src/HOL/HOL.thy
 author wenzelm Wed Dec 05 03:19:14 2001 +0100 (2001-12-05) changeset 12386 9c38ec9eca1c parent 12354 5f5ee25513c5 child 12436 a2df07fefed7 permissions -rw-r--r--
tuned declarations (rules, sym, etc.);
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   ~=            :: "['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   "~="          :: "['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   "op ~="       :: "['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   "op ~="       :: "['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 local
168 typed_print_translation {*
169   let
170     fun tr' c = (c, fn show_sorts => fn T => fn ts =>
171       if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
172       else Syntax.const Syntax.constrainC \$ Syntax.const c \$ Syntax.term_of_typ show_sorts T);
173   in [tr' "0", tr' "1"] end;
174 *} -- {* show types that are presumably too general *}
177 consts
178   abs           :: "'a::minus => 'a"
179   inverse       :: "'a::inverse => 'a"
180   divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
182 syntax (xsymbols)
183   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
184 syntax (HTML output)
185   abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
187 axclass plus_ac0 < plus, zero
188   commute: "x + y = y + x"
189   assoc:   "(x + y) + z = x + (y + z)"
190   zero:    "0 + x = x"
193 subsection {* Theory and package setup *}
195 subsubsection {* Basic lemmas *}
197 use "HOL_lemmas.ML"
198 theorems case_split = case_split_thm [case_names True False]
201 subsubsection {* Intuitionistic Reasoning *}
203 lemma impE':
204   (assumes 1: "P --> Q" and 2: "Q ==> R" and 3: "P --> Q ==> P") R
205 proof -
206   from 3 and 1 have P .
207   with 1 have Q by (rule impE)
208   with 2 show R .
209 qed
211 lemma allE':
212   (assumes 1: "ALL x. P x" and 2: "P x ==> ALL x. P x ==> Q") Q
213 proof -
214   from 1 have "P x" by (rule spec)
215   from this and 1 show Q by (rule 2)
216 qed
218 lemma notE': (assumes 1: "~ P" and 2: "~ P ==> P") R
219 proof -
220   from 2 and 1 have P .
221   with 1 show R by (rule notE)
222 qed
224 lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
225   and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
226   and [CPure.elim 2] = allE notE' impE'
227   and [CPure.intro] = exI disjI2 disjI1
229 lemmas [trans] = trans
230   and [sym] = sym not_sym
231   and [CPure.elim?] = iffD1 iffD2 impE
234 subsubsection {* Atomizing meta-level connectives *}
236 lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
237 proof
238   assume "!!x. P x"
239   show "ALL x. P x" by (rule allI)
240 next
241   assume "ALL x. P x"
242   thus "!!x. P x" by (rule allE)
243 qed
245 lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
246 proof
247   assume r: "A ==> B"
248   show "A --> B" by (rule impI) (rule r)
249 next
250   assume "A --> B" and A
251   thus B by (rule mp)
252 qed
254 lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
255 proof
256   assume "x == y"
257   show "x = y" by (unfold prems) (rule refl)
258 next
259   assume "x = y"
260   thus "x == y" by (rule eq_reflection)
261 qed
263 lemma atomize_conj [atomize]:
264   "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
265 proof
266   assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
267   show "A & B" by (rule conjI)
268 next
269   fix C
270   assume "A & B"
271   assume "A ==> B ==> PROP C"
272   thus "PROP C"
273   proof this
274     show A by (rule conjunct1)
275     show B by (rule conjunct2)
276   qed
277 qed
279 lemmas [symmetric, rulify] = atomize_all atomize_imp
282 subsubsection {* Classical Reasoner setup *}
285 setup hypsubst_setup
287 ML_setup {*
288   Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
289 *}
291 setup Classical.setup
292 setup clasetup
294 lemmas [intro?] = ext
295   and [elim?] = ex1_implies_ex
297 use "blastdata.ML"
298 setup Blast.setup
301 subsubsection {* Simplifier setup *}
303 lemma meta_eq_to_obj_eq: "x == y ==> x = y"
304 proof -
305   assume r: "x == y"
306   show "x = y" by (unfold r) (rule refl)
307 qed
309 lemma eta_contract_eq: "(%s. f s) = f" ..
