by (iprover intro: classical p1 p2 notE)
624 subsubsection {*Unique existence*}
627 assumes "P a" "!!x. P(x) ==> x=a"
629 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
631 text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
633 assumes ex_prem: "EX x. P(x)"
634 and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
636 by (iprover intro: ex_prem [THEN exE] ex1I eq)
639 assumes major: "EX! x. P(x)"
640 and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
642 apply (rule major [unfolded Ex1_def, THEN exE])
644 apply (iprover intro: minor)
647 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
654 subsubsection {*THE: definite description operator*}
658 and premx: "!!x. P x ==> x=a"
659 shows "(THE x. P x) = a"
660 apply (rule trans [OF _ the_eq_trivial])
661 apply (rule_tac f = "The" in arg_cong)
665 apply (erule ssubst, rule prema)
669 assumes "P a" and "!!x. P x ==> x=a"
670 shows "P (THE x. P x)"
671 by (iprover intro: assms the_equality [THEN ssubst])
673 lemma theI': "EX! x. P x ==> P (THE x. P x)"
681 (*Easier to apply than theI: only one occurrence of P*)
683 assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
684 shows "Q (THE x. P x)"
685 by (iprover intro: assms theI)
687 lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
688 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
691 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
692 apply (rule the_equality)
695 apply (erule all_dupE)
704 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
705 apply (rule the_equality)
711 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
714 assumes "~Q ==> P" shows "P|Q"
715 apply (rule classical)
716 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
719 lemma excluded_middle: "~P | P"
720 by (iprover intro: disjCI)
723 case distinction as a natural deduction rule.
724 Note that @{term "~P"} is the second case, not the first
726 lemma case_split_thm:
727 assumes prem1: "P ==> Q"
728 and prem2: "~P ==> Q"
730 apply (rule excluded_middle [THEN disjE])
734 lemmas case_split = case_split_thm [case_names True False]
736 (*Classical implies (-->) elimination. *)
738 assumes major: "P-->Q"
739 and minor: "~P ==> R" "Q ==> R"
741 apply (rule excluded_middle [of P, THEN disjE])
742 apply (iprover intro: minor major [THEN mp])+
745 (*This version of --> elimination works on Q before P. It works best for
746 those cases in which P holds "almost everywhere". Can't install as
747 default: would break old proofs.*)
749 assumes major: "P-->Q"
750 and minor: "Q ==> R" "~P ==> R"
752 apply (rule excluded_middle [of P, THEN disjE])
753 apply (iprover intro: minor major [THEN mp])+
756 (*Classical <-> elimination. *)
759 and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
761 apply (rule major [THEN iffE])
762 apply (iprover intro: minor elim: impCE notE)
766 assumes "ALL x. ~P(x) ==> P(a)"
769 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
773 subsubsection {* Intuitionistic Reasoning *}
778 and 3: "P --> Q ==> P"
781 from 3 and 1 have P .
782 with 1 have Q by (rule impE)
787 assumes 1: "ALL x. P x"
788 and 2: "P x ==> ALL x. P x ==> Q"
791 from 1 have "P x" by (rule spec)
792 from this and 1 show Q by (rule 2)
800 from 2 and 1 have P .
801 with 1 show R by (rule notE)
804 lemma TrueE: "True ==> P ==> P" .
805 lemma notFalseE: "~ False ==> P ==> P" .
807 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
808 and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
809 and [Pure.elim 2] = allE notE' impE'
810 and [Pure.intro] = exI disjI2 disjI1
812 lemmas [trans] = trans
813 and [sym] = sym not_sym
814 and [Pure.elim?] = iffD1 iffD2 impE
819 subsubsection {* Atomizing meta-level connectives *}
821 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
824 then show "ALL x. P x" ..
