src/HOL/HOL.thy
 author wenzelm Mon Aug 05 21:16:36 2002 +0200 (2002-08-05) changeset 13456 42601eb7553f parent 13438 527811f00c56 child 13550 5a176b8dda84 permissions -rw-r--r--
special syntax for index "1" (plain numeral hidden by "1" symbol in HOL);
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 syntax (output)
79   "="           :: "['a, 'a] => bool"                    (infix 50)
80   "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
82 syntax (xsymbols)
83   Not           :: "bool => bool"                        ("\<not> _"  40)
84   "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
85   "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
86   "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
87   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
88   "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
89   "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
90   "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
91   "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
92 (*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \\<orelse> _")*)
94 syntax (xsymbols output)
95   "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
97 syntax (HTML output)
98   Not           :: "bool => bool"                        ("\<not> _"  40)
100 syntax (HOL)
101   "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
102   "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
103   "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
106 subsubsection {* Axioms and basic definitions *}
108 axioms
109   eq_reflection: "(x=y) ==> (x==y)"
111   refl:         "t = (t::'a)"
112   subst:        "[| s = t; P(s) |] ==> P(t::'a)"
114   ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
115     -- {* Extensionality is built into the meta-logic, and this rule expresses *}
116     -- {* a related property.  It is an eta-expanded version of the traditional *}
117     -- {* rule, and similar to the ABS rule of HOL *}
119   the_eq_trivial: "(THE x. x = a) = (a::'a)"
121   impI:         "(P ==> Q) ==> P-->Q"
122   mp:           "[| P-->Q;  P |] ==> Q"
124 defs
125   True_def:     "True      == ((%x::bool. x) = (%x. x))"
126   All_def:      "All(P)    == (P = (%x. True))"
127   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
128   False_def:    "False     == (!P. P)"
129   not_def:      "~ P       == P-->False"
130   and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
131   or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
132   Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
134 axioms
135   iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
136   True_or_False:  "(P=True) | (P=False)"
138 defs
139   Let_def:      "Let s f == f(s)"
140   if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
142   arbitrary_def:  "False ==> arbitrary == (THE x. False)"
143     -- {* @{term arbitrary} is completely unspecified, but is made to appear as a
144     definition syntactically *}
147 subsubsection {* Generic algebraic operations *}
149 axclass zero < type
150 axclass one < type
151 axclass plus < type
152 axclass minus < type
153 axclass times < type
154 axclass inverse < type
156 global
158 consts
159   "0"           :: "'a::zero"                       ("0")
160   "1"           :: "'a::one"                        ("1")
161   "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
162   -             :: "['a::minus, 'a] => 'a"          (infixl 65)
163   uminus        :: "['a::minus] => 'a"              ("- _"  80)
164   *             :: "['a::times, 'a] => 'a"          (infixl 70)
166 syntax
167   "_index1"  :: index    ("\<^sub>1")
168 translations
169   (index) "\<^sub>1" == "_index 1"
171 local
173 typed_print_translation {*
174   let
175     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
176       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
177       else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
178   in [tr' "0", tr' "1"] end;
179 *} -- {* show types that are presumably too general *}
182 consts
183   abs           :: "'a::minus => 'a"
184   inverse       :: "'a::inverse => 'a"
185   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
187 syntax (xsymbols)
188   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
189 syntax (HTML output)
190   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
192 axclass plus_ac0 < plus, zero
193   commute: "x + y = y + x"
194   assoc:   "(x + y) + z = x + (y + z)"
195   zero:    "0 + x = x"
198 subsection {* Theory and package setup *}
200 subsubsection {* Basic lemmas *}
202 use "HOL_lemmas.ML"
203 theorems case_split = case_split_thm [case_names True False]
206 subsubsection {* Intuitionistic Reasoning *}
208 lemma impE':
209   assumes 1: "P --> Q"
210     and 2: "Q ==> R"
211     and 3: "P --> Q ==> P"
212   shows R
213 proof -
214   from 3 and 1 have P .
215   with 1 have Q by (rule impE)
216   with 2 show R .
