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
50 Trueprop :: "bool => prop" ("(_)" 5)
53 Not :: "bool => bool" ("~ _" [40] 40)
58 The :: "('a => bool) => 'a"
59 All :: "('a => bool) => bool" (binder "ALL " 10)
60 Ex :: "('a => bool) => bool" (binder "EX " 10)
61 Ex1 :: "('a => bool) => bool" (binder "EX! " 10)
62 Let :: "['a, 'a => 'b] => 'b"
64 "op =" :: "['a, 'a] => bool" (infixl "=" 50)
65 "op &" :: "[bool, bool] => bool" (infixr "&" 35)
66 "op |" :: "[bool, bool] => bool" (infixr "|" 30)
67 "op -->" :: "[bool, bool] => bool" (infixr "-->" 25)
72 If :: "[bool, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10)
75 subsubsection {* Additional concrete syntax *}
81 not_equal :: "['a, 'a] => bool" (infixl "~=" 50) where
85 not_equal (infix "~=" 50)
88 Not ("\<not> _" [40] 40) and
89 "op &" (infixr "\<and>" 35) and
90 "op |" (infixr "\<or>" 30) and
91 "op -->" (infixr "\<longrightarrow>" 25) and
92 not_equal (infix "\<noteq>" 50)
94 notation (HTML output)
95 Not ("\<not> _" [40] 40) and
96 "op &" (infixr "\<and>" 35) and
97 "op |" (infixr "\<or>" 30) and
98 not_equal (infix "\<noteq>" 50)
101 iff :: "[bool, bool] => bool" (infixr "<->" 25) where
105 iff (infixr "\<longleftrightarrow>" 25)
113 "_The" :: "[pttrn, bool] => 'a" ("(3THE _./ _)" [0, 10] 10)
115 "_bind" :: "[pttrn, 'a] => letbind" ("(2_ =/ _)" 10)
116 "" :: "letbind => letbinds" ("_")
117 "_binds" :: "[letbind, letbinds] => letbinds" ("_;/ _")
118 "_Let" :: "[letbinds, 'a] => 'a" ("(let (_)/ in (_))" 10)
120 "_case_syntax":: "['a, cases_syn] => 'b" ("(case _ of/ _)" 10)
121 "_case1" :: "['a, 'b] => case_syn" ("(2_ =>/ _)" 10)
122 "" :: "case_syn => cases_syn" ("_")
123 "_case2" :: "[case_syn, cases_syn] => cases_syn" ("_/ | _")
126 "THE x. P" == "The (%x. P)"
127 "_Let (_binds b bs) e" == "_Let b (_Let bs e)"
128 "let x = a in e" == "Let a (%x. e)"
131 (* To avoid eta-contraction of body: *)
132 [("The", fn [Abs abs] =>
133 let val (x,t) = atomic_abs_tr' abs
134 in Syntax.const "_The" $ x $ t end)]
138 "_case1" :: "['a, 'b] => case_syn" ("(2_ \<Rightarrow>/ _)" 10)
141 All (binder "\<forall>" 10) and
142 Ex (binder "\<exists>" 10) and
143 Ex1 (binder "\<exists>!" 10)
145 notation (HTML output)
146 All (binder "\<forall>" 10) and
147 Ex (binder "\<exists>" 10) and
148 Ex1 (binder "\<exists>!" 10)
151 All (binder "! " 10) and
152 Ex (binder "? " 10) and
153 Ex1 (binder "?! " 10)
156 subsubsection {* Axioms and basic definitions *}
159 eq_reflection: "(x=y) ==> (x==y)"
163 ext: "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
164 -- {*Extensionality is built into the meta-logic, and this rule expresses
165 a related property. It is an eta-expanded version of the traditional
166 rule, and similar to the ABS rule of HOL*}
168 the_eq_trivial: "(THE x. x = a) = (a::'a)"
170 impI: "(P ==> Q) ==> P-->Q"
171 mp: "[| P-->Q; P |] ==> Q"
175 True_def: "True == ((%x::bool. x) = (%x. x))"
176 All_def: "All(P) == (P = (%x. True))"
177 Ex_def: "Ex(P) == !Q. (!x. P x --> Q) --> Q"
178 False_def: "False == (!P. P)"
179 not_def: "~ P == P-->False"
180 and_def: "P & Q == !R. (P-->Q-->R) --> R"
181 or_def: "P | Q == !R. (P-->R) --> (Q-->R) --> R"
182 Ex1_def: "Ex1(P) == ? x. P(x) & (! y. P(y) --> y=x)"
185 iff: "(P-->Q) --> (Q-->P) --> (P=Q)"
186 True_or_False: "(P=True) | (P=False)"
189 Let_def: "Let s f == f(s)"
190 if_def: "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
202 undefined_fun: "undefined x = undefined"
205 subsubsection {* Generic classes and algebraic operations *}
207 class default = type +
211 fixes zero :: 'a ("\<^loc>0")
214 fixes one :: 'a ("\<^loc>1")
216 hide (open) const zero one
219 fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>+" 65)
222 fixes uminus :: "'a \<Rightarrow> 'a"
223 and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>-" 65)
226 fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>*" 70)
228 class inverse = type +
229 fixes inverse :: "'a \<Rightarrow> 'a"
230 and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>'/" 70)
233 fixes abs :: "'a \<Rightarrow> 'a"
236 fixes sgn :: "'a \<Rightarrow> 'a"
239 uminus ("- _" [81] 80)
242 abs ("\<bar>_\<bar>")
243 notation (HTML output)
