src/HOL/HOL.thy
 author paulson Tue Sep 23 15:42:01 2003 +0200 (2003-09-23) changeset 14201 7ad7ab89c402 parent 13764 3e180bf68496 child 14208 144f45277d5a permissions -rw-r--r--
some basic new lemmas
1 (*  Title:      HOL/HOL.thy
2     ID:         \$Id\$
3     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
5 *)
7 header {* The basis of Higher-Order Logic *}
9 theory HOL = CPure
10 files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
13 subsection {* Primitive logic *}
15 subsubsection {* Core syntax *}
17 classes type < logic
18 defaultsort type
20 global
22 typedecl bool
24 arities
25   bool :: type
26   fun :: (type, type) type
28 judgment
29   Trueprop      :: "bool => prop"                   ("(_)" 5)
31 consts
32   Not           :: "bool => bool"                   ("~ _"  40)
33   True          :: bool
34   False         :: bool
35   If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
36   arbitrary     :: 'a
38   The           :: "('a => bool) => 'a"
39   All           :: "('a => bool) => bool"           (binder "ALL " 10)
40   Ex            :: "('a => bool) => bool"           (binder "EX " 10)
41   Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
42   Let           :: "['a, 'a => 'b] => 'b"
44   "="           :: "['a, 'a] => bool"               (infixl 50)
45   &             :: "[bool, bool] => bool"           (infixr 35)
46   "|"           :: "[bool, bool] => bool"           (infixr 30)
47   -->           :: "[bool, bool] => bool"           (infixr 25)
49 local
52 subsubsection {* Additional concrete syntax *}
54 nonterminals
55   letbinds  letbind
56   case_syn  cases_syn
58 syntax
59   "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
60   "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
62   "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
63   ""            :: "letbind => letbinds"                 ("_")
64   "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
65   "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
67   "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
68   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
69   ""            :: "case_syn => cases_syn"               ("_")
70   "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
72 translations
73   "x ~= y"                == "~ (x = y)"
74   "THE x. P"              == "The (%x. P)"
75   "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
76   "let x = a in e"        == "Let a (%x. e)"
78 print_translation {*
79 (* To avoid eta-contraction of body: *)
80 [("The", fn [Abs abs] =>
81      let val (x,t) = atomic_abs_tr' abs
82      in Syntax.const "_The" \$ x \$ t end)]
83 *}
85 syntax (output)
86   "="           :: "['a, 'a] => bool"                    (infix 50)
87   "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
89 syntax (xsymbols)
90   Not           :: "bool => bool"                        ("\<not> _"  40)
91   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
92   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
93   "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
94   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
95   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
96   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
97   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
98   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
99 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \\<orelse> _")*)
101 syntax (xsymbols output)
102   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
104 syntax (HTML output)
105   Not           :: "bool => bool"                        ("\<not> _"  40)
107 syntax (HOL)
108   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
109   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
110   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
113 subsubsection {* Axioms and basic definitions *}
115 axioms
116   eq_reflection: "(x=y) ==> (x==y)"
118   refl:         "t = (t::'a)"
119   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
121   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
122     -- {* Extensionality is built into the meta-logic, and this rule expresses *}
123     -- {* a related property.  It is an eta-expanded version of the traditional *}
124     -- {* rule, and similar to the ABS rule of HOL *}
126   the_eq_trivial: "(THE x. x = a) = (a::'a)"
128   impI:         "(P ==> Q) ==> P-->Q"
129   mp:           "[| P-->Q;  P |] ==> Q"
131 defs
132   True_def:     "True      == ((%x::bool. x) = (%x. x))"
133   All_def:      "All(P)    == (P = (%x. True))"
134   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
135   False_def:    "False     == (!P. P)"
136   not_def:      "~ P       == P-->False"
137   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
