src/HOL/HOL.thy
 author berghofe Mon Dec 10 15:16:49 2001 +0100 (2001-12-10) changeset 12436 a2df07fefed7 parent 12386 9c38ec9eca1c child 12633 ad9277743664 permissions -rw-r--r--
Replaced several occurrences of "blast" by "rules".
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     "(P ~= Q) = (P = (~Q))"
314     "(P | ~P) = True"    "(~P | P) = True"
315     "((~P) = (~Q)) = (P=Q)"
316     "(x = x) = True"
317     "(~True) = False"  "(~False) = True"
318     "(~P) ~= P"  "P ~= (~P)"
319     "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
320     "(True --> P) = P"  "(False --> P) = True"
321     "(P --> True) = True"  "(P --> P) = True"
322     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
323     "(P & True) = P"  "(True & P) = P"
324     "(P & False) = False"  "(False & P) = False"
325     "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
326     "(P & ~P) = False"    "(~P & P) = False"
327     "(P | True) = True"  "(True | P) = True"
328     "(P | False) = P"  "(False | P) = P"
329     "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
330     "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
331     -- {* needed for the one-point-rule quantifier simplification procs *}
332     -- {* essential for termination!! *} and
333     "!!P. (EX x. x=t & P(x)) = P(t)"
334     "!!P. (EX x. t=x & P(x)) = P(t)"
335     "!!P. (ALL x. x=t --> P(x)) = P(t)"
336     "!!P. (ALL x. t=x --> P(x)) = P(t)")
337   by (blast, blast, blast, blast, blast, rules+)
339 lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
340   by rules
342 lemma ex_simps:
343   "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
344   "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
345   "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
346   "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
347   "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
348   "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
349   -- {* Miniscoping: pushing in existential quantifiers. *}
350   by (rules | blast)+
352 lemma all_simps:
353   "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
354   "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
355   "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
356   "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
357   "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
358   "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
359   -- {* Miniscoping: pushing in universal quantifiers. *}
360   by (rules | blast)+
362 lemma eq_ac:
363  (eq_commute: "(a=b) = (b=a)" and
364   eq_left_commute: "(P=(Q=R)) = (Q=(P=R))" and
365   eq_assoc: "((P=Q)=R) = (P=(Q=R))") by (rules, blast+)
366 lemma neq_commute: "(a~=b) = (b~=a)" by rules
368 lemma conj_comms:
369  (conj_commute: "(P&Q) = (Q&P)" and
370   conj_left_commute: "(P&(Q&R)) = (Q&(P&R))") by rules+
371 lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
373 lemma disj_comms:
374  (disj_commute: "(P|Q) = (Q|P)" and
375   disj_left_commute: "(P|(Q|R)) = (Q|(P|R))") by rules+
376 lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
378 lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
379 lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
381 lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
382 lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
384 lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
385 lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
386 lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
388 text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
389 lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
390 lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
392 lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
393 lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
395 lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
396 lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
397 lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
398 lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
399 lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
400 lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
401   by blast
402 lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
404 lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
407 lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
408   -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
409   -- {* cases boil down to the same thing. *}
410   by blast
412 lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
413 lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
414 lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
415 lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
417 lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
418 lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
420 text {*
421   \medskip The @{text "&"} congruence rule: not included by default!
422   May slow rewrite proofs down by as much as 50\% *}
424 lemma conj_cong:
425     "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
426   by rules
428 lemma rev_conj_cong:
429     "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
430   by rules
432 text {* The @{text "|"} congruence rule: not included by default! *}
434 lemma disj_cong:
435     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
436   by blast
438 lemma eq_sym_conv: "(x = y) = (y = x)"
439   by rules
442 text {* \medskip if-then-else rules *}
444 lemma if_True: "(if True then x else y) = x"
445   by (unfold if_def) blast
447 lemma if_False: "(if False then x else y) = y"
448   by (unfold if_def) blast
450 lemma if_P: "P ==> (if P then x else y) = x"
451   by (unfold if_def) blast
453 lemma if_not_P: "~P ==> (if P then x else y) = y"
454   by (unfold if_def) blast
456 lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
457   apply (rule case_split [of Q])
458    apply (subst if_P)
459     prefer 3 apply (subst if_not_P)
460      apply blast+
461   done
463 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
464   apply (subst split_if)
465   apply blast
466   done
468 lemmas if_splits = split_if split_if_asm
470 lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