217 qed
219 lemma allE':
220   assumes 1: "ALL x. P x"
221     and 2: "P x ==> ALL x. P x ==> Q"
222   shows Q
223 proof -
224   from 1 have "P x" by (rule spec)
225   from this and 1 show Q by (rule 2)
226 qed
228 lemma notE':
229   assumes 1: "~ P"
230     and 2: "~ P ==> P"
231   shows R
232 proof -
233   from 2 and 1 have P .
234   with 1 show R by (rule notE)
235 qed
237 lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
238   and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
239   and [CPure.elim 2] = allE notE' impE'
240   and [CPure.intro] = exI disjI2 disjI1
242 lemmas [trans] = trans
243   and [sym] = sym not_sym
244   and [CPure.elim?] = iffD1 iffD2 impE
247 subsubsection {* Atomizing meta-level connectives *}
249 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
250 proof
251   assume "!!x. P x"
252   show "ALL x. P x" by (rule allI)
253 next
254   assume "ALL x. P x"
255   thus "!!x. P x" by (rule allE)
256 qed
258 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
259 proof
260   assume r: "A ==> B"
261   show "A --> B" by (rule impI) (rule r)
262 next
263   assume "A --> B" and A
264   thus B by (rule mp)
265 qed
267 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
268 proof
269   assume "x == y"
270   show "x = y" by (unfold prems) (rule refl)
271 next
272   assume "x = y"
273   thus "x == y" by (rule eq_reflection)
274 qed
276 lemma atomize_conj [atomize]:
277   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
278 proof
279   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
280   show "A & B" by (rule conjI)
281 next
282   fix C
283   assume "A & B"
284   assume "A ==> B ==> PROP C"
285   thus "PROP C"
286   proof this
287     show A by (rule conjunct1)
288     show B by (rule conjunct2)
289   qed
290 qed
292 lemmas [symmetric, rulify] = atomize_all atomize_imp
295 subsubsection {* Classical Reasoner setup *}
298 setup hypsubst_setup
300 ML_setup {*
301   Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
302 *}
304 setup Classical.setup
305 setup clasetup
307 lemmas [intro?] = ext
308   and [elim?] = ex1_implies_ex
310 use "blastdata.ML"
311 setup Blast.setup
314 subsubsection {* Simplifier setup *}
316 lemma meta_eq_to_obj_eq: "x == y ==> x = y"
317 proof -
318   assume r: "x == y"
319   show "x = y" by (unfold r) (rule refl)
320 qed
322 lemma eta_contract_eq: "(%s. f s) = f" ..
324 lemma simp_thms:
325   shows not_not: "(~ ~ P) = P"
326   and
327     "(P ~= Q) = (P = (~Q))"
328     "(P | ~P) = True"    "(~P | P) = True"
329     "((~P) = (~Q)) = (P=Q)"
330     "(x = x) = True"
331     "(~True) = False"  "(~False) = True"
332     "(~P) ~= P"  "P ~= (~P)"
333     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
334     "(True --> P) = P"  "(False --> P) = True"
335     "(P --> True) = True"  "(P --> P) = True"
336     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
337     "(P & True) = P"  "(True & P) = P"
338     "(P & False) = False"  "(False & P) = False"
339     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
340     "(P & ~P) = False"    "(~P & P) = False"
341     "(P | True) = True"  "(True | P) = True"
342     "(P | False) = P"  "(False | P) = P"
343     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
344     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
345     -- {* needed for the one-point-rule quantifier simplification procs *}
346     -- {* essential for termination!! *} and
347     "!!P. (EX x. x=t & P(x)) = P(t)"
348     "!!P. (EX x. t=x & P(x)) = P(t)"
349     "!!P. (ALL x. x=t --> P(x)) = P(t)"
350     "!!P. (ALL x. t=x --> P(x)) = P(t)"
351   by (blast, blast, blast, blast, blast, rules+)
353 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
354   by rules
356 lemma ex_simps:
357   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
358   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
359   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
360   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
361   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
362   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
363   -- {* Miniscoping: pushing in existential quantifiers. *}
364   by (rules | blast)+
366 lemma all_simps:
367   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
368   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
369   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
370   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
371   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
372   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
373   -- {* Miniscoping: pushing in universal quantifiers. *}
374   by (rules | blast)+
376 lemma eq_ac:
377   shows eq_commute: "(a=b) = (b=a)"
378     and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
379     and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
380 lemma neq_commute: "(a~=b) = (b~=a)" by rules
382 lemma conj_comms:
383   shows conj_commute: "(P&Q) = (Q&P)"
384     and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
385 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
387 lemma disj_comms:
388   shows disj_commute: "(P|Q) = (Q|P)"
389     and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
390 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
392 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
393 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
395 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
396 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
398 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
399 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
400 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
402 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
403 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
404 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
406 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
407 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
409 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
410 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
411 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
412 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
413 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
414 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
415   by blast
416 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
418 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
421 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
422   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
423   -- {* cases boil down to the same thing. *}
424   by blast
426 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
427 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
428 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
429 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
431 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
432 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
434 text {*
435   \medskip The @{text "&"} congruence rule: not included by default!