311 lemma simp_thms:
312   (not_not: "(~ ~ P) = P" and
313     "(x = x) = True"
314     "(~True) = False"  "(~False) = True"
315     "(~P) ~= P"  "P ~= (~P)"  "(P ~= Q) = (P = (~Q))"
316     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
317     "(True --> P) = P"  "(False --> P) = True"
318     "(P --> True) = True"  "(P --> P) = True"
319     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
320     "(P & True) = P"  "(True & P) = P"
321     "(P & False) = False"  "(False & P) = False"
322     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
323     "(P & ~P) = False"    "(~P & P) = False"
324     "(P | True) = True"  "(True | P) = True"
325     "(P | False) = P"  "(False | P) = P"
326     "(P | P) = P"  "(P | (P | Q)) = (P | Q)"
327     "(P | ~P) = True"    "(~P | P) = True"
328     "((~P) = (~Q)) = (P=Q)" and
329     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
330     -- {* needed for the one-point-rule quantifier simplification procs *}
331     -- {* essential for termination!! *} and
332     "!!P. (EX x. x=t & P(x)) = P(t)"
333     "!!P. (EX x. t=x & P(x)) = P(t)"
334     "!!P. (ALL x. x=t --> P(x)) = P(t)"
335     "!!P. (ALL x. t=x --> P(x)) = P(t)")
336   by blast+
338 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
339   by rules
341 lemma ex_simps:
342   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
343   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
344   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
345   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
346   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
347   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
348   -- {* Miniscoping: pushing in existential quantifiers. *}
349   by blast+
351 lemma all_simps:
352   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
353   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
354   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
355   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
356   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
357   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
358   -- {* Miniscoping: pushing in universal quantifiers. *}
359   by blast+
361 lemma eq_ac:
362  (eq_commute: "(a=b) = (b=a)" and
363   eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" and
364   eq_assoc: "((P=Q)=R) = (P=(Q=R))") by blast+
365 lemma neq_commute: "(a~=b) = (b~=a)" by blast
367 lemma conj_comms:
368  (conj_commute: "(P&Q) = (Q&P)" and
369   conj_left_commute: "(P&(Q&R)) = (Q&(P&R))") by blast+
370 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by blast
372 lemma disj_comms:
373  (disj_commute: "(P|Q) = (Q|P)" and
374   disj_left_commute: "(P|(Q|R)) = (Q|(P|R))") by blast+
375 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by blast
377 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by blast
378 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by blast
380 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by blast
381 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by blast
383 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by blast
384 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by blast
385 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by blast
387 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
388 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
389 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
391 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
392 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
394 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by blast
395 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
396 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
397 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
398 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
399 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
400   by blast
401 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
403 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by blast
406 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
407   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
408   -- {* cases boil down to the same thing. *}
409   by blast
411 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
412 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
413 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by blast
414 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by blast
416 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by blast
417 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by blast
419 text {*
420   \medskip The @{text "&"} congruence rule: not included by default!
421   May slow rewrite proofs down by as much as 50\% *}
423 lemma conj_cong:
424     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
425   by rules
427 lemma rev_conj_cong:
428     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
429   by rules
431 text {* The @{text "|"} congruence rule: not included by default! *}
433 lemma disj_cong:
434     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
435   by blast
437 lemma eq_sym_conv: "(x = y) = (y = x)"
438   by rules
441 text {* \medskip if-then-else rules *}
443 lemma if_True: "(if True then x else y) = x"
444   by (unfold if_def) blast
446 lemma if_False: "(if False then x else y) = y"
447   by (unfold if_def) blast
449 lemma if_P: "P ==> (if P then x else y) = x"
450   by (unfold if_def) blast
452 lemma if_not_P: "~P ==> (if P then x else y) = y"
453   by (unfold if_def) blast
455 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
456   apply (rule case_split [of Q])
457    apply (subst if_P)
458     prefer 3 apply (subst if_not_P)
459      apply blast+
460   done
462 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
463   apply (subst split_if)
464   apply blast
465   done
467 lemmas if_splits = split_if split_if_asm
469 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
470   by (rule split_if)
472 lemma if_cancel: "(if c then x else x) = x"
473   apply (subst split_if)
474   apply blast