471   by (rule split_if)
473 lemma if_cancel: "(if c then x else x) = x"
474   apply (subst split_if)
475   apply blast
476   done
478 lemma if_eq_cancel: "(if x = y then y else x) = x"
479   apply (subst split_if)
480   apply blast
481   done
483 lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
484   -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
485   by (rule split_if)
487 lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
488   -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
489   apply (subst split_if)
490   apply blast
491   done
493 lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
494 lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
496 use "simpdata.ML"
497 setup Simplifier.setup
498 setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
499 setup Splitter.setup setup Clasimp.setup
502 subsubsection {* Generic cases and induction *}
504 constdefs
505   induct_forall :: "('a => bool) => bool"
506   "induct_forall P == \<forall>x. P x"
507   induct_implies :: "bool => bool => bool"
508   "induct_implies A B == A --> B"
509   induct_equal :: "'a => 'a => bool"
510   "induct_equal x y == x = y"
511   induct_conj :: "bool => bool => bool"
512   "induct_conj A B == A & B"
514 lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
515   by (simp only: atomize_all induct_forall_def)
517 lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
518   by (simp only: atomize_imp induct_implies_def)
520 lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
521   by (simp only: atomize_eq induct_equal_def)
523 lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
524     induct_conj (induct_forall A) (induct_forall B)"
525   by (unfold induct_forall_def induct_conj_def) rules
527 lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
528     induct_conj (induct_implies C A) (induct_implies C B)"
529   by (unfold induct_implies_def induct_conj_def) rules
531 lemma induct_conj_curry: "(induct_conj A B ==> C) == (A ==> B ==> C)"
532   by (simp only: atomize_imp atomize_eq induct_conj_def) (rules intro: equal_intr_rule)
534 lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
535   by (simp add: induct_implies_def)
537 lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
538 lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
539 lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
540 lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
542 hide const induct_forall induct_implies induct_equal induct_conj
545 text {* Method setup. *}
547 ML {*
548   structure InductMethod = InductMethodFun
549   (struct
550     val dest_concls = HOLogic.dest_concls;
551     val cases_default = thm "case_split";
552     val local_impI = thm "induct_impliesI";
553     val conjI = thm "conjI";
554     val atomize = thms "induct_atomize";
555     val rulify1 = thms "induct_rulify1";
556     val rulify2 = thms "induct_rulify2";
557     val localize = [Thm.symmetric (thm "induct_implies_def")];
558   end);
559 *}
561 setup InductMethod.setup
564 subsection {* Order signatures and orders *}
566 axclass
567   ord < type
569 syntax
570   "op <"        :: "['a::ord, 'a] => bool"             ("op <")
571   "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
573 global
575 consts
576   "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
577   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
579 local
581 syntax (xsymbols)
582   "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
583   "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
585 (*Tell blast about overloading of < and <= to reduce the risk of
586   its applying a rule for the wrong type*)
587 ML {*
588 Blast.overloaded ("op <" , domain_type);
589 Blast.overloaded ("op <=", domain_type);
590 *}
593 subsubsection {* Monotonicity *}
595 constdefs
596   mono :: "['a::ord => 'b::ord] => bool"
597   "mono f == ALL A B. A <= B --> f A <= f B"
599 lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
600   by (unfold mono_def) rules
602 lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
603   by (unfold mono_def) rules
605 constdefs
606   min :: "['a::ord, 'a] => 'a"
607   "min a b == (if a <= b then a else b)"
608   max :: "['a::ord, 'a] => 'a"
609   "max a b == (if a <= b then b else a)"
611 lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
612   by (simp add: min_def)
614 lemma min_of_mono:
615     "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
616   by (simp add: min_def)
618 lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
619   by (simp add: max_def)
621 lemma max_of_mono:
622     "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
623   by (simp add: max_def)
626 subsubsection "Orders"
628 axclass order < ord
629   order_refl [iff]: "x <= x"
630   order_trans: "x <= y ==> y <= z ==> x <= z"
631   order_antisym: "x <= y ==> y <= x ==> x = y"
632   order_less_le: "(x < y) = (x <= y & x ~= y)"
635 text {* Reflexivity. *}
637 lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
638     -- {* This form is useful with the classical reasoner. *}
639   apply (erule ssubst)
640   apply (rule order_refl)
641   done
643 lemma order_less_irrefl [simp]: "~ x < (x::'a::order)"
644   by (simp add: order_less_le)
646 lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
647     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
648   apply (simp add: order_less_le)
649   apply blast
650   done
652 lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
654 lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
655   by (simp add: order_less_le)
658 text {* Asymmetry. *}
660 lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
661   by (simp add: order_less_le order_antisym)
663 lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
664   apply (drule order_less_not_sym)
665   apply (erule contrapos_np)
666   apply simp
667   done
670 text {* Transitivity. *}
672 lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
673   apply (simp add: order_less_le)
674   apply (blast intro: order_trans order_antisym)
675   done
677 lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
678   apply (simp add: order_less_le)
679   apply (blast intro: order_trans order_antisym)
680   done