436   May slow rewrite proofs down by as much as 50\% *}
438 lemma conj_cong:
439     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
440   by rules
442 lemma rev_conj_cong:
443     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
444   by rules
446 text {* The @{text "|"} congruence rule: not included by default! *}
448 lemma disj_cong:
449     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
450   by blast
452 lemma eq_sym_conv: "(x = y) = (y = x)"
453   by rules
456 text {* \medskip if-then-else rules *}
458 lemma if_True: "(if True then x else y) = x"
459   by (unfold if_def) blast
461 lemma if_False: "(if False then x else y) = y"
462   by (unfold if_def) blast
464 lemma if_P: "P ==> (if P then x else y) = x"
465   by (unfold if_def) blast
467 lemma if_not_P: "~P ==> (if P then x else y) = y"
468   by (unfold if_def) blast
470 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
471   apply (rule case_split [of Q])
472    apply (subst if_P)
473     prefer 3 apply (subst if_not_P)
474      apply blast+
475   done
477 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
478   apply (subst split_if)
479   apply blast
480   done
482 lemmas if_splits = split_if split_if_asm
484 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
485   by (rule split_if)
487 lemma if_cancel: "(if c then x else x) = x"
488   apply (subst split_if)
489   apply blast
490   done
492 lemma if_eq_cancel: "(if x = y then y else x) = x"
493   apply (subst split_if)
494   apply blast
495   done
497 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
498   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
499   by (rule split_if)
501 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
502   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
503   apply (subst split_if)
504   apply blast
505   done
507 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
508 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
510 use "simpdata.ML"
511 setup Simplifier.setup
512 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
513 setup Splitter.setup setup Clasimp.setup
515 text{*Needs only HOL-lemmas:*}
516 lemma mk_left_commute:
517   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
518           c: "\<And>x y. f x y = f y x"
519   shows "f x (f y z) = f y (f x z)"
520 by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
523 subsubsection {* Generic cases and induction *}
525 constdefs
526   induct_forall :: "('a => bool) => bool"
527   "induct_forall P == \<forall>x. P x"
528   induct_implies :: "bool => bool => bool"
529   "induct_implies A B == A --> B"
530   induct_equal :: "'a => 'a => bool"
531   "induct_equal x y == x = y"
532   induct_conj :: "bool => bool => bool"
533   "induct_conj A B == A & B"
535 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
536   by (simp only: atomize_all induct_forall_def)
538 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
539   by (simp only: atomize_imp induct_implies_def)
541 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
542   by (simp only: atomize_eq induct_equal_def)
544 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
545     induct_conj (induct_forall A) (induct_forall B)"
546   by (unfold induct_forall_def induct_conj_def) rules
548 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
549     induct_conj (induct_implies C A) (induct_implies C B)"
550   by (unfold induct_implies_def induct_conj_def) rules
552 lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
553   by (simp only: atomize_imp atomize_eq induct_conj_def) (rules intro: equal_intr_rule)
555 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
556   by (simp add: induct_implies_def)
558 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
559 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
560 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
561 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
563 hide const induct_forall induct_implies induct_equal induct_conj
566 text {* Method setup. *}
568 ML {*
569   structure InductMethod = InductMethodFun
570   (struct
571     val dest_concls = HOLogic.dest_concls;
572     val cases_default = thm "case_split";
573     val local_impI = thm "induct_impliesI";
574     val conjI = thm "conjI";
575     val atomize = thms "induct_atomize";
576     val rulify1 = thms "induct_rulify1";
577     val rulify2 = thms "induct_rulify2";
578     val localize = [Thm.symmetric (thm "induct_implies_def")];
579   end);
580 *}
582 setup InductMethod.setup
585 subsection {* Order signatures and orders *}
587 axclass
588   ord < type
590 syntax
591   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
592   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
594 global
596 consts
597   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
598   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
600 local
602 syntax (xsymbols)