827 then show "!!x. P x" by (rule allE)
830 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
833 show "A --> B" by (rule impI) (rule r)
835 assume "A --> B" and A
836 then show B by (rule mp)
839 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
841 assume r: "A ==> False"
842 show "~A" by (rule notI) (rule r)
845 then show False by (rule notE)
848 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
851 show "x = y" by (unfold `x == y`) (rule refl)
854 then show "x == y" by (rule eq_reflection)
857 lemma atomize_conj [atomize]:
858 includes meta_conjunction_syntax
859 shows "(A && B) == Trueprop (A & B)"
861 assume conj: "A && B"
864 from conj show A by (rule conjunctionD1)
865 from conj show B by (rule conjunctionD2)
876 lemmas [symmetric, rulify] = atomize_all atomize_imp
877 and [symmetric, defn] = atomize_all atomize_imp atomize_eq
880 subsection {* Package setup *}
882 subsubsection {* Classical Reasoner setup *}
885 "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
888 structure Hypsubst = HypsubstFun(
890 structure Simplifier = Simplifier
891 val dest_eq = HOLogic.dest_eq
892 val dest_Trueprop = HOLogic.dest_Trueprop
893 val dest_imp = HOLogic.dest_imp
894 val eq_reflection = @{thm HOL.eq_reflection}
895 val rev_eq_reflection = @{thm HOL.meta_eq_to_obj_eq}
896 val imp_intr = @{thm HOL.impI}
897 val rev_mp = @{thm HOL.rev_mp}
898 val subst = @{thm HOL.subst}
899 val sym = @{thm HOL.sym}
900 val thin_refl = @{thm thin_refl};
904 structure Classical = ClassicalFun(
906 val mp = @{thm HOL.mp}
907 val not_elim = @{thm HOL.notE}
908 val classical = @{thm HOL.classical}
909 val sizef = Drule.size_of_thm
910 val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
913 structure BasicClassical: BASIC_CLASSICAL = Classical;
916 ML_Context.value_antiq "claset"
917 (Scan.succeed ("claset", "Classical.local_claset_of (ML_Context.the_local_context ())"));
919 structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
921 structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
924 text {*ResBlacklist holds theorems blacklisted to sledgehammer.
925 These theorems typically produce clauses that are prolific (match too many equality or
926 membership literals) and relate to seldom-used facts. Some duplicate other rules.*}
930 (*prevent substitution on bool*)
931 fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
932 Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
933 (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
935 Hypsubst.hypsubst_setup
936 #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
939 #> ResBlacklist.setup
943 declare iffI [intro!]
951 declare iffCE [elim!]
958 declare ex_ex1I [intro!]
960 and the_equality [intro]
966 ML {* val HOL_cs = @{claset} *}
968 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
970 apply (erule (1) meta_mp)
973 declare ex_ex1I [rule del, intro! 2]
976 lemmas [intro?] = ext
977 and [elim?] = ex1_implies_ex
979 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
980 lemma alt_ex1E [elim!]:
981 assumes major: "\<exists>!x. P x"
982 and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
984 apply (rule ex1E [OF major])
986 apply (tactic {* ares_tac @{thms allI} 1 *})+
987 apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
992 structure Blast = BlastFun
994 type claset = Classical.claset
995 val equality_name = @{const_name "op ="}
996 val not_name = @{const_name Not}
997 val notE = @{thm HOL.notE}
998 val ccontr = @{thm HOL.ccontr}
999 val contr_tac = Classical.contr_tac
1000 val dup_intr = Classical.dup_intr
1001 val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
1002 val claset = Classical.claset
1003 val rep_cs = Classical.rep_cs
1004 val cla_modifiers = Classical.cla_modifiers
1005 val cla_meth' = Classical.cla_meth'
1007 val Blast_tac = Blast.Blast_tac;
1008 val blast_tac = Blast.blast_tac;
1014 subsubsection {* Simplifier *}
1016 lemma eta_contract_eq: "(%s. f s) = f" ..