244 abs ("\<bar>_\<bar>")
247 fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
248 and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
252 less_eq ("op \<^loc><=") and
253 less_eq ("(_/ \<^loc><= _)" [51, 51] 50) and
254 less ("op \<^loc><") and
255 less ("(_/ \<^loc>< _)" [51, 51] 50)
258 less_eq ("op \<^loc>\<le>") and
259 less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50)
261 notation (HTML output)
262 less_eq ("op \<^loc>\<le>") and
263 less_eq ("(_/ \<^loc>\<le> _)" [51, 51] 50)
266 greater_eq (infix "\<^loc>>=" 50) where
267 "x \<^loc>>= y \<equiv> y \<^loc><= x"
270 greater_eq (infix "\<^loc>\<ge>" 50)
273 greater (infix "\<^loc>>" 50) where
274 "x \<^loc>> y \<equiv> y \<^loc>< x"
277 Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "\<^loc>LEAST " 10)
279 "Least P == (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<^loc>\<le> y))"
284 less_eq ("op <=") and
285 less_eq ("(_/ <= _)" [51, 51] 50) and
287 less ("(_/ < _)" [51, 51] 50)
290 less_eq ("op \<le>") and
291 less_eq ("(_/ \<le> _)" [51, 51] 50)
293 notation (HTML output)
294 less_eq ("op \<le>") and
295 less_eq ("(_/ \<le> _)" [51, 51] 50)
298 greater_eq (infix "\<ge>" 50)
301 "_index1" :: index ("\<^sub>1")
303 (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
305 typed_print_translation {*
307 fun tr' c = (c, fn show_sorts => fn T => fn ts =>
308 if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
309 else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
310 in map tr' [@{const_syntax HOL.one}, @{const_syntax HOL.zero}] end;
311 *} -- {* show types that are presumably too general *}
314 subsection {* Fundamental rules *}
316 subsubsection {* Equality *}
318 text {* Thanks to Stephan Merz *}
320 assumes eq: "s = t" and p: "P s"
323 from eq have meta: "s \<equiv> t"
324 by (rule eq_reflection)
329 lemma sym: "s = t ==> t = s"
330 by (erule subst) (rule refl)
332 lemma ssubst: "t = s ==> P s ==> P t"
333 by (drule sym) (erule subst)
335 lemma trans: "[| r=s; s=t |] ==> r=t"
338 lemma meta_eq_to_obj_eq:
339 assumes meq: "A == B"
341 by (unfold meq) (rule refl)
343 text {* Useful with @{text erule} for proving equalities from known equalities. *}
347 lemma box_equals: "[| a=b; a=c; b=d |] ==> c=d"
354 text {* For calculational reasoning: *}
356 lemma forw_subst: "a = b ==> P b ==> P a"
359 lemma back_subst: "P a ==> a = b ==> P b"
363 subsubsection {*Congruence rules for application*}
365 (*similar to AP_THM in Gordon's HOL*)
366 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
371 (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
372 lemma arg_cong: "x=y ==> f(x)=f(y)"
377 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
378 apply (erule ssubst)+
382 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
388 subsubsection {*Equality of booleans -- iff*}
390 lemma iffI: assumes "P ==> Q" and "Q ==> P" shows "P=Q"
391 by (iprover intro: iff [THEN mp, THEN mp] impI assms)
393 lemma iffD2: "[| P=Q; Q |] ==> P"
396 lemma rev_iffD2: "[| Q; P=Q |] ==> P"
399 lemma iffD1: "Q = P \<Longrightarrow> Q \<Longrightarrow> P"
400 by (drule sym) (rule iffD2)
402 lemma rev_iffD1: "Q \<Longrightarrow> Q = P \<Longrightarrow> P"
403 by (drule sym) (rule rev_iffD2)
407 and minor: "[| P --> Q; Q --> P |] ==> R"
409 by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
412 subsubsection {*True*}
415 unfolding True_def by (rule refl)
417 lemma eqTrueI: "P ==> P = True"
418 by (iprover intro: iffI TrueI)
420 lemma eqTrueE: "P = True ==> P"
421 by (erule iffD2) (rule TrueI)
424 subsubsection {*Universal quantifier*}
426 lemma allI: assumes "!!x::'a. P(x)" shows "ALL x. P(x)"
427 unfolding All_def by (iprover intro: ext eqTrueI assms)
429 lemma spec: "ALL x::'a. P(x) ==> P(x)"
430 apply (unfold All_def)
432 apply (erule fun_cong)
436 assumes major: "ALL x. P(x)"
437 and minor: "P(x) ==> R"
439 by (iprover intro: minor major [THEN spec])
442 assumes major: "ALL x. P(x)"
443 and minor: "[| P(x); ALL x. P(x) |] ==> R"
445 by (iprover intro: minor major major [THEN spec])
448 subsubsection {* False *}
451 Depends upon @{text spec}; it is impossible to do propositional
452 logic before quantifiers!