603   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
604   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
606 (*Tell blast about overloading of < and <= to reduce the risk of
607   its applying a rule for the wrong type*)
608 ML {*
609 Blast.overloaded ("op <" , domain_type);
610 Blast.overloaded ("op <=", domain_type);
611 *}
614 subsubsection {* Monotonicity *}
616 locale mono =
617   fixes f
618   assumes mono: "A <= B ==> f A <= f B"
620 lemmas monoI [intro?] = mono.intro
621   and monoD [dest?] = mono.mono
623 constdefs
624   min :: "['a::ord, 'a] => 'a"
625   "min a b == (if a <= b then a else b)"
626   max :: "['a::ord, 'a] => 'a"
627   "max a b == (if a <= b then b else a)"
629 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
630   by (simp add: min_def)
632 lemma min_of_mono:
633     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
634   by (simp add: min_def)
636 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
637   by (simp add: max_def)
639 lemma max_of_mono:
640     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
641   by (simp add: max_def)
644 subsubsection "Orders"
646 axclass order < ord
647   order_refl [iff]: "x <= x"
648   order_trans: "x <= y ==> y <= z ==> x <= z"
649   order_antisym: "x <= y ==> y <= x ==> x = y"
650   order_less_le: "(x < y) = (x <= y & x ~= y)"
653 text {* Reflexivity. *}
655 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
656     -- {* This form is useful with the classical reasoner. *}
657   apply (erule ssubst)
658   apply (rule order_refl)
659   done
661 lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
662   by (simp add: order_less_le)
664 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
665     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
666   apply (simp add: order_less_le)
667   apply blast
668   done
670 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
672 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
673   by (simp add: order_less_le)
676 text {* Asymmetry. *}
678 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
679   by (simp add: order_less_le order_antisym)
681 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
682   apply (drule order_less_not_sym)
683   apply (erule contrapos_np)
684   apply simp
685   done
688 text {* Transitivity. *}
690 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
691   apply (simp add: order_less_le)
692   apply (blast intro: order_trans order_antisym)
693   done
695 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
696   apply (simp add: order_less_le)
697   apply (blast intro: order_trans order_antisym)
698   done
700 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
701   apply (simp add: order_less_le)
702   apply (blast intro: order_trans order_antisym)
703   done
706 text {* Useful for simplification, but too risky to include by default. *}
708 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
709   by (blast elim: order_less_asym)
711 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
712   by (blast elim: order_less_asym)
714 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
715   by auto
717 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
718   by auto
721 text {* Other operators. *}
723 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
724   apply (simp add: min_def)
725   apply (blast intro: order_antisym)
726   done
728 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
729   apply (simp add: max_def)
730   apply (blast intro: order_antisym)
731   done
734 subsubsection {* Least value operator *}
736 constdefs
737   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
738   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
739     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
741 lemma LeastI2:
742   "[| P (x::'a::order);
743       !!y. P y ==> x <= y;
744       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
745    ==> Q (Least P)"
746   apply (unfold Least_def)
747   apply (rule theI2)
748     apply (blast intro: order_antisym)+
749   done
751 lemma Least_equality:
752     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
753   apply (simp add: Least_def)
754   apply (rule the_equality)
755   apply (auto intro!: order_antisym)
756   done
759 subsubsection "Linear / total orders"
761 axclass linorder < order
762   linorder_linear: "x <= y | y <= x"
764 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
765   apply (simp add: order_less_le)
766   apply (insert linorder_linear)
767   apply blast
768   done
770 lemma linorder_cases [case_names less equal greater]:
771     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
772   apply (insert linorder_less_linear)
773   apply blast
774   done
776 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
777   apply (simp add: order_less_le)
778   apply (insert linorder_linear)
779   apply (blast intro: order_antisym)
780   done
