# HG changeset patch # User wenzelm # Date 1007595535 -3600 # Node ID 2298d5b8e5304d164972d8498355aeb48eb1d7c7 # Parent d6913de7655f17507f7d265e687bf30757ec17ba renamed theory Finite to Finite_Set and converted; diff -r d6913de7655f -r 2298d5b8e530 src/HOL/Finite.ML --- a/src/HOL/Finite.ML Thu Dec 06 00:37:59 2001 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,903 +0,0 @@ -(* Title: HOL/Finite.ML - ID: $Id$ - Author: Lawrence C Paulson & Tobias Nipkow - Copyright 1995 University of Cambridge & TU Muenchen - -Finite sets and their cardinality. -*) - -section "finite"; - -(*Discharging ~ x:y entails extra work*) -val major::prems = Goal - "[| finite F; P({}); \ -\ !!F x. [| finite F; x ~: F; P(F) |] ==> P(insert x F) \ -\ |] ==> P(F)"; -by (rtac (major RS Finites.induct) 1); -by (excluded_middle_tac "a:A" 2); -by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*) -by (REPEAT (ares_tac prems 1)); -qed "finite_induct"; - -val major::subs::prems = Goal - "[| finite F; F <= A; \ -\ P({}); \ -\ !!F a. [| finite F; a:A; a ~: F; P(F) |] ==> P(insert a F) \ -\ |] ==> P(F)"; -by (rtac (subs RS rev_mp) 1); -by (rtac (major RS finite_induct) 1); -by (ALLGOALS (blast_tac (claset() addIs prems))); -qed "finite_subset_induct"; - -Addsimps Finites.intrs; -AddSIs Finites.intrs; - -(*The union of two finite sets is finite*) -Goal "[| finite F; finite G |] ==> finite(F Un G)"; -by (etac finite_induct 1); -by (ALLGOALS Asm_simp_tac); -qed "finite_UnI"; - -(*Every subset of a finite set is finite*) -Goal "finite B ==> ALL A. A<=B --> finite A"; -by (etac finite_induct 1); -by (ALLGOALS (simp_tac (simpset() addsimps [subset_insert_iff]))); -by Safe_tac; - by (eres_inst_tac [("t","A")] (insert_Diff RS subst) 1); - by (ALLGOALS Blast_tac); -val lemma = result(); - -Goal "[| A<=B; finite B |] ==> finite A"; -by (dtac lemma 1); -by (Blast_tac 1); -qed "finite_subset"; - -Goal "finite(F Un G) = (finite F & finite G)"; -by (blast_tac (claset() - addIs [inst "B" "?X Un ?Y" finite_subset, finite_UnI]) 1); -qed "finite_Un"; -AddIffs[finite_Un]; - -(*The converse obviously fails*) -Goal "finite F | finite G ==> finite(F Int G)"; -by (blast_tac (claset() addIs [finite_subset]) 1); -qed "finite_Int"; - -Addsimps [finite_Int]; -AddIs [finite_Int]; - -Goal "finite(insert a A) = finite A"; -by (stac insert_is_Un 1); -by (simp_tac (HOL_ss addsimps [finite_Un]) 1); -by (Blast_tac 1); -qed "finite_insert"; -Addsimps[finite_insert]; - -(*The image of a finite set is finite *) -Goal "finite F ==> finite(h`F)"; -by (etac finite_induct 1); -by (Simp_tac 1); -by (Asm_simp_tac 1); -qed "finite_imageI"; - -Goal "finite (range g) ==> finite (range (%x. f (g x)))"; -by (Simp_tac 1); -by (etac finite_imageI 1); -qed "finite_range_imageI"; - -val major::prems = Goal - "[| finite c; finite b; \ -\ P(b); \ -\ !!x y. [| finite y; x:y; P(y) |] ==> P(y-{x}) \ -\ |] ==> c<=b --> P(b-c)"; -by (rtac (major RS finite_induct) 1); -by (stac Diff_insert 2); -by (ALLGOALS (asm_simp_tac - (simpset() addsimps prems@[Diff_subset RS finite_subset]))); -val lemma = result(); - -val prems = Goal - "[| finite A; \ -\ P(A); \ -\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ -\ |] ==> P({})"; -by (rtac (Diff_cancel RS subst) 1); -by (rtac (lemma RS mp) 1); -by (REPEAT (ares_tac (subset_refl::prems) 1)); -qed "finite_empty_induct"; - - -(* finite B ==> finite (B - Ba) *) -bind_thm ("finite_Diff", Diff_subset RS finite_subset); -Addsimps [finite_Diff]; - -Goal "finite(A - insert a B) = finite(A-B)"; -by (stac Diff_insert 1); -by (case_tac "a : A-B" 1); -by (rtac (finite_insert RS sym RS trans) 1); -by (stac insert_Diff 1); -by (ALLGOALS Asm_full_simp_tac); -qed "finite_Diff_insert"; -AddIffs [finite_Diff_insert]; - -(*lemma merely for classical reasoner in the proof below: force_tac can't - prove it.*) -Goal "finite(A-{}) = finite A"; -by (Simp_tac 1); -val lemma = result(); - -(*Lemma for proving finite_imageD*) -Goal "finite B ==> ALL A. f`A = B --> inj_on f A --> finite A"; -by (etac finite_induct 1); - by (ALLGOALS Asm_simp_tac); -by (Clarify_tac 1); -by (subgoal_tac "EX y:A. f y = x & F = f`(A-{y})" 1); - by (Clarify_tac 1); - by (full_simp_tac (simpset() addsimps [inj_on_def]) 1); - by (blast_tac (claset() addSDs [lemma RS iffD1]) 1); -by (thin_tac "ALL A. ?PP(A)" 1); -by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1); -by (Clarify_tac 1); -by (res_inst_tac [("x","xa")] bexI 1); -by (ALLGOALS - (asm_full_simp_tac (simpset() addsimps [inj_on_image_set_diff]))); -val lemma = result(); - -Goal "[| finite(f`A); inj_on f A |] ==> finite A"; -by (dtac lemma 1); -by (Blast_tac 1); -qed "finite_imageD"; - -(** The finite UNION of finite sets **) - -Goal "finite A ==> (ALL a:A. finite(B a)) --> finite(UN a:A. B a)"; -by (etac finite_induct 1); -by (ALLGOALS Asm_simp_tac); -bind_thm("finite_UN_I", ballI RSN (2, result() RS mp)); - -(*strengthen RHS to - ((ALL x:A. finite (B x)) & finite {x. x:A & B x ~= {}}) ? - we'd need to prove - finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x ~= {}} - by induction*) -Goal "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"; -by (blast_tac (claset() addIs [finite_UN_I, finite_subset]) 1); -qed "finite_UN"; -Addsimps [finite_UN]; - -(** Sigma of finite sets **) - -Goalw [Sigma_def] - "[| finite A; ALL a:A. finite(B a) |] ==> finite(SIGMA a:A. B a)"; -by (blast_tac (claset() addSIs [finite_UN_I]) 1); -bind_thm("finite_SigmaI", ballI RSN (2,result())); -Addsimps [finite_SigmaI]; - -Goal "[| finite (UNIV::'a set); finite (UNIV::'b set)|] ==> finite (UNIV::('a * 'b) set)"; -by (subgoal_tac "(UNIV::('a * 'b) set) = Sigma UNIV (%x. UNIV)" 1); -by (etac ssubst 1); -by (etac finite_SigmaI 1); -by Auto_tac; -qed "finite_Prod_UNIV"; - -Goal "finite (UNIV :: ('a::finite * 'b::finite) set)"; -by (rtac (finite_Prod_UNIV) 1); -by (rtac finite 1); -by (rtac finite 1); -qed "finite_Prod"; - -Goal "finite (UNIV :: unit set)"; -by (subgoal_tac "UNIV = {()}" 1); -by (etac ssubst 1); -by Auto_tac; -qed "finite_unit"; - -(** The powerset of a finite set **) - -Goal "finite(Pow A) ==> finite A"; -by (subgoal_tac "finite ((%x.{x})`A)" 1); -by (rtac finite_subset 2); -by (assume_tac 3); -by (ALLGOALS - (fast_tac (claset() addSDs [rewrite_rule [inj_on_def] finite_imageD]))); -val lemma = result(); - -Goal "finite(Pow A) = finite A"; -by (rtac iffI 1); -by (etac lemma 1); -(*Opposite inclusion: finite A ==> finite (Pow A) *) -by (etac finite_induct 1); -by (ALLGOALS - (asm_simp_tac - (simpset() addsimps [finite_UnI, finite_imageI, Pow_insert]))); -qed "finite_Pow_iff"; -AddIffs [finite_Pow_iff]; - -Goal "finite(r^-1) = finite r"; -by (subgoal_tac "r^-1 = (%(x,y).(y,x))`r" 1); - by (Asm_simp_tac 1); - by (rtac iffI 1); - by (etac (rewrite_rule [inj_on_def] finite_imageD) 1); - by (simp_tac (simpset() addsplits [split_split]) 1); - by (etac finite_imageI 1); -by (simp_tac (simpset() addsimps [converse_def,image_def]) 1); -by Auto_tac; -by (rtac bexI 1); -by (assume_tac 2); -by (Simp_tac 1); -qed "finite_converse"; -AddIffs [finite_converse]; - -Goal "finite (lessThan (k::nat))"; -by (induct_tac "k" 1); -by (simp_tac (simpset() addsimps [lessThan_Suc]) 2); -by Auto_tac; -qed "finite_lessThan"; - -Goal "finite (atMost (k::nat))"; -by (induct_tac "k" 1); -by (simp_tac (simpset() addsimps [atMost_Suc]) 2); -by Auto_tac; -qed "finite_atMost"; -AddIffs [finite_lessThan, finite_atMost]; - -(* A bounded set of natural numbers is finite *) -Goal "(ALL i:N. i<(n::nat)) ==> finite N"; -by (rtac finite_subset 1); - by (rtac finite_lessThan 2); -by Auto_tac; -qed "bounded_nat_set_is_finite"; - -(** Finiteness of transitive closure (thanks to Sidi Ehmety) **) - -(*A finite relation has a finite field ( = domain U range) *) -Goal "finite r ==> finite (Field r)"; -by (etac finite_induct 1); -by (auto_tac (claset(), - simpset() addsimps [Field_def, Domain_insert, Range_insert])); -qed "finite_Field"; - -Goal "r^+ <= Field r <*> Field r"; -by (Clarify_tac 1); -by (etac trancl_induct 1); -by (auto_tac (claset(), simpset() addsimps [Field_def])); -qed "trancl_subset_Field2"; - -Goal "finite (r^+) = finite r"; -by Auto_tac; -by (rtac (trancl_subset_Field2 RS finite_subset) 2); -by (rtac finite_SigmaI 2); -by (blast_tac (claset() addIs [r_into_trancl, finite_subset]) 1); -by (auto_tac (claset(), simpset() addsimps [finite_Field])); -qed "finite_trancl"; - - -section "Finite cardinality -- 'card'"; - -bind_thm ("cardR_emptyE", cardR.mk_cases "({},n) : cardR"); -bind_thm ("cardR_insertE", cardR.mk_cases "(insert a A,n) : cardR"); - -AddSEs [cardR_emptyE]; -AddSIs cardR.intrs; - -Goal "[| (A,n) : cardR |] ==> a : A --> (EX m. n = Suc m)"; -by (etac cardR.induct 1); - by (Blast_tac 1); -by (Blast_tac 1); -qed "cardR_SucD"; - -Goal "(A,m): cardR ==> (ALL n a. m = Suc n --> a:A --> (A-{a},n) : cardR)"; -by (etac cardR.induct 1); - by Auto_tac; -by (asm_simp_tac (simpset() addsimps [insert_Diff_if]) 1); -by Auto_tac; -by (ftac cardR_SucD 1); -by (Blast_tac 1); -val lemma = result(); - -Goal "[| (insert a A, Suc m) : cardR; a ~: A |] ==> (A,m) : cardR"; -by (dtac lemma 1); -by (Asm_full_simp_tac 1); -val lemma = result(); - -Goal "(A,m): cardR ==> (ALL n. (A,n) : cardR --> n=m)"; -by (etac cardR.induct 1); - by (safe_tac (claset() addSEs [cardR_insertE])); -by (rename_tac "B b m" 1 THEN case_tac "a = b" 1); - by (subgoal_tac "A = B" 1); - by (blast_tac (claset() addEs [equalityE]) 2); - by (Blast_tac 1); -by (subgoal_tac "EX C. A = insert b C & B = insert a C" 1); - by (res_inst_tac [("x","A Int B")] exI 2); - by (blast_tac (claset() addEs [equalityE]) 2); -by (forw_inst_tac [("A","B")] cardR_SucD 1); -by (blast_tac (claset() addDs [lemma]) 1); -qed_spec_mp "cardR_determ"; - -Goal "(A,n) : cardR ==> finite(A)"; -by (etac cardR.induct 1); -by Auto_tac; -qed "cardR_imp_finite"; - -Goal "finite(A) ==> EX n. (A, n) : cardR"; -by (etac finite_induct 1); -by Auto_tac; -qed "finite_imp_cardR"; - -Goalw [card_def] "(A,n) : cardR ==> card A = n"; -by (blast_tac (claset() addIs [cardR_determ]) 1); -qed "card_equality"; - -Goalw [card_def] "card {} = 0"; -by (Blast_tac 1); -qed "card_empty"; -Addsimps [card_empty]; - -Goal "x ~: A \ -\ ==> ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)"; -by Auto_tac; -by (res_inst_tac [("A1", "A")] (finite_imp_cardR RS exE) 1); -by (force_tac (claset() addDs [cardR_imp_finite], simpset()) 1); -by (blast_tac (claset() addIs [cardR_determ]) 1); -val lemma = result(); - -Goalw [card_def] - "[| finite A; x ~: A |] ==> card (insert x A) = Suc(card A)"; -by (asm_simp_tac (simpset() addsimps [lemma]) 1); -by (rtac the_equality 1); -by (auto_tac (claset() addIs [finite_imp_cardR], - simpset() addcongs [conj_cong] - addsimps [symmetric card_def, - card_equality])); -qed "card_insert_disjoint"; -Addsimps [card_insert_disjoint]; - -(* Delete rules to do with cardR relation: obsolete *) -Delrules [cardR_emptyE]; -Delrules cardR.intrs; - -Goal "finite A ==> (card A = 0) = (A = {})"; -by Auto_tac; -by (dres_inst_tac [("a","x")] mk_disjoint_insert 1); -by (Clarify_tac 1); -by (rotate_tac ~1 1); -by Auto_tac; -qed "card_0_eq"; -Addsimps[card_0_eq]; - -Goal "finite A ==> card(insert x A) = (if x:A then card A else Suc(card(A)))"; -by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1); -qed "card_insert_if"; - -Goal "[| finite A; x: A |] ==> Suc (card (A-{x})) = card A"; -by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); -by (assume_tac 1); -by (Asm_simp_tac 1); -qed "card_Suc_Diff1"; - -Goal "[| finite A; x: A |] ==> card (A-{x}) = card A - 1"; -by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1 RS sym]) 1); -qed "card_Diff_singleton"; - -Goal "finite A ==> card (A-{x}) = (if x:A then card A - 1 else card A)"; -by (asm_simp_tac (simpset() addsimps [card_Diff_singleton]) 1); -qed "card_Diff_singleton_if"; - -Goal "finite A ==> card(insert x A) = Suc(card(A-{x}))"; -by (asm_simp_tac (simpset() addsimps [card_insert_if,card_Suc_Diff1]) 1); -qed "card_insert"; - -Goal "finite A ==> card A <= card (insert x A)"; -by (asm_simp_tac (simpset() addsimps [card_insert_if]) 1); -qed "card_insert_le"; - -Goal "finite B ==> ALL A. A <= B --> card B <= card A --> A = B"; -by (etac finite_induct 1); - by (Simp_tac 1); -by (Clarify_tac 1); -by (subgoal_tac "finite A & A-{x} <= F" 1); - by (blast_tac (claset() addIs [finite_subset]) 2); -by (dres_inst_tac [("x","A-{x}")] spec 1); -by (asm_full_simp_tac (simpset() addsimps [card_Diff_singleton_if] - addsplits [split_if_asm]) 1); -by (case_tac "card A" 1); -by Auto_tac; -qed_spec_mp "card_seteq"; - -Goalw [psubset_def] "[| finite B; A < B |] ==> card A < card B"; -by (simp_tac (simpset() addsimps [linorder_not_le RS sym]) 1); -by (blast_tac (claset() addDs [card_seteq]) 1); -qed "psubset_card_mono" ; - -Goal "[| finite B; A <= B |] ==> card A <= card B"; -by (case_tac "A=B" 1); - by (Asm_simp_tac 1); -by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1); -by (blast_tac (claset() addDs [card_seteq] addIs [order_less_imp_le]) 1); -qed "card_mono" ; - -Goal "[| finite A; finite B |] \ -\ ==> card A + card B = card (A Un B) + card (A Int B)"; -by (etac finite_induct 1); -by (Simp_tac 1); -by (asm_simp_tac (simpset() addsimps [insert_absorb, Int_insert_left]) 1); -qed "card_Un_Int"; - -Goal "[| finite A; finite B; A Int B = {} |] \ -\ ==> card (A Un B) = card A + card B"; -by (asm_simp_tac (simpset() addsimps [card_Un_Int]) 1); -qed "card_Un_disjoint"; - -Goal "[| finite A; B<=A |] ==> card A - card B = card (A - B)"; -by (subgoal_tac "(A-B) Un B = A" 1); -by (Blast_tac 2); -by (rtac (add_right_cancel RS iffD1) 1); -by (rtac (card_Un_disjoint RS subst) 1); -by (etac ssubst 4); -by (Blast_tac 3); -by (ALLGOALS - (asm_simp_tac - (simpset() addsimps [add_commute, not_less_iff_le, - add_diff_inverse, card_mono, finite_subset]))); -qed "card_Diff_subset"; - -Goal "[| finite A; x: A |] ==> card(A-{x}) < card A"; -by (rtac Suc_less_SucD 1); -by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1]) 1); -qed "card_Diff1_less"; - -Goal "[| finite A; x: A; y: A |] ==> card(A-{x}-{y}) < card A"; -by (case_tac "x=y" 1); -by (asm_simp_tac (simpset() addsimps [card_Diff1_less]) 1); -by (rtac less_trans 1); -by (ALLGOALS (force_tac (claset() addSIs [card_Diff1_less], simpset()))); -qed "card_Diff2_less"; - -Goal "finite A ==> card(A-{x}) <= card A"; -by (case_tac "x: A" 1); -by (ALLGOALS (asm_simp_tac (simpset() addsimps [card_Diff1_less, less_imp_le]))); -qed "card_Diff1_le"; - -Goal "[| finite B; A <= B; card A < card B |] ==> A < B"; -by (etac psubsetI 1); -by (Blast_tac 1); -qed "card_psubset"; - -(*** Cardinality of image ***) - -Goal "finite A ==> card (f ` A) <= card A"; -by (etac finite_induct 1); - by (Simp_tac 1); -by (asm_simp_tac (simpset() addsimps [le_SucI, finite_imageI, - card_insert_if]) 1); -qed "card_image_le"; - -Goal "finite(A) ==> inj_on f A --> card (f ` A) = card A"; -by (etac finite_induct 1); -by (ALLGOALS Asm_simp_tac); -by Safe_tac; -by (rewtac