682 lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
683   apply (simp add: order_less_le)
684   apply (blast intro: order_trans order_antisym)
685   done
688 text {* Useful for simplification, but too risky to include by default. *}
690 lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
691   by (blast elim: order_less_asym)
693 lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
694   by (blast elim: order_less_asym)
696 lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
697   by auto
699 lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
700   by auto
703 text {* Other operators. *}
705 lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
706   apply (simp add: min_def)
707   apply (blast intro: order_antisym)
708   done
710 lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
711   apply (simp add: max_def)
712   apply (blast intro: order_antisym)
713   done
716 subsubsection {* Least value operator *}
718 constdefs
719   Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
720   "Least P == THE x. P x & (ALL y. P y --> x <= y)"
721     -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
723 lemma LeastI2:
724   "[| P (x::'a::order);
725       !!y. P y ==> x <= y;
726       !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
727    ==> Q (Least P)"
728   apply (unfold Least_def)
729   apply (rule theI2)
730     apply (blast intro: order_antisym)+
731   done
733 lemma Least_equality:
734     "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
735   apply (simp add: Least_def)
736   apply (rule the_equality)
737   apply (auto intro!: order_antisym)
738   done
741 subsubsection "Linear / total orders"
743 axclass linorder < order
744   linorder_linear: "x <= y | y <= x"
746 lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
747   apply (simp add: order_less_le)
748   apply (insert linorder_linear)
749   apply blast
750   done
752 lemma linorder_cases [case_names less equal greater]:
753     "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
754   apply (insert linorder_less_linear)
755   apply blast
756   done
758 lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
759   apply (simp add: order_less_le)
760   apply (insert linorder_linear)
761   apply (blast intro: order_antisym)
762   done
764 lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
765   apply (simp add: order_less_le)
766   apply (insert linorder_linear)
767   apply (blast intro: order_antisym)
768   done
770 lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
771   apply (cut_tac x = x and y = y in linorder_less_linear)
772   apply auto
773   done
775 lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
776   apply (simp add: linorder_neq_iff)
777   apply blast
778   done
781 subsubsection "Min and max on (linear) orders"
783 lemma min_same [simp]: "min (x::'a::order) x = x"
784   by (simp add: min_def)
786 lemma max_same [simp]: "max (x::'a::order) x = x"
787   by (simp add: max_def)
789 lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
790   apply (simp add: max_def)
791   apply (insert linorder_linear)
792   apply (blast intro: order_trans)
793   done
795 lemma le_maxI1: "(x::'a::linorder) <= max x y"
796   by (simp add: le_max_iff_disj)
798 lemma le_maxI2: "(y::'a::linorder) <= max x y"
799     -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
800   by (simp add: le_max_iff_disj)
802 lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
803   apply (simp add: max_def order_le_less)
804   apply (insert linorder_less_linear)
805   apply (blast intro: order_less_trans)
806   done
808 lemma max_le_iff_conj [simp]:
809     "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
810   apply (simp add: max_def)
811   apply (insert linorder_linear)
812   apply (blast intro: order_trans)
813   done
815 lemma max_less_iff_conj [simp]:
816     "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
817   apply (simp add: order_le_less max_def)
818   apply (insert linorder_less_linear)
819   apply (blast intro: order_less_trans)
820   done
822 lemma le_min_iff_conj [simp]:
823     "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
824     -- {* @{text "[iff]"} screws up a Q{text blast} in MiniML *}
825   apply (simp add: min_def)
826   apply (insert linorder_linear)
827   apply (blast intro: order_trans)
828   done
830 lemma min_less_iff_conj [simp]:
831     "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
832   apply (simp add: order_le_less min_def)
833   apply (insert linorder_less_linear)
834   apply (blast intro: order_less_trans)
835   done
837 lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
838   apply (simp add: min_def)
839   apply (insert linorder_linear)
840   apply (blast intro: order_trans)
841   done
843 lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
844   apply (simp add: min_def order_le_less)
845   apply (insert linorder_less_linear)
846   apply (blast intro: order_less_trans)
847   done
849 lemma split_min:
850     "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
851   by (simp add: min_def)
853 lemma split_max:
854     "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
855   by (simp add: max_def)
858 subsubsection "Bounded quantifiers"
860 syntax
861   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
862   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
863   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
864   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
866 syntax (xsymbols)
867   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
868   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
869   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
870   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
872 syntax (HOL)
873   "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
874   "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
875   "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
876   "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
878 translations
879  "ALL x<y. P"   =>  "ALL x. x < y --> P"
880  "EX x<y. P"    =>  "EX x. x < y  & P"
881  "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
882  "EX x<=y. P"   =>  "EX x. x <= y & P"
884 end