src/HOL/Fun.thy
 author haftmann Thu Oct 29 11:41:36 2009 +0100 (2009-10-29) changeset 33318 ddd97d9dfbfb parent 33057 764547b68538 child 34101 d689f0b33047 permissions -rw-r--r--
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
1 (*  Title:      HOL/Fun.thy
2     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
3     Copyright   1994  University of Cambridge
4 *)
8 theory Fun
9 imports Complete_Lattice
10 begin
12 text{*As a simplification rule, it replaces all function equalities by
13   first-order equalities.*}
14 lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
15 apply (rule iffI)
16 apply (simp (no_asm_simp))
17 apply (rule ext)
18 apply (simp (no_asm_simp))
19 done
21 lemma apply_inverse:
22   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
23   by auto
26 subsection {* The Identity Function @{text id} *}
28 definition
29   id :: "'a \<Rightarrow> 'a"
30 where
31   "id = (\<lambda>x. x)"
33 lemma id_apply [simp]: "id x = x"
36 lemma image_ident [simp]: "(%x. x) ` Y = Y"
37 by blast
39 lemma image_id [simp]: "id ` Y = Y"
42 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
43 by blast
45 lemma vimage_id [simp]: "id -` A = A"
49 subsection {* The Composition Operator @{text "f \<circ> g"} *}
51 definition
52   comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
53 where
54   "f o g = (\<lambda>x. f (g x))"
56 notation (xsymbols)
57   comp  (infixl "\<circ>" 55)
59 notation (HTML output)
60   comp  (infixl "\<circ>" 55)
62 text{*compatibility*}
63 lemmas o_def = comp_def
65 lemma o_apply [simp]: "(f o g) x = f (g x)"
68 lemma o_assoc: "f o (g o h) = f o g o h"
71 lemma id_o [simp]: "id o g = g"
74 lemma o_id [simp]: "f o id = f"
77 lemma image_compose: "(f o g) ` r = f`(g`r)"
78 by (simp add: comp_def, blast)
80 lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
81   by auto
83 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
84 by (unfold comp_def, blast)
87 subsection {* The Forward Composition Operator @{text fcomp} *}
89 definition
90   fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
91 where
92   "f o> g = (\<lambda>x. g (f x))"
94 lemma fcomp_apply:  "(f o> g) x = g (f x)"
97 lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
100 lemma id_fcomp [simp]: "id o> g = g"
103 lemma fcomp_id [simp]: "f o> id = f"
106 code_const fcomp
107   (Eval infixl 1 "#>")
109 no_notation fcomp (infixl "o>" 60)
112 subsection {* Injectivity and Surjectivity *}
114 constdefs
115   inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
116   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
118 text{*A common special case: functions injective over the entire domain type.*}
120 abbreviation
121   "inj f == inj_on f UNIV"
123 definition
124   bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
125   [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
127 constdefs
128   surj :: "('a => 'b) => bool"                   (*surjective*)
129   "surj f == ! y. ? x. y=f(x)"
131   bij :: "('a => 'b) => bool"                    (*bijective*)
132   "bij f == inj f & surj f"
134 lemma injI:
135   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
136   shows "inj f"
137   using assms unfolding inj_on_def by auto
139 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
140 lemma datatype_injI:
141     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
144 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
145   by (unfold inj_on_def, blast)
147 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
150 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
151 by (force simp add: inj_on_def)
153 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
156 lemma inj_on_id[simp]: "inj_on id A"
159 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
162 lemma surj_id[simp]: "surj id"
165 lemma bij_id[simp]: "bij id"
166 by (simp add: bij_def inj_on_id surj_id)
168 lemma inj_onI:
169     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
172 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
173 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
175 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
176 by (unfold inj_on_def, blast)
178 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
179 by (blast dest!: inj_onD)
181 lemma comp_inj_on:
182      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
183 by (simp add: comp_def inj_on_def)
185 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
187 apply blast
188 done
190 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
191   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
192 apply(unfold inj_on_def)
193 apply blast
194 done
196 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
197 by (unfold inj_on_def, blast)
199 lemma inj_singleton: "inj (%s. {s})"
202 lemma inj_on_empty[iff]: "inj_on f {}"
205 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
206 by (unfold inj_on_def, blast)
208 lemma inj_on_Un:
209  "inj_on f (A Un B) =
210   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
211 apply(unfold inj_on_def)
212 apply (blast intro:sym)
213 done
215 lemma inj_on_insert[iff]:
216   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
217 apply(unfold inj_on_def)
