src/HOL/Algebra/FiniteProduct.thy
 author haftmann Sun Jun 23 21:16:07 2013 +0200 (2013-06-23) changeset 52435 6646bb548c6b parent 46721 f88b187ad8ca child 56142 8bb21318e10b permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
1 (*  Title:      HOL/Algebra/FiniteProduct.thy
2     Author:     Clemens Ballarin, started 19 November 2002
4 This file is largely based on HOL/Finite_Set.thy.
5 *)
7 theory FiniteProduct
8 imports Group
9 begin
11 subsection {* Product Operator for Commutative Monoids *}
13 subsubsection {* Inductive Definition of a Relation for Products over Sets *}
15 text {* Instantiation of locale @{text LC} of theory @{text Finite_Set} is not
16   possible, because here we have explicit typing rules like
17   @{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain
18   @{text D}. *}
20 inductive_set
21   foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
22   for D :: "'a set" and f :: "'b => 'a => 'a" and e :: 'a
23   where
24     emptyI [intro]: "e \<in> D ==> ({}, e) \<in> foldSetD D f e"
25   | insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>
26                       (insert x A, f x y) \<in> foldSetD D f e"
28 inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
30 definition
31   foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
32   where "foldD D f e A = (THE x. (A, x) \<in> foldSetD D f e)"
34 lemma foldSetD_closed:
35   "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D
36       |] ==> z \<in> D";
37   by (erule foldSetD.cases) auto
39 lemma Diff1_foldSetD:
40   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>
41    (A, f x y) \<in> foldSetD D f e"
42   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
43     apply auto
44   done
46 lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"
47   by (induct set: foldSetD) auto
49 lemma finite_imp_foldSetD:
50   "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
51    EX x. (A, x) \<in> foldSetD D f e"
52 proof (induct set: finite)
53   case empty then show ?case by auto
54 next
55   case (insert x F)
56   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
57   with insert have "y \<in> D" by (auto dest: foldSetD_closed)
58   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
59     by (intro foldSetD.intros) auto
60   then show ?case ..
61 qed
64 text {* Left-Commutative Operations *}
66 locale LCD =
67   fixes B :: "'b set"
68   and D :: "'a set"
69   and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
70   assumes left_commute:
71     "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
72   and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"
74 lemma (in LCD) foldSetD_closed [dest]:
75   "(A, z) \<in> foldSetD D f e ==> z \<in> D";
76   by (erule foldSetD.cases) auto
78 lemma (in LCD) Diff1_foldSetD:
79   "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
80   (A, f x y) \<in> foldSetD D f e"
81   apply (subgoal_tac "x \<in> B")
82    prefer 2 apply fast
83   apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
84     apply auto
85   done
87 lemma (in LCD) foldSetD_imp_finite [simp]:
88   "(A, x) \<in> foldSetD D f e ==> finite A"
89   by (induct set: foldSetD) auto
91 lemma (in LCD) finite_imp_foldSetD:
92   "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> foldSetD D f e"
93 proof (induct set: finite)
94   case empty then show ?case by auto
95 next
96   case (insert x F)
97   then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
98   with insert have "y \<in> D" by auto
99   with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
100     by (intro foldSetD.intros) auto
101   then show ?case ..