782 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
783   apply (simp add: order_less_le)
784   apply (insert linorder_linear)
785   apply (blast intro: order_antisym)
786   done
788 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
789   apply (cut_tac x = x and y = y in linorder_less_linear)
790   apply auto
791   done
793 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
794   apply (simp add: linorder_neq_iff)
795   apply blast
796   done
799 subsubsection "Min and max on (linear) orders"
801 lemma min_same [simp]: "min (x::'a::order) x = x"
802   by (simp add: min_def)
804 lemma max_same [simp]: "max (x::'a::order) x = x"
805   by (simp add: max_def)
807 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
808   apply (simp add: max_def)
809   apply (insert linorder_linear)
810   apply (blast intro: order_trans)
811   done
813 lemma le_maxI1: "(x::'a::linorder) <= max x y"
814   by (simp add: le_max_iff_disj)
816 lemma le_maxI2: "(y::'a::linorder) <= max x y"
817     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
818   by (simp add: le_max_iff_disj)
820 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
821   apply (simp add: max_def order_le_less)
822   apply (insert linorder_less_linear)
823   apply (blast intro: order_less_trans)
824   done
826 lemma max_le_iff_conj [simp]:
827     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
828   apply (simp add: max_def)
829   apply (insert linorder_linear)
830   apply (blast intro: order_trans)
831   done
833 lemma max_less_iff_conj [simp]:
834     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
835   apply (simp add: order_le_less max_def)
836   apply (insert linorder_less_linear)
837   apply (blast intro: order_less_trans)
838   done
840 lemma le_min_iff_conj [simp]:
841     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
842     -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
843   apply (simp add: min_def)
844   apply (insert linorder_linear)
845   apply (blast intro: order_trans)
846   done
848 lemma min_less_iff_conj [simp]:
849     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
850   apply (simp add: order_le_less min_def)
851   apply (insert linorder_less_linear)
852   apply (blast intro: order_less_trans)
853   done
855 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
856   apply (simp add: min_def)
857   apply (insert linorder_linear)
858   apply (blast intro: order_trans)
859   done
861 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
862   apply (simp add: min_def order_le_less)
863   apply (insert linorder_less_linear)
864   apply (blast intro: order_less_trans)
865   done
867 declare order_less_irrefl [iff]
869 lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
871 apply(rule conjI)
872 apply(blast intro:order_trans)
874 apply(blast dest: order_less_trans order_le_less_trans)
875 done
877 lemma max_commute: "!!x::'a::linorder. max x y = max y x"
879 apply(rule conjI)
880 apply(blast intro:order_antisym)
882 apply(blast dest: order_less_trans)
883 done
885 lemmas max_ac = max_assoc max_commute
886                 mk_left_commute[of max,OF max_assoc max_commute]
888 lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
890 apply(rule conjI)
891 apply(blast intro:order_trans)
893 apply(blast dest: order_less_trans order_le_less_trans)
894 done
896 lemma min_commute: "!!x::'a::linorder. min x y = min y x"
898 apply(rule conjI)
899 apply(blast intro:order_antisym)
901 apply(blast dest: order_less_trans)
902 done
904 lemmas min_ac = min_assoc min_commute
905                 mk_left_commute[of min,OF min_assoc min_commute]
907 declare order_less_irrefl [iff del]
908 declare order_less_irrefl [simp]
910 lemma split_min:
911     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
912   by (simp add: min_def)
914 lemma split_max:
915     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
916   by (simp add: max_def)
919 subsubsection "Bounded quantifiers"
921 syntax
922   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
923   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
924   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
925   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
927 syntax (xsymbols)
928   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
929   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
930   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
931   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
933 syntax (HOL)
934   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
935   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
936   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
937   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
939 translations
940  "ALL x<y. P"   =>  "ALL x. x < y --> P"
941  "EX x<y. P"    =>  "EX x. x < y  & P"
942  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
943  "EX x<=y. P"   =>  "EX x. x <= y & P"
945 end