218 apply (blast intro:sym)
219 done
221 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
222 apply(unfold inj_on_def)
223 apply (blast)
224 done
226 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
228 apply (blast intro: sym)
229 done
231 lemma surj_range: "surj f ==> range f = UNIV"
232 by (auto simp add: surj_def)
234 lemma surjD: "surj f ==> EX x. y = f x"
237 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
238 by (simp add: surj_def, blast)
240 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
241 apply (simp add: comp_def surj_def, clarify)
242 apply (drule_tac x = y in spec, clarify)
243 apply (drule_tac x = x in spec, blast)
244 done
246 lemma bijI: "[| inj f; surj f |] ==> bij f"
249 lemma bij_is_inj: "bij f ==> inj f"
252 lemma bij_is_surj: "bij f ==> surj f"
255 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
258 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
259 by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
261 lemma bij_betw_trans:
262   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
265 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
266 proof -
267   have i: "inj_on f A" and s: "f ` A = B"
268     using assms by(auto simp:bij_betw_def)
269   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
270   { fix a b assume P: "?P b a"
271     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
272     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
273     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
274   } note g = this
275   have "inj_on ?g B"
276   proof(rule inj_onI)
277     fix x y assume "x:B" "y:B" "?g x = ?g y"
278     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
279     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
280     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
281   qed
282   moreover have "?g ` B = A"
283   proof(auto simp:image_def)
284     fix b assume "b:B"
285     with s obtain a where P: "?P b a" unfolding image_def by blast
286     thus "?g b \<in> A" using g[OF P] by auto
287   next
288     fix a assume "a:A"
289     then obtain b where P: "?P b a" using s unfolding image_def by blast
290     then have "b:B" using s unfolding image_def by blast
291     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
292   qed
293   ultimately show ?thesis by(auto simp:bij_betw_def)
294 qed
296 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
299 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
300 by (simp add: inj_on_def, blast)
302 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
303 apply (unfold surj_def)
304 apply (blast intro: sym)
305 done
307 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
308 by (unfold inj_on_def, blast)
310 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
311 apply (unfold bij_def)
312 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
313 done
315 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
316 by(blast dest: inj_onD)
318 lemma inj_on_image_Int:
319    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
320 apply (simp add: inj_on_def, blast)
321 done
323 lemma inj_on_image_set_diff:
324    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
325 apply (simp add: inj_on_def, blast)
326 done
328 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
329 by (simp add: inj_on_def, blast)
331 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
332 by (simp add: inj_on_def, blast)
334 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
335 by (blast dest: injD)
337 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
338 by (simp add: inj_on_def, blast)
340 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
341 by (blast dest: injD)
343 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
344 lemma image_INT:
345    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
346     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
347 apply (simp add: inj_on_def, blast)
348 done
350 (*Compare with image_INT: no use of inj_on, and if f is surjective then
351   it doesn't matter whether A is empty*)
352 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
354 apply (simp add: inj_on_def surj_def, blast)
355 done
357 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
358 by (auto simp add: surj_def)
360 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
361 by (auto simp add: inj_on_def)
363 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
365 apply (rule equalityI)
366 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
367 done
370 subsection{*Function Updating*}
372 constdefs
373   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
374   "fun_upd f a b == % x. if x=a then b else f x"
376 nonterminals
377   updbinds updbind
378 syntax
379   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
380   ""         :: "updbind => updbinds"             ("_")
381   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
382   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
384 translations
385   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
386   "f(x:=y)"                     == "fun_upd f x y"
388 (* Hint: to define the sum of two functions (or maps), use sum_case.