102 qed
104 lemma (in LCD) foldSetD_determ_aux:
105   "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
106     (\<forall>y. (A, y) \<in> foldSetD D f e --> y = x)"
107   apply (induct n)
108    apply (auto simp add: less_Suc_eq) (* slow *)
109   apply (erule foldSetD.cases)
110    apply blast
111   apply (erule foldSetD.cases)
112    apply blast
113   apply clarify
114   txt {* force simplification of @{text "card A < card (insert ...)"}. *}
115   apply (erule rev_mp)
116   apply (simp add: less_Suc_eq_le)
117   apply (rule impI)
118   apply (rename_tac xa Aa ya xb Ab yb, case_tac "xa = xb")
119    apply (subgoal_tac "Aa = Ab")
120     prefer 2 apply (blast elim!: equalityE)
121    apply blast
122   txt {* case @{prop "xa \<notin> xb"}. *}
123   apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> Ab")
124    prefer 2 apply (blast elim!: equalityE)
125   apply clarify
126   apply (subgoal_tac "Aa = insert xb Ab - {xa}")
127    prefer 2 apply blast
128   apply (subgoal_tac "card Aa \<le> card Ab")
129    prefer 2
130    apply (rule Suc_le_mono [THEN subst])
131    apply (simp add: card_Suc_Diff1)
132   apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
133      apply (blast intro: foldSetD_imp_finite)
134     apply best
135    apply assumption
136   apply (frule (1) Diff1_foldSetD)
137    apply best
138   apply (subgoal_tac "ya = f xb x")
139    prefer 2
140    apply (subgoal_tac "Aa \<subseteq> B")
141     prefer 2 apply best (* slow *)
142    apply (blast del: equalityCE)
143   apply (subgoal_tac "(Ab - {xa}, x) \<in> foldSetD D f e")
144    prefer 2 apply simp
145   apply (subgoal_tac "yb = f xa x")
146    prefer 2
147    apply (blast del: equalityCE dest: Diff1_foldSetD)
148   apply (simp (no_asm_simp))
149   apply (rule left_commute)
150     apply assumption
151    apply best (* slow *)
152   apply best
153   done
155 lemma (in LCD) foldSetD_determ:
156   "[| (A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
157   ==> y = x"
158   by (blast intro: foldSetD_determ_aux [rule_format])
160 lemma (in LCD) foldD_equality:
161   "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"
162   by (unfold foldD_def) (blast intro: foldSetD_determ)
164 lemma foldD_empty [simp]:
165   "e \<in> D ==> foldD D f e {} = e"
166   by (unfold foldD_def) blast
168 lemma (in LCD) foldD_insert_aux:
169   "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
170     ((insert x A, v) \<in> foldSetD D f e) =
171     (EX y. (A, y) \<in> foldSetD D f e & v = f x y)"
172   apply auto
173   apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
174      apply (fastforce dest: foldSetD_imp_finite)
175     apply assumption
176    apply assumption
177   apply (blast intro: foldSetD_determ)
178   done
180 lemma (in LCD) foldD_insert:
181     "[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
182      foldD D f e (insert x A) = f x (foldD D f e A)"
183   apply (unfold foldD_def)
184   apply (simp add: foldD_insert_aux)
185   apply (rule the_equality)
186    apply (auto intro: finite_imp_foldSetD
187      cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
188   done
190 lemma (in LCD) foldD_closed [simp]:
191   "[| finite A; e \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
192 proof (induct set: finite)
193   case empty then show ?case by simp
194 next
195   case insert then show ?case by (simp add: foldD_insert)
196 qed
198 lemma (in LCD) foldD_commute:
199   "[| finite A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
200    f x (foldD D f e A) = foldD D f (f x e) A"
201   apply (induct set: finite)
202    apply simp
203   apply (auto simp add: left_commute foldD_insert)
204   done
206 lemma Int_mono2:
207   "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
208   by blast
210 lemma (in LCD) foldD_nest_Un_Int:
211   "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B |] ==>
212    foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
213   apply (induct set: finite)
214    apply simp
215   apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
216     Int_mono2)
217   done
219 lemma (in LCD) foldD_nest_Un_disjoint:
220   "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B |]
221     ==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
222   by (simp add: foldD_nest_Un_Int)
224 -- {* Delete rules to do with @{text foldSetD} relation. *}
226 declare foldSetD_imp_finite [simp del]
227   empty_foldSetDE [rule del]
228   foldSetD.intros [rule del]
229 declare (in LCD)
230   foldSetD_closed [rule del]
233 text {* Commutative Monoids *}
235 text {*
236   We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
237   instead of @{text "'b => 'a => 'a"}.
238 *}
240 locale ACeD =
241   fixes D :: "'a set"
242     and f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
243     and e :: 'a
244   assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"
245     and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"
246     and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
247     and e_closed [simp]: "e \<in> D"
248     and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"
250 lemma (in ACeD) left_commute:
251   "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
252 proof -
253   assume D: "x \<in> D" "y \<in> D" "z \<in> D"
254   then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
255   also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
256   also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
257   finally show ?thesis .