455 lemma FalseE: "False ==> P"
456 apply (unfold False_def)
460 lemma False_neq_True: "False = True ==> P"
461 by (erule eqTrueE [THEN FalseE])
464 subsubsection {* Negation *}
467 assumes "P ==> False"
469 apply (unfold not_def)
470 apply (iprover intro: impI assms)
473 lemma False_not_True: "False ~= True"
475 apply (erule False_neq_True)
478 lemma True_not_False: "True ~= False"
481 apply (erule False_neq_True)
484 lemma notE: "[| ~P; P |] ==> R"
485 apply (unfold not_def)
486 apply (erule mp [THEN FalseE])
490 lemma notI2: "(P \<Longrightarrow> \<not> Pa) \<Longrightarrow> (P \<Longrightarrow> Pa) \<Longrightarrow> \<not> P"
491 by (erule notE [THEN notI]) (erule meta_mp)
494 subsubsection {*Implication*}
497 assumes "P-->Q" "P" "Q ==> R"
499 by (iprover intro: assms mp)
501 (* Reduces Q to P-->Q, allowing substitution in P. *)
502 lemma rev_mp: "[| P; P --> Q |] ==> Q"
503 by (iprover intro: mp)
509 by (iprover intro: notI minor major [THEN notE])
511 (*not used at all, but we already have the other 3 combinations *)
514 and minor: "P ==> ~Q"
516 by (iprover intro: notI minor major notE)
518 lemma not_sym: "t ~= s ==> s ~= t"
519 by (erule contrapos_nn) (erule sym)
521 lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
522 by (erule subst, erule ssubst, assumption)
524 (*still used in HOLCF*)
526 assumes pq: "P ==> Q"
529 apply (rule nq [THEN contrapos_nn])
533 subsubsection {*Existential quantifier*}
535 lemma exI: "P x ==> EX x::'a. P x"
536 apply (unfold Ex_def)
537 apply (iprover intro: allI allE impI mp)
541 assumes major: "EX x::'a. P(x)"
542 and minor: "!!x. P(x) ==> Q"
544 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
545 apply (iprover intro: impI [THEN allI] minor)
549 subsubsection {*Conjunction*}
551 lemma conjI: "[| P; Q |] ==> P&Q"
552 apply (unfold and_def)
553 apply (iprover intro: impI [THEN allI] mp)
556 lemma conjunct1: "[| P & Q |] ==> P"
557 apply (unfold and_def)
558 apply (iprover intro: impI dest: spec mp)
561 lemma conjunct2: "[| P & Q |] ==> Q"
562 apply (unfold and_def)
563 apply (iprover intro: impI dest: spec mp)
568 and minor: "[| P; Q |] ==> R"
571 apply (rule major [THEN conjunct1])
572 apply (rule major [THEN conjunct2])
576 assumes "P" "P ==> Q" shows "P & Q"
577 by (iprover intro: conjI assms)
580 subsubsection {*Disjunction*}
582 lemma disjI1: "P ==> P|Q"
583 apply (unfold or_def)
584 apply (iprover intro: allI impI mp)
587 lemma disjI2: "Q ==> P|Q"
588 apply (unfold or_def)
589 apply (iprover intro: allI impI mp)
594 and minorP: "P ==> R"
595 and minorQ: "Q ==> R"
597 by (iprover intro: minorP minorQ impI
598 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
601 subsubsection {*Classical logic*}
604 assumes prem: "~P ==> P"
606 apply (rule True_or_False [THEN disjE, THEN eqTrueE])
608 apply (rule notI [THEN prem, THEN eqTrueI])
613 lemmas ccontr = FalseE [THEN classical, standard]
615 (*notE with premises exchanged; it discharges ~R so that it can be used to
616 make elimination rules*)
619 and premnot: "~R ==> ~P"
622 apply (erule notE [OF premnot premp])
625 (*Double negation law*)
626 lemma notnotD: "~~P ==> P"
627 apply (rule classical)
636 by (iprover intro: classical p1 p2 notE)
639 subsubsection {*Unique existence*}
642 assumes "P a" "!!x. P(x) ==> x=a"
644 by (unfold Ex1_def, iprover intro: assms exI conjI allI impI)
646 text{*Sometimes easier to use: the premises have no shared variables. Safe!*}
648 assumes ex_prem: "EX x. P(x)"
649 and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
651 by (iprover intro: ex_prem [THEN exE] ex1I eq)
654 assumes major: "EX! x. P(x)"
655 and minor: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R"
657 apply (rule major [unfolded Ex1_def, THEN exE])
659 apply (iprover intro: minor)
662 lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
669 subsubsection {*THE: definite description operator*}
673 and premx: "!!x. P x ==> x=a"
674 shows "(THE x. P x) = a"
675 apply (rule trans [OF _ the_eq_trivial])
676 apply (rule_tac f = "The" in arg_cong)
680 apply (erule ssubst, rule prema)
684 assumes "P a" and "!!x. P x ==> x=a"
685 shows "P (THE x. P x)"
686 by (iprover intro: assms the_equality [THEN ssubst])
688 lemma theI': "EX! x. P x ==> P (THE x. P x)"
696 (*Easier to apply than theI: only one occurrence of P*)
698 assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
699 shows "Q (THE x. P x)"
700 by (iprover intro: assms theI)
702 lemma the1I2: assumes "EX! x. P x" "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
703 by(iprover intro:assms(2) theI2[where P=P and Q=Q] ex1E[OF assms(1)]
706 lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
707 apply (rule the_equality)
710 apply (erule all_dupE)
719 lemma the_sym_eq_trivial: "(THE y. x=y) = x"
720 apply (rule the_equality)
726 subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
729 assumes "~Q ==> P" shows "P|Q"
730 apply (rule classical)
731 apply (iprover intro: assms disjI1 disjI2 notI elim: notE)
734 lemma excluded_middle: "~P | P"
735 by (iprover intro: disjCI)
738 case distinction as a natural deduction rule.