138   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
139   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
141 axioms
142   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
143   True_or_False:  "(P=True) | (P=False)"
145 defs
146   Let_def:      "Let s f == f(s)"
147   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
149   arbitrary_def:  "False ==> arbitrary == (THE x. False)"
150     -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
151     definition syntactically *}
154 subsubsection {* Generic algebraic operations *}
156 axclass zero < type
157 axclass one < type
158 axclass plus < type
159 axclass minus < type
160 axclass times < type
161 axclass inverse < type
163 global
165 consts
166   "0"           :: "'a::zero"                       ("0")
167   "1"           :: "'a::one"                        ("1")
168   "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
169   -             :: "['a::minus, 'a] => 'a"          (infixl 65)
170   uminus        :: "['a::minus] => 'a"              ("- _"  80)
171   *             :: "['a::times, 'a] => 'a"          (infixl 70)
173 syntax
174   "_index1"  :: index    ("\<^sub>1")
175 translations
176   (index) "\<^sub>1" == "_index 1"
178 local
180 typed_print_translation {*
181   let
182     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
183       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
184       else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
185   in [tr' "0", tr' "1"] end;
186 *} -- {* show types that are presumably too general *}
189 consts
190   abs           :: "'a::minus => 'a"
191   inverse       :: "'a::inverse => 'a"
192   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
194 syntax (xsymbols)
195   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
196 syntax (HTML output)
197   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
199 axclass plus_ac0 < plus, zero
200   commute: "x + y = y + x"
201   assoc:   "(x + y) + z = x + (y + z)"
202   zero:    "0 + x = x"
205 subsection {* Theory and package setup *}
207 subsubsection {* Basic lemmas *}
209 use "HOL_lemmas.ML"
210 theorems case_split = case_split_thm [case_names True False]
213 subsubsection {* Intuitionistic Reasoning *}
215 lemma impE':
216   assumes 1: "P --> Q"
217     and 2: "Q ==> R"
218     and 3: "P --> Q ==> P"
219   shows R
220 proof -
221   from 3 and 1 have P .
222   with 1 have Q by (rule impE)
223   with 2 show R .
224 qed
226 lemma allE':
227   assumes 1: "ALL x. P x"
228     and 2: "P x ==> ALL x. P x ==> Q"
229   shows Q
230 proof -
231   from 1 have "P x" by (rule spec)
232   from this and 1 show Q by (rule 2)
233 qed
235 lemma notE':
236   assumes 1: "~ P"
237     and 2: "~ P ==> P"
238   shows R
239 proof -
240   from 2 and 1 have P .
241   with 1 show R by (rule notE)
242 qed
244 lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
245   and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
246   and [CPure.elim 2] = allE notE' impE'
247   and [CPure.intro] = exI disjI2 disjI1
249 lemmas [trans] = trans
250   and [sym] = sym not_sym
251   and [CPure.elim?] = iffD1 iffD2 impE
254 subsubsection {* Atomizing meta-level connectives *}
256 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
257 proof
258   assume "!!x. P x"
259   show "ALL x. P x" by (rule allI)
260 next
261   assume "ALL x. P x"
262   thus "!!x. P x" by (rule allE)
263 qed
265 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
266 proof
267   assume r: "A ==> B"
268   show "A --> B" by (rule impI) (rule r)
269 next
270   assume "A --> B" and A
271   thus B by (rule mp)
272 qed
274 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
275 proof
276   assume "x == y"
277   show "x = y" by (unfold prems) (rule refl)
278 next
279   assume "x = y"
280   thus "x == y" by (rule eq_reflection)
281 qed
283 lemma atomize_conj [atomize]:
284   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
285 proof
286   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
287   show "A & B" by (rule conjI)
288 next
289   fix C
290   assume "A & B"
291   assume "A ==> B ==> PROP C"
292   thus "PROP C"
293   proof this
294     show A by (rule conjunct1)
295     show B by (rule conjunct2)
296   qed
297 qed
299 lemmas [symmetric, rulify] = atomize_all atomize_imp
302 subsubsection {* Classical Reasoner setup *}
305 setup hypsubst_setup
307 ML_setup {*
308   Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
309 *}
311 setup Classical.setup
312 setup clasetup
314 lemmas [intro?] = ext
315   and [elim?] = ex1_implies_ex
317 use "blastdata.ML"
318 setup Blast.setup
321 subsubsection {* Simplifier setup *}
323 lemma meta_eq_to_obj_eq: "x == y ==> x = y"
324 proof -
325   assume r: "x == y"
326   show "x = y" by (unfold r) (rule refl)
327 qed
329 lemma eta_contract_eq: "(%s. f s) = f" ..