1019 shows not_not: "(~ ~ P) = P"
1020 and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
1022 "(P ~= Q) = (P = (~Q))"
1023 "(P | ~P) = True" "(~P | P) = True"
1025 and not_True_eq_False: "(\<not> True) = False"
1026 and not_False_eq_True: "(\<not> False) = True"
1028 "(~P) ~= P" "P ~= (~P)"
1030 and eq_True: "(P = True) = P"
1031 and "(False=P) = (~P)"
1032 and eq_False: "(P = False) = (\<not> P)"
1034 "(True --> P) = P" "(False --> P) = True"
1035 "(P --> True) = True" "(P --> P) = True"
1036 "(P --> False) = (~P)" "(P --> ~P) = (~P)"
1037 "(P & True) = P" "(True & P) = P"
1038 "(P & False) = False" "(False & P) = False"
1039 "(P & P) = P" "(P & (P & Q)) = (P & Q)"
1040 "(P & ~P) = False" "(~P & P) = False"
1041 "(P | True) = True" "(True | P) = True"
1042 "(P | False) = P" "(False | P) = P"
1043 "(P | P) = P" "(P | (P | Q)) = (P | Q)" and
1044 "(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
1045 -- {* needed for the one-point-rule quantifier simplification procs *}
1046 -- {* essential for termination!! *} and
1047 "!!P. (EX x. x=t & P(x)) = P(t)"
1048 "!!P. (EX x. t=x & P(x)) = P(t)"
1049 "!!P. (ALL x. x=t --> P(x)) = P(t)"
1050 "!!P. (ALL x. t=x --> P(x)) = P(t)"
1051 by (blast, blast, blast, blast, blast, iprover+)
1053 lemma disj_absorb: "(A | A) = A"
1056 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
1059 lemma conj_absorb: "(A & A) = A"
1062 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
1066 shows eq_commute: "(a=b) = (b=a)"
1067 and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
1068 and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
1069 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
1072 shows conj_commute: "(P&Q) = (Q&P)"
1073 and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
1074 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
1076 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
1079 shows disj_commute: "(P|Q) = (Q|P)"
1080 and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
1081 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
1083 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
1085 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
1086 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
1088 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
1089 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
1091 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
1092 lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
1093 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
1095 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
1096 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
1097 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
1099 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
1100 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
1102 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
1105 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
1106 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
1107 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
1108 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
1109 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
1110 lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
1112 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
1114 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
1117 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
1118 -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
1119 -- {* cases boil down to the same thing. *}
1122 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
1123 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
1124 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
1125 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
1126 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
1128 declare All_def [noatp]
1130 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
1131 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
1134 \medskip The @{text "&"} congruence rule: not included by default!
1135 May slow rewrite proofs down by as much as 50\% *}
1138 "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
1141 lemma rev_conj_cong:
1142 "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
1145 text {* The @{text "|"} congruence rule: not included by default! *}
1148 "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
1152 text {* \medskip if-then-else rules *}
1154 lemma if_True: "(if True then x else y) = x"
1155 by (unfold if_def) blast
1157 lemma if_False: "(if False then x else y) = y"
1158 by (unfold if_def) blast
1160 lemma if_P: "P ==> (if P then x else y) = x"
1161 by (unfold if_def) blast
1163 lemma if_not_P: "~P ==> (if P then x else y) = y"
1164 by (unfold if_def) blast
1166 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
1167 apply (rule case_split [of Q])
1168 apply (simplesubst if_P)
1169 prefer 3 apply (simplesubst if_not_P, blast+)
1172 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