258 qed
260 lemmas (in ACeD) AC = assoc commute left_commute
262 lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"
263 proof -
264   assume "x \<in> D"
265   then have "x \<cdot> e = x" by (rule ident)
266   with `x \<in> D` show ?thesis by (simp add: commute)
267 qed
269 lemma (in ACeD) foldD_Un_Int:
270   "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
271     foldD D f e A \<cdot> foldD D f e B =
272     foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
273   apply (induct set: finite)
274    apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
275   apply (simp add: AC insert_absorb Int_insert_left
276     LCD.foldD_insert [OF LCD.intro [of D]]
277     LCD.foldD_closed [OF LCD.intro [of D]]
278     Int_mono2)
279   done
281 lemma (in ACeD) foldD_Un_disjoint:
282   "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>
283     foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
284   by (simp add: foldD_Un_Int
285     left_commute LCD.foldD_closed [OF LCD.intro [of D]])
288 subsubsection {* Products over Finite Sets *}
290 definition
291   finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
292   where "finprod G f A =
293    (if finite A
294     then foldD (carrier G) (mult G o f) \<one>\<^bsub>G\<^esub> A
295     else undefined)"
297 syntax
298   "_finprod" :: "index => idt => 'a set => 'b => 'b"
299       ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
300 syntax (xsymbols)
301   "_finprod" :: "index => idt => 'a set => 'b => 'b"
302       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
303 syntax (HTML output)
304   "_finprod" :: "index => idt => 'a set => 'b => 'b"
305       ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
306 translations
307   "\<Otimes>\<index>i:A. b" == "CONST finprod \<struct>\<index> (%i. b) A"
308   -- {* Beware of argument permutation! *}
310 lemma (in comm_monoid) finprod_empty [simp]:
311   "finprod G f {} = \<one>"
312   by (simp add: finprod_def)
314 declare funcsetI [intro]
315   funcset_mem [dest]
317 context comm_monoid begin
319 lemma finprod_insert [simp]:
320   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
321    finprod G f (insert a F) = f a \<otimes> finprod G f F"
322   apply (rule trans)
323    apply (simp add: finprod_def)
324   apply (rule trans)
325    apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
326          apply simp
327          apply (rule m_lcomm)
328            apply fast
329           apply fast
330          apply assumption
331         apply fastforce
332        apply simp+
333    apply fast
334   apply (auto simp add: finprod_def)
335   done
337 lemma finprod_one [simp]:
338   "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
339 proof (induct set: finite)
340   case empty show ?case by simp
341 next
342   case (insert a A)
343   have "(%i. \<one>) \<in> A -> carrier G" by auto
344   with insert show ?case by simp
345 qed
347 lemma finprod_closed [simp]:
348   fixes A
349   assumes fin: "finite A" and f: "f \<in> A -> carrier G"
350   shows "finprod G f A \<in> carrier G"
351 using fin f
352 proof induct
353   case empty show ?case by simp
354 next
355   case (insert a A)
356   then have a: "f a \<in> carrier G" by fast
357   from insert have A: "f \<in> A -> carrier G" by fast
358   from insert A a show ?case by simp
359 qed
361 lemma funcset_Int_left [simp, intro]:
362   "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
363   by fast
365 lemma funcset_Un_left [iff]:
366   "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
367   by fast
369 lemma finprod_Un_Int:
370   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
371      finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
372      finprod G g A \<otimes> finprod G g B"
373 -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
374 proof (induct set: finite)
375   case empty then show ?case by simp
376 next
377   case (insert a A)
378   then have a: "g a \<in> carrier G" by fast
379   from insert have A: "g \<in> A -> carrier G" by fast
380   from insert A a show ?case
381     by (simp add: m_ac Int_insert_left insert_absorb Int_mono2)
382 qed
384 lemma finprod_Un_disjoint:
385   "[| finite A; finite B; A Int B = {};
386       g \<in> A -> carrier G; g \<in> B -> carrier G |]
387    ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
388   apply (subst finprod_Un_Int [symmetric])
389       apply auto
390   done
392 lemma finprod_multf:
393   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
394    finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
395 proof (induct set: finite)
396   case empty show ?case by simp