739 Note that @{term "~P"} is the second case, not the first
741 lemma case_split_thm:
742 assumes prem1: "P ==> Q"
743 and prem2: "~P ==> Q"
745 apply (rule excluded_middle [THEN disjE])
749 lemmas case_split = case_split_thm [case_names True False]
751 (*Classical implies (-->) elimination. *)
753 assumes major: "P-->Q"
754 and minor: "~P ==> R" "Q ==> R"
756 apply (rule excluded_middle [of P, THEN disjE])
757 apply (iprover intro: minor major [THEN mp])+
760 (*This version of --> elimination works on Q before P. It works best for
761 those cases in which P holds "almost everywhere". Can't install as
762 default: would break old proofs.*)
764 assumes major: "P-->Q"
765 and minor: "Q ==> R" "~P ==> R"
767 apply (rule excluded_middle [of P, THEN disjE])
768 apply (iprover intro: minor major [THEN mp])+
771 (*Classical <-> elimination. *)
774 and minor: "[| P; Q |] ==> R" "[| ~P; ~Q |] ==> R"
776 apply (rule major [THEN iffE])
777 apply (iprover intro: minor elim: impCE notE)
781 assumes "ALL x. ~P(x) ==> P(a)"
784 apply (iprover intro: assms exI allI notI notE [of "\<exists>x. P x"])
788 subsubsection {* Intuitionistic Reasoning *}
793 and 3: "P --> Q ==> P"
796 from 3 and 1 have P .
797 with 1 have Q by (rule impE)
802 assumes 1: "ALL x. P x"
803 and 2: "P x ==> ALL x. P x ==> Q"
806 from 1 have "P x" by (rule spec)
807 from this and 1 show Q by (rule 2)
815 from 2 and 1 have P .
816 with 1 show R by (rule notE)
819 lemma TrueE: "True ==> P ==> P" .
820 lemma notFalseE: "~ False ==> P ==> P" .
822 lemmas [Pure.elim!] = disjE iffE FalseE conjE exE TrueE notFalseE
823 and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
824 and [Pure.elim 2] = allE notE' impE'
825 and [Pure.intro] = exI disjI2 disjI1
827 lemmas [trans] = trans
828 and [sym] = sym not_sym
829 and [Pure.elim?] = iffD1 iffD2 impE
834 subsubsection {* Atomizing meta-level connectives *}
836 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
839 then show "ALL x. P x" ..
842 then show "!!x. P x" by (rule allE)
845 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
848 show "A --> B" by (rule impI) (rule r)
850 assume "A --> B" and A
851 then show B by (rule mp)
854 lemma atomize_not: "(A ==> False) == Trueprop (~A)"
856 assume r: "A ==> False"
857 show "~A" by (rule notI) (rule r)
860 then show False by (rule notE)
863 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
866 show "x = y" by (unfold `x == y`) (rule refl)
869 then show "x == y" by (rule eq_reflection)
872 lemma atomize_conj [atomize]:
873 includes meta_conjunction_syntax
874 shows "(A && B) == Trueprop (A & B)"
876 assume conj: "A && B"
879 from conj show A by (rule conjunctionD1)
880 from conj show B by (rule conjunctionD2)
891 lemmas [symmetric, rulify] = atomize_all atomize_imp
892 and [symmetric, defn] = atomize_all atomize_imp atomize_eq
895 subsection {* Package setup *}
897 subsubsection {* Classical Reasoner setup *}
900 "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
903 structure Hypsubst = HypsubstFun(
905 structure Simplifier = Simplifier
906 val dest_eq = HOLogic.dest_eq
907 val dest_Trueprop = HOLogic.dest_Trueprop
908 val dest_imp = HOLogic.dest_imp
909 val eq_reflection = @{thm HOL.eq_reflection}
910 val rev_eq_reflection = @{thm HOL.meta_eq_to_obj_eq}
911 val imp_intr = @{thm HOL.impI}
912 val rev_mp = @{thm HOL.rev_mp}
913 val subst = @{thm HOL.subst}
914 val sym = @{thm HOL.sym}
915 val thin_refl = @{thm thin_refl};
919 structure Classical = ClassicalFun(
921 val mp = @{thm HOL.mp}
922 val not_elim = @{thm HOL.notE}
923 val classical = @{thm HOL.classical}
924 val sizef = Drule.size_of_thm
925 val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]
928 structure BasicClassical: BASIC_CLASSICAL = Classical;
931 ML_Context.value_antiq "claset"
932 (Scan.succeed ("claset", "Classical.local_claset_of (ML_Context.the_local_context ())"));
934 structure ResAtpset = NamedThmsFun(val name = "atp" val description = "ATP rules");
936 structure ResBlacklist = NamedThmsFun(val name = "noatp" val description = "Theorems blacklisted for ATP");
939 (*ResBlacklist holds theorems blacklisted to sledgehammer.
940 These theorems typically produce clauses that are prolific (match too many equality or
941 membership literals) and relate to seldom-used facts. Some duplicate other rules.*)
945 (*prevent substitution on bool*)
946 fun hyp_subst_tac' i thm = if i <= Thm.nprems_of thm andalso
947 Term.exists_Const (fn ("op =", Type (_, [T, _])) => T <> Type ("bool", []) | _ => false)
948 (nth (Thm.prems_of thm) (i - 1)) then Hypsubst.hyp_subst_tac i thm else no_tac thm;
950 Hypsubst.hypsubst_setup
951 #> ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac)
954 #> ResBlacklist.setup
958 declare iffI [intro!]