1173 by (simplesubst split_if, blast)
1175 lemmas if_splits [noatp] = split_if split_if_asm
1177 lemma if_cancel: "(if c then x else x) = x"
1178 by (simplesubst split_if, blast)
1180 lemma if_eq_cancel: "(if x = y then y else x) = x"
1181 by (simplesubst split_if, blast)
1183 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
1184 -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
1187 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
1188 -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
1189 apply (simplesubst split_if, blast)
1192 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
1193 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
1195 text {* \medskip let rules for simproc *}
1197 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
1200 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
1204 The following copy of the implication operator is useful for
1205 fine-tuning congruence rules. It instructs the simplifier to simplify
1210 simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
1211 "simp_implies \<equiv> op ==>"
1213 lemma simp_impliesI:
1214 assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1215 shows "PROP P =simp=> PROP Q"
1216 apply (unfold simp_implies_def)
1221 lemma simp_impliesE:
1222 assumes PQ: "PROP P =simp=> PROP Q"
1224 and QR: "PROP Q \<Longrightarrow> PROP R"
1227 apply (rule PQ [unfolded simp_implies_def])
1231 lemma simp_implies_cong:
1232 assumes PP' :"PROP P == PROP P'"
1233 and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
1234 shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
1235 proof (unfold simp_implies_def, rule equal_intr_rule)
1236 assume PQ: "PROP P \<Longrightarrow> PROP Q"
1238 from PP' [symmetric] and P' have "PROP P"
1239 by (rule equal_elim_rule1)
1240 then have "PROP Q" by (rule PQ)
1241 with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1243 assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1245 from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1246 then have "PROP Q'" by (rule P'Q')
1247 with P'QQ' [OF P', symmetric] show "PROP Q"
1248 by (rule equal_elim_rule1)
1252 assumes "P \<longrightarrow> Q \<longrightarrow> R"
1253 shows "P \<and> Q \<longrightarrow> R"
1254 using assms by blast
1257 assumes "\<And>x. P x = Q x"
1258 shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1259 using assms by blast
1262 assumes "\<And>x. P x = Q x"
1263 shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1264 using assms by blast
1267 "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1271 "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1275 ML {* open Simpdata *}
1278 Simplifier.method_setup Splitter.split_modifiers
1279 #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
1285 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
1287 simproc_setup neq ("x = y") = {* fn _ =>
1289 val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
1290 fun is_neq eq lhs rhs thm =
1291 (case Thm.prop_of thm of
1292 _ $ (Not $ (eq' $ l' $ r')) =>
1293 Not = HOLogic.Not andalso eq' = eq andalso
1294 r' aconv lhs andalso l' aconv rhs
1297 (case Thm.term_of ct of
1299 (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
1300 SOME thm => SOME (thm RS neq_to_EQ_False)
1306 simproc_setup let_simp ("Let x f") = {*
1308 val (f_Let_unfold, x_Let_unfold) =
1309 let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
1310 in (cterm_of @{theory} f, cterm_of @{theory} x) end
1311 val (f_Let_folded, x_Let_folded) =
1312 let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
1313 in (cterm_of @{theory} f, cterm_of @{theory} x) end;
1315 let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
1319 val ctxt = Simplifier.the_context ss;
1320 val thy = ProofContext.theory_of ctxt;
1321 val t = Thm.term_of ct;
1322 val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
1323 in Option.map (hd o Variable.export ctxt' ctxt o single)
1324 (case t' of Const ("Let",_) $ x $ f => (* x and f are already in normal form *)
1325 if is_Free x orelse is_Bound x orelse is_Const x
1326 then SOME @{thm Let_def}
1329 val n = case f of (Abs (x,_,_)) => x | _ => "x";
1330 val cx = cterm_of thy x;
1331 val {T=xT,...} = rep_cterm cx;
1332 val cf = cterm_of thy f;
1333 val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
1334 val (_$_$g) = prop_of fx_g;
1335 val g' = abstract_over (x,g);
1340 cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
1341 in SOME (rl OF [fx_g]) end
1342 else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
1344 val abs_g'= Abs (n,xT,g');
1346 val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
1347 val rl = cterm_instantiate
1348 [(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
1349 (g_Let_folded,cterm_of thy abs_g')]
1351 in SOME (rl OF [transitive fx_g g_g'x])
1359 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1361 assume "True \<Longrightarrow> PROP P"
1362 from this [OF TrueI] show "PROP P" .