397 next
398   case (insert a A) then
399   have fA: "f \<in> A -> carrier G" by fast
400   from insert have fa: "f a \<in> carrier G" by fast
401   from insert have gA: "g \<in> A -> carrier G" by fast
402   from insert have ga: "g a \<in> carrier G" by fast
403   from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"
404     by (simp add: Pi_def)
405   show ?case
406     by (simp add: insert fA fa gA ga fgA m_ac)
407 qed
409 lemma finprod_cong':
410   "[| A = B; g \<in> B -> carrier G;
411       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
412 proof -
413   assume prems: "A = B" "g \<in> B -> carrier G"
414     "!!i. i \<in> B ==> f i = g i"
415   show ?thesis
416   proof (cases "finite B")
417     case True
418     then have "!!A. [| A = B; g \<in> B -> carrier G;
419       !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
420     proof induct
421       case empty thus ?case by simp
422     next
423       case (insert x B)
424       then have "finprod G f A = finprod G f (insert x B)" by simp
425       also from insert have "... = f x \<otimes> finprod G f B"
426       proof (intro finprod_insert)
427         show "finite B" by fact
428       next
429         show "x ~: B" by fact
430       next
431         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
432           "g \<in> insert x B \<rightarrow> carrier G"
433         thus "f \<in> B -> carrier G" by fastforce
434       next
435         assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
436           "g \<in> insert x B \<rightarrow> carrier G"
437         thus "f x \<in> carrier G" by fastforce
438       qed
439       also from insert have "... = g x \<otimes> finprod G g B" by fastforce
440       also from insert have "... = finprod G g (insert x B)"
441       by (intro finprod_insert [THEN sym]) auto
442       finally show ?case .
443     qed
444     with prems show ?thesis by simp
445   next
446     case False with prems show ?thesis by (simp add: finprod_def)
447   qed
448 qed
450 lemma finprod_cong:
451   "[| A = B; f \<in> B -> carrier G = True;
452       !!i. i \<in> B =simp=> f i = g i |] ==> finprod G f A = finprod G g B"
453   (* This order of prems is slightly faster (3%) than the last two swapped. *)
454   by (rule finprod_cong') (auto simp add: simp_implies_def)
456 text {*Usually, if this rule causes a failed congruence proof error,
457   the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
458   Adding @{thm [source] Pi_def} to the simpset is often useful.
459   For this reason, @{thm [source] comm_monoid.finprod_cong}
460   is not added to the simpset by default.
461 *}
463 end
465 declare funcsetI [rule del]
466   funcset_mem [rule del]
468 context comm_monoid begin
470 lemma finprod_0 [simp]:
471   "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"
472 by (simp add: Pi_def)
474 lemma finprod_Suc [simp]:
475   "f \<in> {..Suc n} -> carrier G ==>
476    finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
477 by (simp add: Pi_def atMost_Suc)
479 lemma finprod_Suc2:
480   "f \<in> {..Suc n} -> carrier G ==>
481    finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
482 proof (induct n)
483   case 0 thus ?case by (simp add: Pi_def)
484 next
485   case Suc thus ?case by (simp add: m_assoc Pi_def)
486 qed
488 lemma finprod_mult [simp]:
489   "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
490      finprod G (%i. f i \<otimes> g i) {..n::nat} =
491      finprod G f {..n} \<otimes> finprod G g {..n}"
492   by (induct n) (simp_all add: m_ac Pi_def)
494 (* The following two were contributed by Jeremy Avigad. *)
496 lemma finprod_reindex:
497   assumes fin: "finite A"
498     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
499         inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"
500   using fin
501   by induct (auto simp add: Pi_def)
503 lemma finprod_const:
504   assumes fin [simp]: "finite A"
505       and a [simp]: "a : carrier G"
506     shows "finprod G (%x. a) A = a (^) card A"
507   using fin apply induct
508   apply force
509   apply (subst finprod_insert)
510   apply auto
511   apply (subst m_comm)
512   apply auto
513   done
515 (* The following lemma was contributed by Jesus Aransay. *)
517 lemma finprod_singleton:
518   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
519   shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
520   using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]
521     fin_A f_Pi finprod_one [of "A - {i}"]
522     finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"]
523   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
525 end
527 end