inj_on_def); -by (Blast_tac 1); -by (stac card_insert_disjoint 1); -by (etac finite_imageI 1); -by (Blast_tac 1); -by (Blast_tac 1); -qed_spec_mp "card_image"; - -Goal "[| finite A; f`A <= A; inj_on f A |] ==> f`A = A"; -by (asm_simp_tac (simpset() addsimps [card_seteq, card_image]) 1); -qed "endo_inj_surj"; - -(*** Cardinality of the Powerset ***) - -Goal "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A"; (* FIXME numeral 2 (!?) *) -by (etac finite_induct 1); -by (ALLGOALS (asm_simp_tac (simpset() addsimps [Pow_insert]))); -by (stac card_Un_disjoint 1); -by (EVERY (map (blast_tac (claset() addIs [finite_imageI])) [3,2,1])); -by (subgoal_tac "inj_on (insert x) (Pow F)" 1); -by (asm_simp_tac (simpset() addsimps [card_image, Pow_insert]) 1); -by (rewtac inj_on_def); -by (blast_tac (claset() addSEs [equalityE]) 1); -qed "card_Pow"; - - -(*Relates to equivalence classes. Based on a theorem of F. Kammueller's. - The "finite C" premise is redundant*) -Goal "finite C ==> finite (Union C) --> \ -\ (ALL c : C. k dvd card c) --> \ -\ (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) \ -\ --> k dvd card(Union C)"; -by (etac finite_induct 1); -by (ALLGOALS Asm_simp_tac); -by (Clarify_tac 1); -by (stac card_Un_disjoint 1); -by (ALLGOALS - (asm_full_simp_tac (simpset() - addsimps [dvd_add, disjoint_eq_subset_Compl]))); -by (thin_tac "ALL c:F. ?PP(c)" 1); -by (thin_tac "ALL c:F. ?PP(c) & ?QQ(c)" 1); -by (Clarify_tac 1); -by (ball_tac 1); -by (Blast_tac 1); -qed_spec_mp "dvd_partition"; - - -(*** foldSet ***) - -bind_thm ("empty_foldSetE", foldSet.mk_cases "({}, x) : foldSet f e"); - -AddSEs [empty_foldSetE]; -AddIs foldSet.intrs; - -Goal "[| (A-{x},y) : foldSet f e; x: A |] ==> (A, f x y) : foldSet f e"; -by (etac (insert_Diff RS subst) 1 THEN resolve_tac foldSet.intrs 1); -by Auto_tac; -qed "Diff1_foldSet"; - -Goal "(A, x) : foldSet f e ==> finite(A)"; -by (eresolve_tac [foldSet.induct] 1); -by Auto_tac; -qed "foldSet_imp_finite"; - -Addsimps [foldSet_imp_finite]; - - -Goal "finite(A) ==> EX x. (A, x) : foldSet f e"; -by (etac finite_induct 1); -by Auto_tac; -qed "finite_imp_foldSet"; - - -Open_locale "LC"; - -val f_lcomm = thm "lcomm"; - - -Goal "ALL A x. card(A) < n --> (A, x) : foldSet f e --> \ -\ (ALL y. (A, y) : foldSet f e --> y=x)"; -by (induct_tac "n" 1); -by (auto_tac (claset(), simpset() addsimps [less_Suc_eq])); -by (etac foldSet.elim 1); -by (Blast_tac 1); -by (etac foldSet.elim 1); -by (Blast_tac 1); -by (Clarify_tac 1); -(*force simplification of "card A < card (insert ...)"*) -by (etac rev_mp 1); -by (asm_simp_tac (simpset() addsimps [less_Suc_eq_le]) 1); -by (rtac impI 1); -(** LEVEL 10 **) -by (rename_tac "Aa xa ya Ab xb yb" 1 THEN case_tac "xa=xb" 1); - by (subgoal_tac "Aa = Ab" 1); - by (blast_tac (claset() addSEs [equalityE]) 2); - by (Blast_tac 1); -(*case xa ~= xb*) -by (subgoal_tac "Aa-{xb} = Ab-{xa} & xb : Aa & xa : Ab" 1); - by (blast_tac (claset() addSEs [equalityE]) 2); -by (Clarify_tac 1); -by (subgoal_tac "Aa = insert xb Ab - {xa}" 1); - by (Blast_tac 2); -(** LEVEL 20 **) -by (subgoal_tac "card Aa <= card Ab" 1); - by (rtac (Suc_le_mono RS subst) 2); - by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1]) 2); -by (res_inst_tac [("A1", "Aa-{xb}"), ("f1","f")] - (finite_imp_foldSet RS exE) 1); -by (blast_tac (claset() addIs [foldSet_imp_finite, finite_Diff]) 1); -by (ftac Diff1_foldSet 1 THEN assume_tac 1); -by (subgoal_tac "ya = f xb x" 1); - by (blast_tac (claset() delrules [equalityCE]) 2); -by (subgoal_tac "(Ab - {xa}, x) : foldSet f e" 1); - by (Asm_full_simp_tac 2); -by (subgoal_tac "yb = f xa x" 1); - by (blast_tac (claset() delrules [equalityCE] - addDs [Diff1_foldSet]) 2); -by (asm_simp_tac (simpset() addsimps [f_lcomm]) 1); -val lemma = result(); - - -Goal "[| (A, x) : foldSet f e; (A, y) : foldSet f e |] ==> y=x"; -by (blast_tac (claset() addIs [ObjectLogic.rulify lemma]) 1); -qed "foldSet_determ"; - -Goalw [fold_def] "(A,y) : foldSet f e ==> fold f e A = y"; -by (blast_tac (claset() addIs [foldSet_determ]) 1); -qed "fold_equality"; - -Goalw [fold_def] "fold f e {} = e"; -by (Blast_tac 1); -qed "fold_empty"; -Addsimps [fold_empty]; - - -Goal "x ~: A ==> \ -\ ((insert x A, v) : foldSet f e) = \ -\ (EX y. (A, y) : foldSet f e & v = f x y)"; -by Auto_tac; -by (res_inst_tac [("A1", "A"), ("f1","f")] (finite_imp_foldSet RS exE) 1); -by (force_tac (claset() addDs [foldSet_imp_finite], simpset()) 1); -by (blast_tac (claset() addIs [foldSet_determ]) 1); -val lemma = result(); - -Goalw [fold_def] - "[| finite A; x ~: A |] ==> fold f e (insert x A) = f x (fold f e A)"; -by (asm_simp_tac (simpset() addsimps [lemma]) 1); -by (rtac the_equality 1); -by (auto_tac (claset() addIs [finite_imp_foldSet], - simpset() addcongs [conj_cong] - addsimps [symmetric fold_def, - fold_equality])); -qed "fold_insert"; - -Goal "finite A ==> ALL e. f x (fold f e A) = fold f (f x e) A"; -by (etac finite_induct 1); -by (Simp_tac 1); -by (asm_simp_tac (simpset() addsimps [f_lcomm, fold_insert]) 1); -qed_spec_mp "fold_commute"; - -Goal "[| finite A; finite B |] \ -\ ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"; -by (etac finite_induct 1); -by (Simp_tac 1); -by (asm_simp_tac (simpset() addsimps [fold_insert, fold_commute, - Int_insert_left, insert_absorb]) 1); -qed "fold_nest_Un_Int"; - -Goal "[| finite A; finite B; A Int B = {} |] \ -\ ==> fold f e (A Un B) = fold f (fold f e B) A"; -by (asm_simp_tac (simpset() addsimps [fold_nest_Un_Int]) 1); -qed "fold_nest_Un_disjoint"; - -(* Delete rules to do with foldSet relation: obsolete *) -Delsimps [foldSet_imp_finite]; -Delrules [empty_foldSetE]; -Delrules foldSet.intrs; - -Close_locale "LC"; - -Open_locale "ACe"; - -(*We enter a more restrictive framework, with f :: ['a,'a] => 'a - instead of ['b,'a] => 'a - At present, none of these results are used!*) - -val f_ident = thm "ident"; -val f_commute = thm "commute"; -val f_assoc = thm "assoc"; - - -Goal "f x (f y z) = f y (f x z)"; -by (rtac (f_commute RS trans) 1); -by (rtac (f_assoc RS trans) 1); -by (rtac (f_commute RS arg_cong) 1); -qed "f_left_commute"; - -val f_ac = [f_assoc, f_commute, f_left_commute]; - -Goal "f e x = x"; -by (stac f_commute 1); -by (rtac f_ident 1); -qed "f_left_ident"; - -val f_idents = [f_left_ident, f_ident]; - -Goal "[| finite A; finite B |] \ -\ ==> f (fold f e A) (fold f e B) = \ -\ f (fold f e (A Un B)) (fold f e (A Int B))"; -by (etac finite_induct 1); -by (simp_tac (simpset() addsimps f_idents) 1); -by (asm_simp_tac (simpset() addsimps f_ac @ f_idents @ - [export fold_insert,insert_absorb, Int_insert_left]) 1); -qed "fold_Un_Int"; - -Goal "[| finite A; finite B; A Int B = {} |] \ -\ ==> fold f e (A Un B) = f (fold f e A) (fold f e B)"; -by (asm_simp_tac (simpset() addsimps fold_Un_Int::f_idents) 1); -qed "fold_Un_disjoint"; - -Goal - "[| finite A; finite B |] ==> A Int B = {} --> \ -\ fold (f o g) e (A Un B) = f (fold (f o g) e A) (fold (f o g) e B)"; -by (etac finite_induct 1); -by (simp_tac (simpset() addsimps f_idents) 1); -by (asm_full_simp_tac (simpset() addsimps f_ac @ f_idents @ - [export fold_insert,insert_absorb, Int_insert_left]) 1); -qed "fold_Un_disjoint2"; - -Close_locale "ACe"; - - -(*** setsum: generalized summation over a set ***) - -Goalw [setsum_def] "setsum f {} = 0"; -by (Simp_tac 1); -qed "setsum_empty"; -Addsimps [setsum_empty]; - -Goalw [setsum_def] - "!!f. [| finite F; a ~: F |] ==> \ -\ setsum f (insert a F) = f a + setsum f F"; -by (asm_simp_tac (simpset() addsimps [export fold_insert, - thm "plus_ac0_left_commute"]) 1); -qed "setsum_insert"; -Addsimps [setsum_insert]; - -Goal "setsum (%i. 0) A = 0"; -by (case_tac "finite A" 1); - by (asm_simp_tac (simpset() addsimps [setsum_def]) 2); -by (etac finite_induct 1); -by Auto_tac; -qed "setsum_0"; - -Goal "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"; -by (etac finite_induct 1); -by Auto_tac; -qed "setsum_eq_0_iff"; -Addsimps [setsum_eq_0_iff]; - -Goal "setsum f A = Suc n ==> EX a:A. 0 < f a"; -by (case_tac "finite A" 1); - by (asm_full_simp_tac (simpset() addsimps [setsum_def]) 2); -by (etac rev_mp 1); -by (etac finite_induct 1); -by Auto_tac; -qed "setsum_SucD"; - -(*Could allow many "card" proofs to be simplified*) -Goal "finite A ==> card A = setsum (%x. 1) A"; -by (etac finite_induct 1); -by Auto_tac; -qed "card_eq_setsum"; - -(*The reversed orientation looks more natural, but LOOPS as a simprule!*) -Goal "!!g. [| finite A; finite B |] \ -\ ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"; -by (etac finite_induct 1); -by (Simp_tac 1); -by (asm_full_simp_tac (simpset() addsimps (thms "plus_ac0") @ - [Int_insert_left, insert_absorb]) 1); -qed "setsum_Un_Int"; - -Goal "[| finite A; finite B; A Int B = {} |] \ -\ ==> setsum g (A Un B) = setsum g A + setsum g B"; -by (stac (setsum_Un_Int RS sym) 1); -by Auto_tac; -qed "setsum_Un_disjoint"; - -Goal "!!f::'a=>'b::plus_ac0. finite I \ -\ ==> (ALL i:I. finite (A i)) --> \ -\ (ALL i:I. ALL j:I. i~=j --> A i Int A j = {}) --> \ -\ setsum f (UNION I A) = setsum (%i. setsum f (A i)) I"; -by (etac finite_induct 1); - by (Simp_tac 1); -by (Clarify_tac 1); -by (subgoal_tac "ALL i:F. x ~= i" 1); - by (Blast_tac 2); -by (subgoal_tac "A x Int UNION F A = {}" 1); - by (Blast_tac 2); -by (asm_simp_tac (simpset() addsimps [setsum_Un_disjoint]) 1); -qed_spec_mp "setsum_UN_disjoint"; - -Goal "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"; -by (case_tac "finite A" 1); - by (asm_full_simp_tac (simpset() addsimps [setsum_def]) 2); -by (etac finite_induct 1); -by Auto_tac; -by (simp_tac (simpset() addsimps (thms "plus_ac0")) 1); -qed "setsum_addf"; - -(** For the natural numbers, we have subtraction **) - -Goal "[| finite A; finite B |] \ -\ ==> (setsum f (A Un B) :: nat) = \ -\ setsum f A + setsum f B - setsum f (A Int B)"; -by (stac (setsum_Un_Int RS sym) 1); -by Auto_tac; -qed "setsum_Un"; - -Goal "(setsum f (A-{a}) :: nat) = \ -\ (if a:A then setsum f A - f a else setsum f A)"; -by (case_tac "finite A" 