966 declare iffCE [elim!]
973 declare ex_ex1I [intro!]
975 and the_equality [intro]
981 ML {* val HOL_cs = @{claset} *}
983 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
985 apply (erule (1) meta_mp)
988 declare ex_ex1I [rule del, intro! 2]
991 lemmas [intro?] = ext
992 and [elim?] = ex1_implies_ex
994 (*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
995 lemma alt_ex1E [elim!]:
996 assumes major: "\<exists>!x. P x"
997 and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
999 apply (rule ex1E [OF major])
1001 apply (tactic {* ares_tac @{thms allI} 1 *})+
1002 apply (tactic {* etac (Classical.dup_elim @{thm allE}) 1 *})
1007 structure Blast = BlastFun(
1009 type claset = Classical.claset
1010 val equality_name = @{const_name "op ="}
1011 val not_name = @{const_name Not}
1012 val notE = @{thm HOL.notE}
1013 val ccontr = @{thm HOL.ccontr}
1014 val contr_tac = Classical.contr_tac
1015 val dup_intr = Classical.dup_intr
1016 val hyp_subst_tac = Hypsubst.blast_hyp_subst_tac
1017 val claset = Classical.claset
1018 val rep_cs = Classical.rep_cs
1019 val cla_modifiers = Classical.cla_modifiers
1020 val cla_meth' = Classical.cla_meth'
1022 val Blast_tac = Blast.Blast_tac;
1023 val blast_tac = Blast.blast_tac;
1029 subsubsection {* Simplifier *}
1031 lemma eta_contract_eq: "(%s. f s) = f" ..
1034 shows not_not: "(~ ~ P) = P"
1035 and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
1037 "(P ~= Q) = (P = (~Q))"
1038 "(P | ~P) = True" "(~P | P) = True"
1040 and not_True_eq_False: "(\<not> True) = False"
1041 and not_False_eq_True: "(\<not> False) = True"
1043 "(~P) ~= P" "P ~= (~P)"
1045 and eq_True: "(P = True) = P"
1046 and "(False=P) = (~P)"
1047 and eq_False: "(P = False) = (\<not> P)"
1049 "(True --> P) = P" "(False --> P) = True"
1050 "(P --> True) = True" "(P --> P) = True"
1051 "(P --> False) = (~P)" "(P --> ~P) = (~P)"
1052 "(P & True) = P" "(True & P) = P"
1053 "(P & False) = False" "(False & P) = False"
1054 "(P & P) = P" "(P & (P & Q)) = (P & Q)"
1055 "(P & ~P) = False" "(~P & P) = False"
1056 "(P | True) = True" "(True | P) = True"
1057 "(P | False) = P" "(False | P) = P"
1058 "(P | P) = P" "(P | (P | Q)) = (P | Q)" and
1059 "(ALL x. P) = P" "(EX x. P) = P" "EX x. x=t" "EX x. t=x"
1060 -- {* needed for the one-point-rule quantifier simplification procs *}
1061 -- {* essential for termination!! *} and
1062 "!!P. (EX x. x=t & P(x)) = P(t)"
1063 "!!P. (EX x. t=x & P(x)) = P(t)"
1064 "!!P. (ALL x. x=t --> P(x)) = P(t)"
1065 "!!P. (ALL x. t=x --> P(x)) = P(t)"
1066 by (blast, blast, blast, blast, blast, iprover+)
1068 lemma disj_absorb: "(A | A) = A"
1071 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
1074 lemma conj_absorb: "(A & A) = A"
1077 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
1081 shows eq_commute: "(a=b) = (b=a)"
1082 and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
1083 and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
1084 lemma neq_commute: "(a~=b) = (b~=a)" by iprover
1087 shows conj_commute: "(P&Q) = (Q&P)"
1088 and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
1089 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
1091 lemmas conj_ac = conj_commute conj_left_commute conj_assoc
1094 shows disj_commute: "(P|Q) = (Q|P)"
1095 and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
1096 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
1098 lemmas disj_ac = disj_commute disj_left_commute disj_assoc
1100 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
1101 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
1103 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
1104 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
1106 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
1107 lemma imp_conjL: "((P&Q) -->R) = (P --> (Q --> R))" by iprover
1108 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
1110 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
1111 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
1112 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
1114 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
1115 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
1117 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
1120 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
1121 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
1122 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
1123 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
1124 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
1125 lemma disj_not2: "(P | ~Q) = (Q --> P)" -- {* changes orientation :-( *}
1127 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
1129 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
1132 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
1133 -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
1134 -- {* cases boil down to the same thing. *}
1137 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
1138 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
1139 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
1140 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
1141 lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" by blast
1143 declare All_def [noatp]
1145 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
1146 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
1149 \medskip The @{text "&"} congruence rule: not included by default!