1365 then show "PROP P" .
1369 "!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
1370 "!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
1371 "!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
1372 "!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
1373 "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
1374 "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
1375 -- {* Miniscoping: pushing in existential quantifiers. *}
1376 by (iprover | blast)+
1379 "!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
1380 "!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
1381 "!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
1382 "!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
1383 "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
1384 "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
1385 -- {* Miniscoping: pushing in universal quantifiers. *}
1386 by (iprover | blast)+
1389 triv_forall_equality (*prunes params*)
1390 True_implies_equals (*prune asms `True'*)
1396 (*In general it seems wrong to add distributive laws by default: they
1397 might cause exponential blow-up. But imp_disjL has been in for a while
1398 and cannot be removed without affecting existing proofs. Moreover,
1399 rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
1400 grounds that it allows simplification of R in the two cases.*)
1418 lemmas [cong] = imp_cong simp_implies_cong
1419 lemmas [split] = split_if
1421 ML {* val HOL_ss = @{simpset} *}
1423 text {* Simplifies x assuming c and y assuming ~c *}
1426 and "c \<Longrightarrow> x = u"
1427 and "\<not> c \<Longrightarrow> y = v"
1428 shows "(if b then x else y) = (if c then u else v)"
1429 unfolding if_def using assms by simp
1431 text {* Prevents simplification of x and y:
1432 faster and allows the execution of functional programs. *}
1433 lemma if_weak_cong [cong]:
1435 shows "(if b then x else y) = (if c then x else y)"
1436 using assms by (rule arg_cong)
1438 text {* Prevents simplification of t: much faster *}
1439 lemma let_weak_cong:
1441 shows "(let x = a in t x) = (let x = b in t x)"
1442 using assms by (rule arg_cong)
1444 text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
1447 shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1451 "f (if c then x else y) = (if c then f x else f y)"
1454 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
1455 side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *}
1456 lemma restrict_to_left:
1458 shows "(x = z) = (y = z)"
1462 subsubsection {* Generic cases and induction *}
1464 text {* Rule projections: *}
1467 structure ProjectRule = ProjectRuleFun
1469 val conjunct1 = @{thm conjunct1};
1470 val conjunct2 = @{thm conjunct2};
1476 induct_forall where "induct_forall P == \<forall>x. P x"
1477 induct_implies where "induct_implies A B == A \<longrightarrow> B"
1478 induct_equal where "induct_equal x y == x = y"
1479 induct_conj where "induct_conj A B == A \<and> B"
1481 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
1482 by (unfold atomize_all induct_forall_def)
1484 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
1485 by (unfold atomize_imp induct_implies_def)
1487 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
1488 by (unfold atomize_eq induct_equal_def)
1490 lemma induct_conj_eq:
1491 includes meta_conjunction_syntax
1492 shows "(A && B) == Trueprop (induct_conj A B)"
1493 by (unfold atomize_conj induct_conj_def)
1495 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
1496 lemmas induct_rulify [symmetric, standard] = induct_atomize
1497 lemmas induct_rulify_fallback =
1498 induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1501 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1502 induct_conj (induct_forall A) (induct_forall B)"
1503 by (unfold induct_forall_def induct_conj_def) iprover
1505 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1506 induct_conj (induct_implies C A) (induct_implies C B)"
1507 by (unfold induct_implies_def induct_conj_def) iprover
1509 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
1511 assume r: "induct_conj A B ==> PROP C" and A B
1512 show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
1514 assume r: "A ==> B ==> PROP C" and "induct_conj A B"
1515 show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
1518 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1520 hide const induct_forall induct_implies induct_equal induct_conj
1522 text {* Method setup. *}
1525 structure Induct = InductFun
1527 val cases_default = @{thm case_split}
1528 val atomize = @{thms induct_atomize}
1529 val rulify = @{thms induct_rulify}
1530 val rulify_fallback = @{thms induct_rulify_fallback}
1537 subsection {* Other simple lemmas and lemma duplicates *}
1539 lemma Let_0 [simp]: "Let 0 f = f 0"
1540 unfolding Let_def ..