475   done
477 lemma if_eq_cancel: "(if x = y then y else x) = x"
478   apply (subst split_if)
479   apply blast
480   done
482 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
483   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
484   by (rule split_if)
486 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
487   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
488   apply (subst split_if)
489   apply blast
490   done
492 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) blast
493 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) blast
495 use "simpdata.ML"
496 setup Simplifier.setup
497 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
498 setup Splitter.setup setup Clasimp.setup
501 subsubsection {* Generic cases and induction *}
503 constdefs
504   induct_forall :: "('a => bool) => bool"
505   "induct_forall P == \<forall>x. P x"
506   induct_implies :: "bool => bool => bool"
507   "induct_implies A B == A --> B"
508   induct_equal :: "'a => 'a => bool"
509   "induct_equal x y == x = y"
510   induct_conj :: "bool => bool => bool"
511   "induct_conj A B == A & B"
513 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
514   by (simp only: atomize_all induct_forall_def)
516 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
517   by (simp only: atomize_imp induct_implies_def)
519 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
520   by (simp only: atomize_eq induct_equal_def)
522 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
523     induct_conj (induct_forall A) (induct_forall B)"
524   by (unfold induct_forall_def induct_conj_def) rules
526 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
527     induct_conj (induct_implies C A) (induct_implies C B)"
528   by (unfold induct_implies_def induct_conj_def) rules
530 lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
531   by (simp only: atomize_imp atomize_eq induct_conj_def) (rules intro: equal_intr_rule)
533 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
534   by (simp add: induct_implies_def)
536 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
537 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
538 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
539 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
541 hide const induct_forall induct_implies induct_equal induct_conj
544 text {* Method setup. *}
546 ML {*
547   structure InductMethod = InductMethodFun
548   (struct
549     val dest_concls = HOLogic.dest_concls;
550     val cases_default = thm "case_split";
551     val local_impI = thm "induct_impliesI";
552     val conjI = thm "conjI";
553     val atomize = thms "induct_atomize";
554     val rulify1 = thms "induct_rulify1";
555     val rulify2 = thms "induct_rulify2";
556     val localize = [Thm.symmetric (thm "induct_implies_def")];
557   end);
558 *}
560 setup InductMethod.setup
563 subsection {* Order signatures and orders *}
565 axclass
566   ord < type
568 syntax
569   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
570   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
572 global
574 consts
575   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
576   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
578 local
580 syntax (xsymbols)
581   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
582   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
584 (*Tell blast about overloading of < and <= to reduce the risk of
585   its applying a rule for the wrong type*)
586 ML {*
587 Blast.overloaded ("op <" , domain_type);
588 Blast.overloaded ("op <=", domain_type);
589 *}
592 subsubsection {* Monotonicity *}
594 constdefs
595   mono :: "['a::ord => 'b::ord] => bool"
596   "mono f == ALL A B. A <= B --> f A <= f B"
598 lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
599   by (unfold mono_def) rules
601 lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
602   by (unfold mono_def) rules
604 constdefs
605   min :: "['a::ord, 'a] => 'a"
606   "min a b == (if a <= b then a else b)"
607   max :: "['a::ord, 'a] => 'a"
608   "max a b == (if a <= b then b else a)"
610 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
611   by (simp add: min_def)
613 lemma min_of_mono:
614     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
615   by (simp add: min_def)
617 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
618   by (simp add: max_def)
620 lemma max_of_mono:
621     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
622   by (simp add: max_def)
625 subsubsection "Orders"
627 axclass order < ord
628   order_refl [iff]: "x <= x"
629   order_trans: "x <= y ==> y <= z ==> x <= z"
630   order_antisym: "x <= y ==> y <= x ==> x = y"
631   order_less_le: "(x < y) = (x <= y & x ~= y)"
634 text {* Reflexivity. *}
636 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
637     -- {* This form is useful with the classical reasoner. *}
638   apply (erule ssubst)
639   apply (rule order_refl)
640   done
642 lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
643   by (simp add: order_less_le)
645 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
646     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
647   apply (simp add: order_less_le)
648   apply blast
649   done
651 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
653 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
654   by (simp add: order_less_le)
657 text {* Asymmetry. *}
659 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
660   by (simp add: order_less_le order_antisym)
662 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
663   apply (drule order_less_not_sym)
664   apply (erule contrapos_np)
665   apply simp
666   done
669 text {* Transitivity. *}
671 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
672   apply (simp add: order_less_le)
673   apply (blast intro: order_trans order_antisym)
674   done
676 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
677   apply (simp add: order_less_le)
678   apply (blast intro: order_trans order_antisym)
679   done
681 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