1150 May slow rewrite proofs down by as much as 50\% *}
1153 "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
1156 lemma rev_conj_cong:
1157 "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
1160 text {* The @{text "|"} congruence rule: not included by default! *}
1163 "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
1167 text {* \medskip if-then-else rules *}
1169 lemma if_True: "(if True then x else y) = x"
1170 by (unfold if_def) blast
1172 lemma if_False: "(if False then x else y) = y"
1173 by (unfold if_def) blast
1175 lemma if_P: "P ==> (if P then x else y) = x"
1176 by (unfold if_def) blast
1178 lemma if_not_P: "~P ==> (if P then x else y) = y"
1179 by (unfold if_def) blast
1181 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
1182 apply (rule case_split [of Q])
1183 apply (simplesubst if_P)
1184 prefer 3 apply (simplesubst if_not_P, blast+)
1187 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
1188 by (simplesubst split_if, blast)
1190 lemmas if_splits [noatp] = split_if split_if_asm
1192 lemma if_cancel: "(if c then x else x) = x"
1193 by (simplesubst split_if, blast)
1195 lemma if_eq_cancel: "(if x = y then y else x) = x"
1196 by (simplesubst split_if, blast)
1198 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
1199 -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
1202 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
1203 -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
1204 apply (simplesubst split_if, blast)
1207 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
1208 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
1210 text {* \medskip let rules for simproc *}
1212 lemma Let_folded: "f x \<equiv> g x \<Longrightarrow> Let x f \<equiv> Let x g"
1215 lemma Let_unfold: "f x \<equiv> g \<Longrightarrow> Let x f \<equiv> g"
1219 The following copy of the implication operator is useful for
1220 fine-tuning congruence rules. It instructs the simplifier to simplify
1225 simp_implies :: "[prop, prop] => prop" (infixr "=simp=>" 1)
1226 "simp_implies \<equiv> op ==>"
1228 lemma simp_impliesI:
1229 assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
1230 shows "PROP P =simp=> PROP Q"
1231 apply (unfold simp_implies_def)
1236 lemma simp_impliesE:
1237 assumes PQ:"PROP P =simp=> PROP Q"
1239 and QR: "PROP Q \<Longrightarrow> PROP R"
1242 apply (rule PQ [unfolded simp_implies_def])
1246 lemma simp_implies_cong:
1247 assumes PP' :"PROP P == PROP P'"
1248 and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
1249 shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
1250 proof (unfold simp_implies_def, rule equal_intr_rule)
1251 assume PQ: "PROP P \<Longrightarrow> PROP Q"
1253 from PP' [symmetric] and P' have "PROP P"
1254 by (rule equal_elim_rule1)
1255 then have "PROP Q" by (rule PQ)
1256 with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
1258 assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
1260 from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
1261 then have "PROP Q'" by (rule P'Q')
1262 with P'QQ' [OF P', symmetric] show "PROP Q"
1263 by (rule equal_elim_rule1)
1267 assumes "P \<longrightarrow> Q \<longrightarrow> R"
1268 shows "P \<and> Q \<longrightarrow> R"
1269 using assms by blast
1272 assumes "\<And>x. P x = Q x"
1273 shows "(\<forall>x. P x) = (\<forall>x. Q x)"
1274 using assms by blast
1277 assumes "\<And>x. P x = Q x"
1278 shows "(\<exists>x. P x) = (\<exists>x. Q x)"
1279 using assms by blast
1282 "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
1286 "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
1290 ML {* open Simpdata *}
1293 Simplifier.method_setup Splitter.split_modifiers
1294 #> (fn thy => (change_simpset_of thy (fn _ => Simpdata.simpset_simprocs); thy))
1300 text {* Simproc for proving @{text "(y = x) == False"} from premise @{text "~(x = y)"}: *}
1302 simproc_setup neq ("x = y") = {* fn _ =>
1304 val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI};
1305 fun is_neq eq lhs rhs thm =
1306 (case Thm.prop_of thm of
1307 _ $ (Not $ (eq' $ l' $ r')) =>
1308 Not = HOLogic.Not andalso eq' = eq andalso
1309 r' aconv lhs andalso l' aconv rhs
1312 (case Thm.term_of ct of
1314 (case find_first (is_neq eq lhs rhs) (Simplifier.prems_of_ss ss) of
1315 SOME thm => SOME (thm RS neq_to_EQ_False)
1321 simproc_setup let_simp ("Let x f") = {*
1323 val (f_Let_unfold, x_Let_unfold) =
1324 let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold}
1325 in (cterm_of @{theory} f, cterm_of @{theory} x) end
1326 val (f_Let_folded, x_Let_folded) =
1327 let val [(_$(f$x)$_)] = prems_of @{thm Let_folded}
1328 in (cterm_of @{theory} f, cterm_of @{theory} x) end;
1330 let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end;
1334 val ctxt = Simplifier.the_context ss;
1335 val thy = ProofContext.theory_of ctxt;
1336 val t = Thm.term_of ct;
1337 val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
1338 in Option.map (hd o Variable.export ctxt' ctxt o single)
1339 (case t' of Const ("Let",_) $ x $ f => (* x and f are already in normal form *)
1340 if is_Free x orelse is_Bound x orelse is_Const x
1341 then SOME @{thm Let_def}
1344 val n = case f of (Abs (x,_,_)) => x | _ => "x";
1345 val cx = cterm_of thy x;
1346 val {T=xT,...} = rep_cterm cx;
1347 val cf = cterm_of thy f;
1348 val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
1349 val (_$_$g) = prop_of fx_g;
1350 val g' = abstract_over (x,g);
1355 cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold};
1356 in SOME (rl OF [fx_g]) end
1357 else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
1359 val abs_g'= Abs (n,xT,g');
1361 val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x));
1362 val rl = cterm_instantiate
1363 [(f_Let_folded,cterm_of thy f),(x_Let_folded,cx),
1364 (g_Let_folded,cterm_of thy abs_g')]
1366 in SOME (rl OF [transitive fx_g g_g'x])
1374 lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
1376 assume "True \<Longrightarrow> PROP P"
1377 from this [OF TrueI] show "PROP P" .