1542 lemma Let_1 [simp]: "Let 1 f = f 1"
1543 unfolding Let_def ..
1545 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
1548 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
1550 apply (rule_tac a = "%x. THE y. P x y" in ex1I)
1551 apply (fast dest!: theI')
1552 apply (fast intro: ext the1_equality [symmetric])
1557 apply (erule_tac x = "%z. if z = x then y else f z" in allE)
1560 apply (rule_tac P = "xa = x" in case_split_thm)
1561 apply (drule_tac [3] x = x in fun_cong, simp_all)
1564 lemma mk_left_commute:
1565 fixes f (infix "\<otimes>" 60)
1566 assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
1567 c: "\<And>x y. x \<otimes> y = y \<otimes> x"
1568 shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
1569 by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
1571 lemmas eq_sym_conv = eq_commute
1574 "(\<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)"
1575 "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
1576 "(\<not> \<not>(P)) = P"
1580 subsection {* Basic ML bindings *}
1583 val FalseE = @{thm FalseE}
1584 val Let_def = @{thm Let_def}
1585 val TrueI = @{thm TrueI}
1586 val allE = @{thm allE}
1587 val allI = @{thm allI}
1588 val all_dupE = @{thm all_dupE}
1589 val arg_cong = @{thm arg_cong}
1590 val box_equals = @{thm box_equals}
1591 val ccontr = @{thm ccontr}
1592 val classical = @{thm classical}
1593 val conjE = @{thm conjE}
1594 val conjI = @{thm conjI}
1595 val conjunct1 = @{thm conjunct1}
1596 val conjunct2 = @{thm conjunct2}
1597 val disjCI = @{thm disjCI}
1598 val disjE = @{thm disjE}
1599 val disjI1 = @{thm disjI1}
1600 val disjI2 = @{thm disjI2}
1601 val eq_reflection = @{thm eq_reflection}
1602 val ex1E = @{thm ex1E}
1603 val ex1I = @{thm ex1I}
1604 val ex1_implies_ex = @{thm ex1_implies_ex}
1605 val exE = @{thm exE}
1606 val exI = @{thm exI}
1607 val excluded_middle = @{thm excluded_middle}
1608 val ext = @{thm ext}
1609 val fun_cong = @{thm fun_cong}
1610 val iffD1 = @{thm iffD1}
1611 val iffD2 = @{thm iffD2}
1612 val iffI = @{thm iffI}
1613 val impE = @{thm impE}
1614 val impI = @{thm impI}
1615 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
1617 val notE = @{thm notE}
1618 val notI = @{thm notI}
1619 val not_all = @{thm not_all}
1620 val not_ex = @{thm not_ex}
1621 val not_iff = @{thm not_iff}
1622 val not_not = @{thm not_not}
1623 val not_sym = @{thm not_sym}
1624 val refl = @{thm refl}
1625 val rev_mp = @{thm rev_mp}
1626 val spec = @{thm spec}
1627 val ssubst = @{thm ssubst}
1628 val subst = @{thm subst}
1629 val sym = @{thm sym}
1630 val trans = @{thm trans}
1634 subsection {* Code generator basic setup -- see further @{text Code_Setup.thy} *}
1636 setup "CodeName.setup #> CodeTarget.setup #> Nbe.setup"
1638 class eq (attach "op =") = type
1640 code_datatype True False
1643 shows "False \<and> x \<longleftrightarrow> False"
1644 and "True \<and> x \<longleftrightarrow> x"
1645 and "x \<and> False \<longleftrightarrow> False"
1646 and "x \<and> True \<longleftrightarrow> x" by simp_all
1649 shows "False \<or> x \<longleftrightarrow> x"
1650 and "True \<or> x \<longleftrightarrow> True"
1651 and "x \<or> False \<longleftrightarrow> x"
1652 and "x \<or> True \<longleftrightarrow> True" by simp_all
1655 shows "\<not> True \<longleftrightarrow> False"
1656 and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
1658 instance bool :: eq ..