331 lemma simp_thms:
332   shows not_not: "(~ ~ P) = P"
333   and
334     "(P ~= Q) = (P = (~Q))"
335     "(P | ~P) = True"    "(~P | P) = True"
336     "((~P) = (~Q)) = (P=Q)"
337     "(x = x) = True"
338     "(~True) = False"  "(~False) = True"
339     "(~P) ~= P"  "P ~= (~P)"
340     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
341     "(True --> P) = P"  "(False --> P) = True"
342     "(P --> True) = True"  "(P --> P) = True"
343     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
344     "(P & True) = P"  "(True & P) = P"
345     "(P & False) = False"  "(False & P) = False"
346     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
347     "(P & ~P) = False"    "(~P & P) = False"
348     "(P | True) = True"  "(True | P) = True"
349     "(P | False) = P"  "(False | P) = P"
350     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
351     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
352     -- {* needed for the one-point-rule quantifier simplification procs *}
353     -- {* essential for termination!! *} and
354     "!!P. (EX x. x=t & P(x)) = P(t)"
355     "!!P. (EX x. t=x & P(x)) = P(t)"
356     "!!P. (ALL x. x=t --> P(x)) = P(t)"
357     "!!P. (ALL x. t=x --> P(x)) = P(t)"
358   by (blast, blast, blast, blast, blast, rules+)
360 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
361   by rules
363 lemma ex_simps:
364   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
365   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
366   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
367   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
368   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
369   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
370   -- {* Miniscoping: pushing in existential quantifiers. *}
371   by (rules | blast)+
373 lemma all_simps:
374   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
375   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
376   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
377   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
378   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
379   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
380   -- {* Miniscoping: pushing in universal quantifiers. *}
381   by (rules | blast)+
383 lemma disj_absorb: "(A | A) = A"
384   by blast
386 lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
387   by blast
389 lemma conj_absorb: "(A & A) = A"
390   by blast
392 lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
393   by blast
395 lemma eq_ac:
396   shows eq_commute: "(a=b) = (b=a)"
397     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
398     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
399 lemma neq_commute: "(a~=b) = (b~=a)" by rules
401 lemma conj_comms:
402   shows conj_commute: "(P&Q) = (Q&P)"
403     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
404 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
406 lemma disj_comms:
407   shows disj_commute: "(P|Q) = (Q|P)"
408     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
409 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
411 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
412 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
414 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
415 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
417 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
418 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
419 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
421 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
422 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
423 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
425 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
426 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
428 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
429 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
430 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
431 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
432 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
433 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
434   by blast
435 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
437 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
440 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
441   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
442   -- {* cases boil down to the same thing. *}
443   by blast
445 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
446 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
447 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
448 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
450 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
451 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
453 text {*
454   \medskip The @{text "&"} congruence rule: not included by default!