1380 then show "PROP P" .
1384 "!!P Q. (EX x. P x & Q) = ((EX x. P x) & Q)"
1385 "!!P Q. (EX x. P & Q x) = (P & (EX x. Q x))"
1386 "!!P Q. (EX x. P x | Q) = ((EX x. P x) | Q)"
1387 "!!P Q. (EX x. P | Q x) = (P | (EX x. Q x))"
1388 "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
1389 "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
1390 -- {* Miniscoping: pushing in existential quantifiers. *}
1391 by (iprover | blast)+
1394 "!!P Q. (ALL x. P x & Q) = ((ALL x. P x) & Q)"
1395 "!!P Q. (ALL x. P & Q x) = (P & (ALL x. Q x))"
1396 "!!P Q. (ALL x. P x | Q) = ((ALL x. P x) | Q)"
1397 "!!P Q. (ALL x. P | Q x) = (P | (ALL x. Q x))"
1398 "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
1399 "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
1400 -- {* Miniscoping: pushing in universal quantifiers. *}
1401 by (iprover | blast)+
1404 triv_forall_equality (*prunes params*)
1405 True_implies_equals (*prune asms `True'*)
1411 (*In general it seems wrong to add distributive laws by default: they
1412 might cause exponential blow-up. But imp_disjL has been in for a while
1413 and cannot be removed without affecting existing proofs. Moreover,
1414 rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
1415 grounds that it allows simplification of R in the two cases.*)
1433 lemmas [cong] = imp_cong simp_implies_cong
1434 lemmas [split] = split_if
1436 ML {* val HOL_ss = @{simpset} *}
1438 text {* Simplifies x assuming c and y assuming ~c *}
1441 and "c \<Longrightarrow> x = u"
1442 and "\<not> c \<Longrightarrow> y = v"
1443 shows "(if b then x else y) = (if c then u else v)"
1444 unfolding if_def using assms by simp
1446 text {* Prevents simplification of x and y:
1447 faster and allows the execution of functional programs. *}
1448 lemma if_weak_cong [cong]:
1450 shows "(if b then x else y) = (if c then x else y)"
1451 using assms by (rule arg_cong)
1453 text {* Prevents simplification of t: much faster *}
1454 lemma let_weak_cong:
1456 shows "(let x = a in t x) = (let x = b in t x)"
1457 using assms by (rule arg_cong)
1459 text {* To tidy up the result of a simproc. Only the RHS will be simplified. *}
1462 shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
1466 "f (if c then x else y) = (if c then f x else f y)"
1469 text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
1470 side of an equality. Used in @{text "{Integ,Real}/simproc.ML"} *}
1471 lemma restrict_to_left:
1473 shows "(x = z) = (y = z)"
1477 subsubsection {* Generic cases and induction *}
1479 text {* Rule projections: *}
1482 structure ProjectRule = ProjectRuleFun
1484 val conjunct1 = @{thm conjunct1};
1485 val conjunct2 = @{thm conjunct2};
1491 induct_forall where "induct_forall P == \<forall>x. P x"
1492 induct_implies where "induct_implies A B == A \<longrightarrow> B"
1493 induct_equal where "induct_equal x y == x = y"
1494 induct_conj where "induct_conj A B == A \<and> B"
1496 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
1497 by (unfold atomize_all induct_forall_def)
1499 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
1500 by (unfold atomize_imp induct_implies_def)
1502 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
1503 by (unfold atomize_eq induct_equal_def)
1505 lemma induct_conj_eq:
1506 includes meta_conjunction_syntax
1507 shows "(A && B) == Trueprop (induct_conj A B)"
1508 by (unfold atomize_conj induct_conj_def)
1510 lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
1511 lemmas induct_rulify [symmetric, standard] = induct_atomize
1512 lemmas induct_rulify_fallback =
1513 induct_forall_def induct_implies_def induct_equal_def induct_conj_def
1516 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
1517 induct_conj (induct_forall A) (induct_forall B)"
1518 by (unfold induct_forall_def induct_conj_def) iprover
1520 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
1521 induct_conj (induct_implies C A) (induct_implies C B)"
1522 by (unfold induct_implies_def induct_conj_def) iprover
1524 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
1526 assume r: "induct_conj A B ==> PROP C" and A B
1527 show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
1529 assume r: "A ==> B ==> PROP C" and "induct_conj A B"
1530 show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
1533 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
1535 hide const induct_forall induct_implies induct_equal induct_conj
1537 text {* Method setup. *}
1540 structure Induct = InductFun
1542 val cases_default = @{thm case_split}
1543 val atomize = @{thms induct_atomize}
1544 val rulify = @{thms induct_rulify}
1545 val rulify_fallback = @{thms induct_rulify_fallback}
1552 subsection {* Other simple lemmas and lemma duplicates *}
1554 lemma Let_0 [simp]: "Let 0 f = f 0"
1555 unfolding Let_def ..
1557 lemma Let_1 [simp]: "Let 1 f = f 1"
1558 unfolding Let_def ..