682   apply (simp add: order_less_le)
683   apply (blast intro: order_trans order_antisym)
684   done
687 text {* Useful for simplification, but too risky to include by default. *}
689 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
690   by (blast elim: order_less_asym)
692 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
693   by (blast elim: order_less_asym)
695 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
696   by auto
698 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
699   by auto
702 text {* Other operators. *}
704 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
705   apply (simp add: min_def)
706   apply (blast intro: order_antisym)
707   done
709 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
710   apply (simp add: max_def)
711   apply (blast intro: order_antisym)
712   done
715 subsubsection {* Least value operator *}
717 constdefs
718   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
719   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
720     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
722 lemma LeastI2:
723   "[| P (x::'a::order);
724       !!y. P y ==> x <= y;
725       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
726    ==> Q (Least P)"
727   apply (unfold Least_def)
728   apply (rule theI2)
729     apply (blast intro: order_antisym)+
730   done
732 lemma Least_equality:
733     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
734   apply (simp add: Least_def)
735   apply (rule the_equality)
736   apply (auto intro!: order_antisym)
737   done
740 subsubsection "Linear / total orders"
742 axclass linorder < order
743   linorder_linear: "x <= y | y <= x"
745 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
746   apply (simp add: order_less_le)
747   apply (insert linorder_linear)
748   apply blast
749   done
751 lemma linorder_cases [case_names less equal greater]:
752     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
753   apply (insert linorder_less_linear)
754   apply blast
755   done
757 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
758   apply (simp add: order_less_le)
759   apply (insert linorder_linear)
760   apply (blast intro: order_antisym)
761   done
763 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
764   apply (simp add: order_less_le)
765   apply (insert linorder_linear)
766   apply (blast intro: order_antisym)
767   done
769 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
770   apply (cut_tac x = x and y = y in linorder_less_linear)
771   apply auto
772   done
774 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
775   apply (simp add: linorder_neq_iff)
776   apply blast
777   done
780 subsubsection "Min and max on (linear) orders"
782 lemma min_same [simp]: "min (x::'a::order) x = x"
783   by (simp add: min_def)
785 lemma max_same [simp]: "max (x::'a::order) x = x"
786   by (simp add: max_def)
788 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
789   apply (simp add: max_def)
790   apply (insert linorder_linear)
791   apply (blast intro: order_trans)
792   done
794 lemma le_maxI1: "(x::'a::linorder) <= max x y"
795   by (simp add: le_max_iff_disj)
797 lemma le_maxI2: "(y::'a::linorder) <= max x y"
798     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
799   by (simp add: le_max_iff_disj)
801 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
802   apply (simp add: max_def order_le_less)
803   apply (insert linorder_less_linear)
804   apply (blast intro: order_less_trans)
805   done
807 lemma max_le_iff_conj [simp]:
808     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
809   apply (simp add: max_def)
810   apply (insert linorder_linear)
811   apply (blast intro: order_trans)
812   done
814 lemma max_less_iff_conj [simp]:
815     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
816   apply (simp add: order_le_less max_def)
817   apply (insert linorder_less_linear)
818   apply (blast intro: order_less_trans)
819   done
821 lemma le_min_iff_conj [simp]:
822     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
823     -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
824   apply (simp add: min_def)
825   apply (insert linorder_linear)
826   apply (blast intro: order_trans)
827   done
829 lemma min_less_iff_conj [simp]:
830     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
831   apply (simp add: order_le_less min_def)
832   apply (insert linorder_less_linear)
833   apply (blast intro: order_less_trans)
834   done
836 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
837   apply (simp add: min_def)
838   apply (insert linorder_linear)
839   apply (blast intro: order_trans)
840   done
842 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
843   apply (simp add: min_def order_le_less)
844   apply (insert linorder_less_linear)
845   apply (blast intro: order_less_trans)
846   done
848 lemma split_min:
849     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
850   by (simp add: min_def)
852 lemma split_max:
853     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
854   by (simp add: max_def)
857 subsubsection "Bounded quantifiers"
859 syntax
860   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
861   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
862   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
863   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
865 syntax (xsymbols)
866   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
867   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
868   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
869   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
871 syntax (HOL)
872   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
873   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
874   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
875   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
877 translations
878  "ALL x<y. P"   =>  "ALL x. x < y --> P"
879  "EX x<y. P"    =>  "EX x. x < y  & P"
880  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
881  "EX x<=y. P"   =>  "EX x. x <= y & P"
883 end