455   May slow rewrite proofs down by as much as 50\% *}
457 lemma conj_cong:
458     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
459   by rules
461 lemma rev_conj_cong:
462     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
463   by rules
465 text {* The @{text "|"} congruence rule: not included by default! *}
467 lemma disj_cong:
468     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
469   by blast
471 lemma eq_sym_conv: "(x = y) = (y = x)"
472   by rules
475 text {* \medskip if-then-else rules *}
477 lemma if_True: "(if True then x else y) = x"
478   by (unfold if_def) blast
480 lemma if_False: "(if False then x else y) = y"
481   by (unfold if_def) blast
483 lemma if_P: "P ==> (if P then x else y) = x"
484   by (unfold if_def) blast
486 lemma if_not_P: "~P ==> (if P then x else y) = y"
487   by (unfold if_def) blast
489 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
490   apply (rule case_split [of Q])
491    apply (subst if_P)
492     prefer 3 apply (subst if_not_P)
493      apply blast+
494   done
496 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
497   apply (subst split_if)
498   apply blast
499   done
501 lemmas if_splits = split_if split_if_asm
503 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
504   by (rule split_if)
506 lemma if_cancel: "(if c then x else x) = x"
507   apply (subst split_if)
508   apply blast
509   done
511 lemma if_eq_cancel: "(if x = y then y else x) = x"
512   apply (subst split_if)
513   apply blast
514   done
516 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
517   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
518   by (rule split_if)
520 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
521   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
522   apply (subst split_if)
523   apply blast
524   done
526 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
527 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
529 subsubsection {* Actual Installation of the Simplifier *}
531 use "simpdata.ML"
532 setup Simplifier.setup
533 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
534 setup Splitter.setup setup Clasimp.setup
536 declare disj_absorb [simp] conj_absorb [simp]
538 lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
539 by blast+
541 theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
542   apply (rule iffI)
543   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
544   apply (fast dest!: theI')
545   apply (fast intro: ext the1_equality [symmetric])
546   apply (erule ex1E)
547   apply (rule allI)
548   apply (rule ex1I)
549   apply (erule spec)
550   apply (rule ccontr)
551   apply (erule_tac x = "%z. if z = x then y else f z" in allE)
552   apply (erule impE)
553   apply (rule allI)
554   apply (rule_tac P = "xa = x" in case_split_thm)
555   apply (drule_tac  x = x in fun_cong)
556   apply simp_all
557   done
559 text{*Needs only HOL-lemmas:*}
560 lemma mk_left_commute:
561   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
562           c: "\<And>x y. f x y = f y x"
563   shows "f x (f y z) = f y (f x z)"
564 by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
567 subsubsection {* Generic cases and induction *}
569 constdefs
570   induct_forall :: "('a => bool) => bool"
571   "induct_forall P == \<forall>x. P x"
572   induct_implies :: "bool => bool => bool"
573   "induct_implies A B == A --> B"
574   induct_equal :: "'a => 'a => bool"
575   "induct_equal x y == x = y"
576   induct_conj :: "bool => bool => bool"
577   "induct_conj A B == A & B"
579 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
580   by (simp only: atomize_all induct_forall_def)
582 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
583   by (simp only: atomize_imp induct_implies_def)
585 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
586   by (simp only: atomize_eq induct_equal_def)
588 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
589     induct_conj (induct_forall A) (induct_forall B)"
590   by (unfold induct_forall_def induct_conj_def) rules
592 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
593     induct_conj (induct_implies C A) (induct_implies C B)"
594   by (unfold induct_implies_def induct_conj_def) rules
596 lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
597 proof
598   assume r: "induct_conj A B ==> PROP C" and A B
599   show "PROP C" by (rule r) (simp! add: induct_conj_def)
600 next
601   assume r: "A ==> B ==> PROP C" and "induct_conj A B"
602   show "PROP C" by (rule r) (simp! add: induct_conj_def)+
603 qed
605 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
606   by (simp add: induct_implies_def)
608 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
609 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
610 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
611 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
613 hide const induct_forall induct_implies induct_equal induct_conj
616 text {* Method setup. *}
618 ML {*
619   structure InductMethod = InductMethodFun
620   (struct
621     val dest_concls = HOLogic.dest_concls;
622     val cases_default = thm "case_split";