1560 lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
1563 lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
1565 apply (rule_tac a = "%x. THE y. P x y" in ex1I)
1566 apply (fast dest!: theI')
1567 apply (fast intro: ext the1_equality [symmetric])
1572 apply (erule_tac x = "%z. if z = x then y else f z" in allE)
1575 apply (rule_tac P = "xa = x" in case_split_thm)
1576 apply (drule_tac [3] x = x in fun_cong, simp_all)
1579 lemma mk_left_commute:
1580 fixes f (infix "\<otimes>" 60)
1581 assumes a: "\<And>x y z. (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" and
1582 c: "\<And>x y. x \<otimes> y = y \<otimes> x"
1583 shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
1584 by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
1586 lemmas eq_sym_conv = eq_commute
1589 "(\<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)"
1590 "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not>(P = Q)) = ((P \<and> \<not> Q) \<or> (\<not>P \<and> Q))"
1591 "(\<not> \<not>(P)) = P"
1595 subsection {* Basic ML bindings *}
1598 val FalseE = @{thm FalseE}
1599 val Let_def = @{thm Let_def}
1600 val TrueI = @{thm TrueI}
1601 val allE = @{thm allE}
1602 val allI = @{thm allI}
1603 val all_dupE = @{thm all_dupE}
1604 val arg_cong = @{thm arg_cong}
1605 val box_equals = @{thm box_equals}
1606 val ccontr = @{thm ccontr}
1607 val classical = @{thm classical}
1608 val conjE = @{thm conjE}
1609 val conjI = @{thm conjI}
1610 val conjunct1 = @{thm conjunct1}
1611 val conjunct2 = @{thm conjunct2}
1612 val disjCI = @{thm disjCI}
1613 val disjE = @{thm disjE}
1614 val disjI1 = @{thm disjI1}
1615 val disjI2 = @{thm disjI2}
1616 val eq_reflection = @{thm eq_reflection}
1617 val ex1E = @{thm ex1E}
1618 val ex1I = @{thm ex1I}
1619 val ex1_implies_ex = @{thm ex1_implies_ex}
1620 val exE = @{thm exE}
1621 val exI = @{thm exI}
1622 val excluded_middle = @{thm excluded_middle}
1623 val ext = @{thm ext}
1624 val fun_cong = @{thm fun_cong}
1625 val iffD1 = @{thm iffD1}
1626 val iffD2 = @{thm iffD2}
1627 val iffI = @{thm iffI}
1628 val impE = @{thm impE}
1629 val impI = @{thm impI}
1630 val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
1632 val notE = @{thm notE}
1633 val notI = @{thm notI}
1634 val not_all = @{thm not_all}
1635 val not_ex = @{thm not_ex}
1636 val not_iff = @{thm not_iff}
1637 val not_not = @{thm not_not}
1638 val not_sym = @{thm not_sym}
1639 val refl = @{thm refl}
1640 val rev_mp = @{thm rev_mp}
1641 val spec = @{thm spec}
1642 val ssubst = @{thm ssubst}
1643 val subst = @{thm subst}
1644 val sym = @{thm sym}
1645 val trans = @{thm trans}
1649 subsection {* Code generator basic setup -- see further @{text Code_Setup.thy} *}
1651 setup "CodeName.setup #> CodeTarget.setup #> Nbe.setup"
1653 class eq (attach "op =") = type
1655 code_datatype True False
1658 shows "False \<and> x \<longleftrightarrow> False"
1659 and "True \<and> x \<longleftrightarrow> x"
1660 and "x \<and> False \<longleftrightarrow> False"
1661 and "x \<and> True \<longleftrightarrow> x" by simp_all
1664 shows "False \<or> x \<longleftrightarrow> x"
1665 and "True \<or> x \<longleftrightarrow> True"
1666 and "x \<or> False \<longleftrightarrow> x"
1667 and "x \<or> True \<longleftrightarrow> True" by simp_all
1670 shows "\<not> True \<longleftrightarrow> False"
1671 and "\<not> False \<longleftrightarrow> True" by (rule HOL.simp_thms)+
1673 instance bool :: eq ..
1676 shows "False = P \<longleftrightarrow> \<not> P"
1677 and "True = P \<longleftrightarrow> P"
1678 and "P = False \<longleftrightarrow> \<not> P"
1679 and "P = True \<longleftrightarrow> P" by simp_all
1681 code_datatype Trueprop "prop"
1683 code_datatype "TYPE('a)"
1685 lemma Let_case_cert:
1686 assumes "CASE \<equiv> (\<lambda>x. Let x f)"
1687 shows "CASE x \<equiv> f x"
1688 using assms by simp_all
1691 includes meta_conjunction_syntax
1692 assumes "CASE \<equiv> (\<lambda>b. If b f g)"
1693 shows "(CASE True \<equiv> f) && (CASE False \<equiv> g)"
1694 using assms by simp_all
1697 Code.add_case @{thm Let_case_cert}
1698 #> Code.add_case @{thm If_case_cert}
1699 #> Code.add_undefined @{const_name undefined}
1703 subsection {* Legacy tactics and ML bindings *}
1706 fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
1708 (* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
1710 fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
1711 | wrong_prem (Bound _) = true
1712 | wrong_prem _ = false;
1713 val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
1715 fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
1716 fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
1719 val all_conj_distrib = thm "all_conj_distrib";
1720 val all_simps = thms "all_simps";
1721 val atomize_not = thm "atomize_not";
1722 val case_split = thm "case_split";
1723 val case_split_thm = thm "case_split_thm"
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"})