389          A nice infix syntax could be defined (in Datatype.thy or below) by
390 consts
391   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
392 translations
393  "fun_sum" == sum_case
394 *)
396 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
397 apply (simp add: fun_upd_def, safe)
398 apply (erule subst)
399 apply (rule_tac [2] ext, auto)
400 done
402 (* f x = y ==> f(x:=y) = f *)
403 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
405 (* f(x := f x) = f *)
406 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
407 declare fun_upd_triv [iff]
409 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
412 (* fun_upd_apply supersedes these two,   but they are useful
413    if fun_upd_apply is intentionally removed from the simpset *)
414 lemma fun_upd_same: "(f(x:=y)) x = y"
415 by simp
417 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
418 by simp
420 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
423 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
424 by (rule ext, auto)
426 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
427 by(fastsimp simp:inj_on_def image_def)
429 lemma fun_upd_image:
430      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
431 by auto
433 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
434 by(auto intro: ext)
437 subsection {* @{text override_on} *}
439 definition
440   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
441 where
442   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
444 lemma override_on_emptyset[simp]: "override_on f g {} = f"
447 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
450 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
454 subsection {* @{text swap} *}
456 definition
457   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
458 where
459   "swap a b f = f (a := f b, b:= f a)"
461 lemma swap_self: "swap a a f = f"
464 lemma swap_commute: "swap a b f = swap b a f"
465 by (rule ext, simp add: fun_upd_def swap_def)
467 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
468 by (rule ext, simp add: fun_upd_def swap_def)
470 lemma inj_on_imp_inj_on_swap:
471   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
472 by (simp add: inj_on_def swap_def, blast)
474 lemma inj_on_swap_iff [simp]:
475   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
476 proof
477   assume "inj_on (swap a b f) A"
478   with A have "inj_on (swap a b (swap a b f)) A"
479     by (iprover intro: inj_on_imp_inj_on_swap)
480   thus "inj_on f A" by simp
481 next
482   assume "inj_on f A"
483   with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap)
484 qed
486 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
487 apply (simp add: surj_def swap_def, clarify)
488 apply (case_tac "y = f b", blast)
489 apply (case_tac "y = f a", auto)
490 done
492 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
493 proof
494   assume "surj (swap a b f)"
495   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap)
496   thus "surj f" by simp
497 next
498   assume "surj f"
499   thus "surj (swap a b f)" by (rule surj_imp_surj_swap)
500 qed
502 lemma bij_swap_iff: "bij (swap a b f) = bij f"
505 hide (open) const swap
508 subsection {* Inversion of injective functions *}
510 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
511 "the_inv_into A f == %x. THE y. y : A & f y = x"
513 lemma the_inv_into_f_f:
514   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
515 apply (simp add: the_inv_into_def inj_on_def)
516 apply (blast intro: the_equality)
517 done
519 lemma f_the_inv_into_f:
520   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
522 apply (rule the1I2)
523  apply(blast dest: inj_onD)
524 apply blast
525 done
527 lemma the_inv_into_into:
528   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
530 apply (rule the1I2)
531  apply(blast dest: inj_onD)
532 apply blast
533 done
535 lemma the_inv_into_onto[simp]:
536   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
537 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
539 lemma the_inv_into_f_eq:
540   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
541   apply (erule subst)
542   apply (erule the_inv_into_f_f, assumption)
543   done
545 lemma the_inv_into_comp:
546   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
547   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
548 apply (rule the_inv_into_f_eq)
549   apply (fast intro: comp_inj_on)
550  apply (simp add: f_the_inv_into_f the_inv_into_into)
552 done
554 lemma inj_on_the_inv_into:
555   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
556 by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
558 lemma bij_betw_the_inv_into:
559   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
560 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
562 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
563   "the_inv f \<equiv> the_inv_into UNIV f"
565 lemma the_inv_f_f:
566   assumes "inj f"
567   shows "the_inv f (f x) = x" using assms UNIV_I
568   by (rule the_inv_into_f_f)
571 subsection {* Proof tool setup *}
573 text {* simplifies terms of the form
574   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
576 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
577 let
578   fun gen_fun_upd NONE T _ _ = NONE
579     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
580   fun dest_fun_T1 (Type (_, T :: Ts)) = T
581   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
582     let
583       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
584             if v aconv x then SOME g else gen_fun_upd (find g) T v w
585         | find t = NONE
586     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
588   fun proc ss ct =
589     let
590       val ctxt = Simplifier.the_context ss
591       val t = Thm.term_of ct
592     in
593       case find_double t of
594         (T, NONE) => NONE
595       | (T, SOME rhs) =>
596           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
597             (fn _ =>
598               rtac eq_reflection 1 THEN
599               rtac ext 1 THEN
600               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
601     end
602 in proc end
603 *}
606 subsection {* Code generator setup *}
608 types_code
609   "fun"  ("(_ ->/ _)")
610 attach (term_of) {*
611 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
612 *}
613 attach (test) {*
614 fun gen_fun_type aF aT bG bT i =
615   let
616     val tab = Unsynchronized.ref [];
617     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
618       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
619   in
620     (fn x =>
621        case AList.lookup op = (!tab) x of
622          NONE =>
623            let val p as (y, _) = bG i
624            in (tab := (x, p) :: !tab; y) end
625        | SOME (y, _) => y,
626      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
627   end;
628 *}
630 code_const "op \<circ>"
631   (SML infixl 5 "o")