623     val local_impI = thm "induct_impliesI";
624     val conjI = thm "conjI";
625     val atomize = thms "induct_atomize";
626     val rulify1 = thms "induct_rulify1";
627     val rulify2 = thms "induct_rulify2";
628     val localize = [Thm.symmetric (thm "induct_implies_def")];
629   end);
630 *}
632 setup InductMethod.setup
635 subsection {* Order signatures and orders *}
637 axclass
638   ord < type
640 syntax
641   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
642   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
644 global
646 consts
647   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
648   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
650 local
652 syntax (xsymbols)
653   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
654   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
657 subsubsection {* Monotonicity *}
659 locale mono =
660   fixes f
661   assumes mono: "A <= B ==> f A <= f B"
663 lemmas monoI [intro?] = mono.intro
664   and monoD [dest?] = mono.mono
666 constdefs
667   min :: "['a::ord, 'a] => 'a"
668   "min a b == (if a <= b then a else b)"
669   max :: "['a::ord, 'a] => 'a"
670   "max a b == (if a <= b then b else a)"
672 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
673   by (simp add: min_def)
675 lemma min_of_mono:
676     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
677   by (simp add: min_def)
679 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
680   by (simp add: max_def)
682 lemma max_of_mono:
683     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
684   by (simp add: max_def)
687 subsubsection "Orders"
689 axclass order < ord
690   order_refl [iff]: "x <= x"
691   order_trans: "x <= y ==> y <= z ==> x <= z"
692   order_antisym: "x <= y ==> y <= x ==> x = y"
693   order_less_le: "(x < y) = (x <= y & x ~= y)"
696 text {* Reflexivity. *}
698 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
699     -- {* This form is useful with the classical reasoner. *}
700   apply (erule ssubst)
701   apply (rule order_refl)
702   done
704 lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
705   by (simp add: order_less_le)
707 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
708     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
709   apply (simp add: order_less_le)
710   apply blast
711   done
713 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
715 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
716   by (simp add: order_less_le)
719 text {* Asymmetry. *}
721 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
722   by (simp add: order_less_le order_antisym)
724 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
725   apply (drule order_less_not_sym)
726   apply (erule contrapos_np)
727   apply simp
728   done
731 text {* Transitivity. *}
733 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
734   apply (simp add: order_less_le)
735   apply (blast intro: order_trans order_antisym)
736   done
738 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
739   apply (simp add: order_less_le)
740   apply (blast intro: order_trans order_antisym)
741   done
743 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
744   apply (simp add: order_less_le)
745   apply (blast intro: order_trans order_antisym)
746   done
749 text {* Useful for simplification, but too risky to include by default. *}
751 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
752   by (blast elim: order_less_asym)
754 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
755   by (blast elim: order_less_asym)
757 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
758   by auto
760 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
761   by auto
764 text {* Other operators. *}
766 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
767   apply (simp add: min_def)
768   apply (blast intro: order_antisym)
769   done
771 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
772   apply (simp add: max_def)
773   apply (blast intro: order_antisym)
774   done
777 subsubsection {* Least value operator *}
779 constdefs
780   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
781   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
782     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
784 lemma LeastI2:
785   "[| P (x::'a::order);
786       !!y. P y ==> x <= y;
787       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
788    ==> Q (Least P)"
789   apply (unfold Least_def)
790   apply (rule theI2)
791     apply (blast intro: order_antisym)+
792   done
794 lemma Least_equality:
795     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
796   apply (simp add: Least_def)
797   apply (rule the_equality)
798   apply (auto intro!: order_antisym)
799   done
802 subsubsection "Linear / total orders"
804 axclass linorder < order
805   linorder_linear: "x <= y | y <= x"
807 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
808   apply (simp add: order_less_le)
809   apply (insert linorder_linear)
810   apply blast
811   done
813 lemma linorder_cases [case_names less equal greater]:
814     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
815   apply (insert linorder_less_linear)
816   apply blast
817   done
819 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
820   apply (simp add: order_less_le)
821   apply (insert linorder_linear)
822   apply (blast intro: order_antisym)
823   done
825 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
826   apply (simp add: order_less_le)
827   apply (insert linorder_linear)
828   apply (blast intro: order_antisym)
829   done
831 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
832   apply (cut_tac x = x and y = y in linorder_less_linear)
833   apply auto
834   done
836 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
837   apply (simp add: linorder_neq_iff)
838   apply blast
839   done
842 subsubsection "Min and max on (linear) orders"
844 lemma min_same [simp]: "min (x::'a::order) x = x"
845   by (simp add: min_def)
847 lemma max_same [simp]: "max (x::'a::order) x = x"
848   by (simp add: max_def)
850 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
851   apply (simp add: max_def)
852   apply (insert linorder_linear)
853   apply (blast intro: order_trans)
854   done
856 lemma le_maxI1: "(x::'a::linorder) <= max x y"
857   by (simp add: le_max_iff_disj)
859 lemma le_maxI2: "(y::'a::linorder) <= max x y"
860     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
861   by (simp add: le_max_iff_disj)
863 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
864   apply (simp add: max_def order_le_less)
865   apply (insert linorder_less_linear)
866   apply (blast intro: order_less_trans)
867   done
869 lemma max_le_iff_conj [simp]:
870     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
871   apply (simp add: max_def)
872   apply (insert linorder_linear)
873   apply (blast intro: order_trans)
874   done
876 lemma max_less_iff_conj [simp]:
877     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
878   apply (simp add: order_le_less max_def)
879   apply (insert linorder_less_linear)
880   apply (blast intro: order_less_trans)
881   done
883 lemma le_min_iff_conj [simp]:
884     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
885     -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
886   apply (simp add: min_def)
887   apply (insert linorder_linear)
888   apply (blast intro: order_trans)
889   done
891 lemma min_less_iff_conj [simp]:
892     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
893   apply (simp add: order_le_less min_def)
894   apply (insert linorder_less_linear)
895   apply (blast intro: order_less_trans)
896   done
898 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
899   apply (simp add: min_def)
900   apply (insert linorder_linear)
901   apply (blast intro: order_trans)
902   done
904 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
905   apply (simp add: min_def order_le_less)
906   apply (insert linorder_less_linear)
907   apply (blast intro: order_less_trans)
908   done
910 lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
912 apply(rule conjI)
913 apply(blast intro:order_trans)
915 apply(blast dest: order_less_trans order_le_less_trans)
916 done
918 lemma max_commute: "!!x::'a::linorder. max x y = max y x"
920 apply(rule conjI)
921 apply(blast intro:order_antisym)
923 apply(blast dest: order_less_trans)
924 done
926 lemmas max_ac = max_assoc max_commute
927                 mk_left_commute[of max,OF max_assoc max_commute]
929 lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
931 apply(rule conjI)
932 apply(blast intro:order_trans)
934 apply(blast dest: order_less_trans order_le_less_trans)
935 done
937 lemma min_commute: "!!x::'a::linorder. min x y = min y x"
939 apply(rule conjI)
940 apply(blast intro:order_antisym)
942 apply(blast dest: order_less_trans)
943 done
945 lemmas min_ac = min_assoc min_commute
946                 mk_left_commute[of min,OF min_assoc min_commute]
948 lemma split_min:
949     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
950   by (simp add: min_def)
952 lemma split_max:
953     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
954   by (simp add: max_def)
957 subsubsection "Bounded quantifiers"
959 syntax
960   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
961   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
962   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
963   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
965 syntax (xsymbols)
966   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
967   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
968   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
969   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
971 syntax (HOL)
972   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
973   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
974   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
975   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
977 translations
978  "ALL x<y. P"   =>  "ALL x. x < y --> P"
979  "EX x<y. P"    =>  "EX x. x < y  & P"
980  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
981  "EX x<=y. P"   =>  "EX x. x <= y & P"
983 end