1661 shows "False = P \<longleftrightarrow> \<not> P"
1662 and "True = P \<longleftrightarrow> P"
1663 and "P = False \<longleftrightarrow> \<not> P"
1664 and "P = True \<longleftrightarrow> P" by simp_all
1666 code_datatype Trueprop "prop"
1668 code_datatype "TYPE('a)"
1670 lemma Let_case_cert:
1671 assumes "CASE \<equiv> (\<lambda>x. Let x f)"
1672 shows "CASE x \<equiv> f x"
1673 using assms by simp_all
1676 includes meta_conjunction_syntax
1677 assumes "CASE \<equiv> (\<lambda>b. If b f g)"
1678 shows "(CASE True \<equiv> f) && (CASE False \<equiv> g)"
1679 using assms by simp_all
1682 Code.add_case @{thm Let_case_cert}
1683 #> Code.add_case @{thm If_case_cert}
1684 #> Code.add_undefined @{const_name undefined}
1688 subsection {* Legacy tactics and ML bindings *}
1691 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
1693 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1695 fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1696 | wrong_prem (Bound _) = true
1697 | wrong_prem _ = false;
1698 val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
1700 fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
1701 fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
1704 val all_conj_distrib = thm "all_conj_distrib";
1705 val all_simps = thms "all_simps";
1706 val atomize_not = thm "atomize_not";
1707 val case_split = thm "case_split";
1708 val case_split_thm = thm "case_split_thm"
1709 val cases_simp = thm "cases_simp";
1710 val choice_eq = thm "choice_eq"
1711 val cong = thm "cong"
1712 val conj_comms = thms "conj_comms";
1713 val conj_cong = thm "conj_cong";
1714 val de_Morgan_conj = thm "de_Morgan_conj";
1715 val de_Morgan_disj = thm "de_Morgan_disj";
1716 val disj_assoc = thm "disj_assoc";
1717 val disj_comms = thms "disj_comms";
1718 val disj_cong = thm "disj_cong";
1719 val eq_ac = thms "eq_ac";
1720 val eq_cong2 = thm "eq_cong2"
1721 val Eq_FalseI = thm "Eq_FalseI";
1722 val Eq_TrueI = thm "Eq_TrueI";
1723 val Ex1_def = thm "Ex1_def"
1724 val ex_disj_distrib = thm "ex_disj_distrib";
1725 val ex_simps = thms "ex_simps";
1726 val if_cancel = thm "if_cancel";
1727 val if_eq_cancel = thm "if_eq_cancel";
1728 val if_False = thm "if_False";
1729 val iff_conv_conj_imp = thm "iff_conv_conj_imp";
1731 val if_splits = thms "if_splits";
1732 val if_True = thm "if_True";
1733 val if_weak_cong = thm "if_weak_cong"
1734 val imp_all = thm "imp_all";
1735 val imp_cong = thm "imp_cong";
1736 val imp_conjL = thm "imp_conjL";
1737 val imp_conjR = thm "imp_conjR";
1738 val imp_conv_disj = thm "imp_conv_disj";
1739 val simp_implies_def = thm "simp_implies_def";
1740 val simp_thms = thms "simp_thms";
1741 val split_if = thm "split_if";
1742 val the1_equality = thm "the1_equality"
1743 val theI = thm "theI"
1744 val theI' = thm "theI'"
1745 val True_implies_equals = thm "True_implies_equals";
1746 val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms @ @{thms "nnf_simps"})