1); - by (asm_full_simp_tac (simpset() addsimps [setsum_def]) 2); -by (etac finite_induct 1); -by (auto_tac (claset(), simpset() addsimps [insert_Diff_if])); -by (dres_inst_tac [("a","a")] mk_disjoint_insert 1); -by Auto_tac; -qed_spec_mp "setsum_diff1"; - -val prems = Goal - "[| A = B; !!x. x:B ==> f x = g x|] \ -\ ==> setsum f A = setsum g B"; -by (case_tac "finite B" 1); - by (asm_simp_tac (simpset() addsimps [setsum_def]@prems) 2); -by (simp_tac (simpset() addsimps prems) 1); -by (subgoal_tac - "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C" 1); - by (asm_simp_tac (simpset() addsimps prems) 1); -by (etac finite_induct 1); - by (Simp_tac 1); -by (asm_simp_tac (simpset() addsimps subset_insert_iff::prems) 1); -by (Clarify_tac 1); -by (subgoal_tac "finite C" 1); - by (blast_tac (claset() addDs [rotate_prems 1 finite_subset]) 2); -by (subgoal_tac "C = insert x (C-{x})" 1); - by (Blast_tac 2); -by (etac ssubst 1); -by (dtac spec 1); -by (mp_tac 1); -by (asm_full_simp_tac (simpset() addsimps Ball_def::prems) 1); -qed "setsum_cong"; - - -(*** Basic theorem about "choose". By Florian Kammueller, tidied by LCP ***) - -Goal "finite A ==> card {B. B <= A & card B = 0} = 1"; -by (asm_simp_tac (simpset() addcongs [conj_cong] - addsimps [finite_subset RS card_0_eq]) 1); -by (simp_tac (simpset() addcongs [rev_conj_cong]) 1); -qed "card_s_0_eq_empty"; - -Goal "[| finite M; x ~: M |] \ -\ ==> {s. s <= insert x M & card(s) = Suc k} \ -\ = {s. s <= M & card(s) = Suc k} Un \ -\ {s. EX t. t <= M & card(t) = k & s = insert x t}"; -by Safe_tac; -by (auto_tac (claset() addIs [finite_subset RS card_insert_disjoint], - simpset())); -by (dres_inst_tac [("x","xa - {x}")] spec 1); -by (subgoal_tac ("x ~: xa") 1); -by Auto_tac; -by (etac rev_mp 1 THEN stac card_Diff_singleton 1); -by (auto_tac (claset() addIs [finite_subset], simpset())); -qed "choose_deconstruct"; - -Goal "[| finite(A); finite(B); f`A <= B; inj_on f A |] \ -\ ==> card A <= card B"; -by (auto_tac (claset() addIs [card_mono], - simpset() addsimps [card_image RS sym])); -qed "card_inj_on_le"; - -Goal "[| finite A; finite B; \ -\ f`A <= B; inj_on f A; g`B <= A; inj_on g B |] \ -\ ==> card(A) = card(B)"; -by (auto_tac (claset() addIs [le_anti_sym,card_inj_on_le], simpset())); -qed "card_bij_eq"; - -Goal "[| finite A; x ~: A |] \ -\ ==> card{B. EX C. C <= A & card(C) = k & B = insert x C} = \ -\ card {B. B <= A & card(B) = k}"; -by (res_inst_tac [("f", "%s. s - {x}"), ("g","insert x")] card_bij_eq 1); -by (res_inst_tac [("B","Pow(insert x A)")] finite_subset 1); -by (res_inst_tac [("B","Pow(A)")] finite_subset 3); -by (REPEAT(Fast_tac 1)); -(* arity *) -by (auto_tac (claset() addSEs [equalityE], simpset() addsimps [inj_on_def])); -by (stac Diff_insert0 1); -by Auto_tac; -qed "constr_bij"; - -(* Main theorem: combinatorial theorem about number of subsets of a set *) -Goal "(ALL A. finite A --> card {B. B <= A & card B = k} = (card A choose k))"; -by (induct_tac "k" 1); - by (simp_tac (simpset() addsimps [card_s_0_eq_empty]) 1); -(* first 0 case finished *) -(* prepare finite set induction *) -by (rtac allI 1 THEN rtac impI 1); -(* second induction *) -by (etac finite_induct 1); -by (ALLGOALS - (asm_simp_tac (simpset() addcongs [conj_cong] - addsimps [card_s_0_eq_empty, choose_deconstruct]))); -by (stac card_Un_disjoint 1); - by (force_tac (claset(), simpset() addsimps [constr_bij]) 4); - by (Force_tac 3); - by (blast_tac (claset() addIs [finite_Pow_iff RS iffD2, - inst "B" "Pow (insert ?x ?F)" finite_subset]) 2); -by (blast_tac (claset() addIs [finite_Pow_iff RS iffD2 - RSN (2, finite_subset)]) 1); -qed "n_sub_lemma"; - -Goal "finite A ==> card {B. B <= A & card(B) = k} = ((card A) choose k)"; -by (asm_simp_tac (simpset() addsimps [n_sub_lemma]) 1); -qed "n_subsets"; diff -r d6913de7655f -r 2298d5b8e530 src/HOL/Finite.thy --- a/src/HOL/Finite.thy Thu Dec 06 00:37:59 2001 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,87 +0,0 @@ -(* Title: HOL/Finite.thy - ID: $Id$ - Author: Lawrence C Paulson & Tobias Nipkow - Copyright 1995 University of Cambridge & TU Muenchen - -Finite sets, their cardinality, and a fold functional. -*) - -Finite = Divides + Power + Inductive + SetInterval + - -consts Finites :: 'a set set - -inductive "Finites" - intrs - emptyI "{} : Finites" - insertI "A : Finites ==> insert a A : Finites" - -syntax finite :: 'a set => bool -translations "finite A" == "A : Finites" - -axclass finite < type - finite "finite UNIV" - -(* This definition, although traditional, is ugly to work with -constdefs - card :: 'a set => nat - "card A == LEAST n. ? f. A = {f i |i. i (insert a A, Suc n) : cardR" - -constdefs - card :: 'a set => nat - "card A == THE n. (A,n) : cardR" - -(* -A "fold" functional for finite sets. For n non-negative we have - fold f e {x1,...,xn} = f x1 (... (f xn e)) -where f is at least left-commutative. -*) - -consts foldSet :: "[['b,'a] => 'a, 'a] => ('b set * 'a) set" - -inductive "foldSet f e" - intrs - emptyI "({}, e) : foldSet f e" - - insertI "[| x ~: A; (A,y) : foldSet f e |] - ==> (insert x A, f x y) : foldSet f e" - -constdefs - fold :: "[['b,'a] => 'a, 'a, 'b set] => 'a" - "fold f e A == THE x. (A,x) : foldSet f e" - - setsum :: "('a => 'b) => 'a set => 'b::plus_ac0" - "setsum f A == if finite A then fold (op+ o f) 0 A else 0" - -syntax - "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\\_:_. _" [0, 51, 10] 10) -syntax (xsymbols) - "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\\_\\_. _" [0, 51, 10] 10) -translations - "\\i:A. b" == "setsum (%i. b) A" (* Beware of argument permutation! *) - - -locale LC = - fixes - f :: ['b,'a] => 'a - assumes - lcomm "f x (f y z) = f y (f x z)" - -locale ACe = - fixes - f :: ['a,'a] => 'a - e :: 'a - assumes - ident "f x e = x" - commute "f x y = f y x" - assoc "f (f x y) z = f x (f y z)" - -end diff -r d6913de7655f -r 2298d5b8e530 src/HOL/Finite_Set.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Finite_Set.ML Thu Dec 06 00:38:55 2001 +0100 @@ -0,0 +1,128 @@ + +(* legacy ML bindings *) + +structure Finites = +struct + val intrs = thms "Finites.intros"; + val elims = thms "Finites.cases"; + val elim = thm "Finites.cases"; + val induct = thm "Finites.induct"; + val mk_cases = InductivePackage.the_mk_cases (the_context ()) "Finite_Set.Finites"; + val [emptyI, insertI] = thms "Finites.intros"; +end; + +structure cardR = +struct + val intrs = thms "cardR.intros"; + val elims = thms "cardR.cases"; + val elim = thm "cardR.cases"; + val induct = thm "cardR.induct"; + val mk_cases = InductivePackage.the_mk_cases (the_context ()) "Finite_Set.cardR"; + val [EmptyI, InsertI] = thms "cardR.intros"; +end; + +structure foldSet = +struct + val intrs = thms "foldSet.intros"; + val elims = thms "foldSet.cases"; + val elim = thm "foldSet.cases"; + val induct = thm "foldSet.induct"; + val mk_cases = InductivePackage.the_mk_cases (the_context ()) "Finite_Set.foldSet"; + val [emptyI, insertI] = thms "foldSet.intros"; +end; + +val Diff1_foldSet = thm "Diff1_foldSet"; +val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite"; +val cardR_SucD = thm "cardR_SucD"; +val cardR_determ = thm "cardR_determ"; +val cardR_emptyE = thm "cardR_emptyE"; +val cardR_imp_finite = thm "cardR_imp_finite"; +val cardR_insertE = thm "cardR_insertE"; +val card_0_eq = thm "card_0_eq"; +val card_Diff1_le = thm "card_Diff1_le"; +val card_Diff1_less = thm "card_Diff1_less"; +val card_Diff2_less = thm "card_Diff2_less"; +val card_Diff_singleton = thm "card_Diff_singleton"; +val card_Diff_singleton_if = thm "card_Diff_singleton_if"; +val card_Diff_subset = thm "card_Diff_subset"; +val card_Pow = thm "card_Pow"; +val card_Suc_Diff1 = thm "card_Suc_Diff1"; +val card_Un_Int = thm "card_Un_Int"; +val card_Un_disjoint = thm "card_Un_disjoint"; +val card_bij_eq = thm "card_bij_eq"; +val card_def = thm "card_def"; +val card_empty = thm "card_empty"; +val card_eq_setsum = thm "card_eq_setsum"; +val card_equality = thm "card_equality"; +val card_image = thm "card_image"; +val card_image_le = thm "card_image_le"; +val card_inj_on_le = thm "card_inj_on_le"; +val card_insert = thm "card_insert"; +val card_insert_disjoint = thm "card_insert_disjoint"; +val card_insert_if = thm "card_insert_if"; +val card_insert_le = thm "card_insert_le"; +val card_mono = thm "card_mono"; +val card_psubset = thm "card_psubset"; +val card_s_0_eq_empty = thm "card_s_0_eq_empty"; +val card_seteq = thm "card_seteq"; +val choose_deconstruct = thm "choose_deconstruct"; +val constr_bij = thm "constr_bij"; +val dvd_partition = thm "dvd_partition"; +val empty_foldSetE = thm "empty_foldSetE"; +val endo_inj_surj = thm "endo_inj_surj"; +val finite = thm "finite"; +val finiteI = thm "finiteI"; +val finite_Diff = thm "finite_Diff"; +val finite_Diff_insert = thm "finite_Diff_insert"; +val finite_Field = thm "finite_Field"; +val finite_Int = thm "finite_Int"; +val finite_Pow_iff = thm "finite_Pow_iff"; +val finite_Prod_UNIV = thm "finite_Prod_UNIV"; +val finite_SigmaI = thm "finite_SigmaI"; +val finite_UN = thm "finite_UN"; +val finite_UN_I = thm "finite_UN_I"; +val finite_Un = thm "finite_Un"; +val finite_UnI = thm "finite_UnI"; +val finite_atMost = thm "finite_atMost"; +val finite_converse = thm "finite_converse"; +val finite_empty_induct = thm "finite_empty_induct"; +val finite_imageD = thm "finite_imageD"; +val finite_imageI = thm "finite_imageI"; +val finite_imp_cardR = thm "finite_imp_cardR"; +val finite_imp_foldSet = thm "finite_imp_foldSet"; +val finite_induct = thm "finite_induct"; +val finite_insert = thm "finite_insert"; +val finite_lessThan = thm "finite_lessThan"; +val finite_range_imageI = thm "finite_range_imageI"; +val finite_subset = thm "finite_subset"; +val finite_subset_induct = thm "finite_subset_induct"; +val finite_trancl = thm "finite_trancl"; +val foldSet_determ = thm "foldSet_determ"; +val foldSet_imp_finite = thm "foldSet_imp_finite"; +val fold_Un_Int = thm "fold_Un_Int"; +val fold_Un_disjoint = thm "fold_Un_disjoint"; +val fold_Un_disjoint2 = thm "fold_Un_disjoint2"; +val fold_commute = thm "fold_commute"; +val fold_def = thm "fold_def"; +val fold_empty = thm "fold_empty"; +val fold_equality = thm "fold_equality"; +val fold_insert = thm "fold_insert"; +val fold_nest_Un_Int = thm "fold_nest_Un_Int"; +val fold_nest_Un_disjoint = thm "fold_nest_Un_disjoint"; +val n_sub_lemma = thm "n_sub_lemma"; +val n_subsets = thm "n_subsets"; +val psubset_card_mono = thm "psubset_card_mono"; +val setsum_0 = thm "setsum_0"; +val setsum_SucD = thm "setsum_SucD"; +val setsum_UN_disjoint = thm "setsum_UN_disjoint"; +val setsum_Un = thm "setsum_Un"; +val setsum_Un_Int = thm "setsum_Un_Int"; +val setsum_Un_disjoint = thm "setsum_Un_disjoint"; +val setsum_addf = thm "setsum_addf"; +val setsum_cong = thm "setsum_cong"; +val setsum_def = thm "setsum_def"; +val setsum_diff1 = thm "setsum_diff1"; +val setsum_empty = thm "setsum_empty"; +val setsum_eq_0_iff = thm "setsum_eq_0_iff"; +val setsum_insert = thm "setsum_insert"; +val trancl_subset_Field2 = thm "trancl_subset_Field2"; diff -r d6913de7655f -r 2298d5b8e530 src/HOL/Finite_Set.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Finite_Set.thy Thu Dec 06 00:38:55 2001 +0100 @@ -0,0 +1,947 @@ +(* Title: HOL/Finite_Set.thy + ID: $Id$ + Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel + License: GPL (GNU GENERAL PUBLIC LICENSE) +*) + +header {* Finite sets *} + +theory Finite_Set = Divides + Power + Inductive + SetInterval: + +subsection {* Collection of finite sets *} + +consts Finites :: "'a set set" + +inductive Finites + intros + emptyI [simp, intro!]: "{} : Finites" + insertI [simp, intro!]: "A : Finites ==> insert a A : Finites" + +syntax + finite :: "'a set => bool" +translations + "finite A" == "A : Finites" + +axclass finite \ type + finite: "finite UNIV" + + +lemma finite_induct [case_names empty insert, induct set: Finites]: + "finite F ==> + P {} ==> (!!F x. finite F ==> x \ F ==> P F ==> P (insert x F)) ==> P F" + -- {* Discharging @{text "x \ F"} entails extra work. *} +proof - + assume "P {}" and insert: "!!F x. finite F ==> x \ F ==> P F ==> P (insert x F)" + assume "finite F" + thus "P F" + proof induct + show "P {}" . + fix F x assume F: "finite F" and P: "P F" + show "P (insert x F)" + proof cases + assume "x \ F" + hence "insert x F = F" by (rule insert_absorb) + with P show ?thesis by (simp only:) + next + assume "x \ F" + from F this P show ?thesis by (rule insert) + qed + qed +qed + +lemma finite_subset_induct [consumes 2, case_names empty insert]: + "finite F ==> F \ A ==> + P {} ==> (!!F a. finite F ==> a \ A ==> a \ F ==> P F ==> P (insert a F)) ==> + P F" +proof - + assume "P {}" and insert: "!!F a. finite F ==> a \ A ==> a \ F ==> P F ==> P (insert a F)" + assume "finite F" + thus "F \ A ==> P F" + proof induct + show "P {}" . + fix F x assume "finite F" and "x \ F" + and P: "F \ A ==> P F" and i: "insert x F \ A" + show "P (insert x F)" + proof (rule insert) + from i show "x \ A" by blast + from i have "F \ A" by blast + with P show "P F" . + qed + qed +qed + +lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" + -- {* The union of two finite sets is finite. *} + by (induct set: Finites) simp_all + +lemma finite_subset: "A \ B ==> finite B ==> finite A" + -- {* Every subset of a finite set is finite. *} +proof - + assume "finite B" + thus "!!A. A \ B ==> finite A" + proof induct + case empty + thus ?case by simp + next + case (insert F x A) + have A: "A \ insert x F" and r: "A - {x} \ F ==> finite (A - {x})" . + show "finite A" + proof cases + assume x: "x \ A" + with A have "A - {x} \ F" by (simp add: subset_insert_iff) + with r have "finite (A - {x})" . + hence "finite (insert x (A - {x}))" .. + also have "insert x (A - {x}) = A" by (rule insert_Diff) + finally show ?thesis . + next + show "A \ F ==> ?thesis" . + assume "x \ A" + with A show "A \ F" by (simp add: subset_insert_iff) + qed + qed +qed + +lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" + by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) + +lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" + -- {* The converse obviously fails. *} + by (blast intro: finite_subset) + +lemma finite_insert [simp]: "finite (insert a A) = finite A" + apply (subst insert_is_Un) + apply (simp only: finite_Un) + apply blast + done + +lemma finite_imageI: "finite F ==> finite (h ` F)" + -- {* The image of a finite set is finite. *} + by (induct set: Finites) simp_all + +lemma finite_range_imageI: + "finite (range g) ==> finite (range (%x. f (g x)))" + apply (drule finite_imageI) + apply simp + done + +lemma finite_empty_induct: + "finite A ==> + P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}" +proof - + assume "finite A" + and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" + have "P (A - A)" + proof - + fix c b :: "'a set" + presume c: "finite c" and b: "finite b" + and P1: "P b" and P2: "!!x y. finite y ==> x \ y ==> P y ==> P (y - {x})" + from c show "c \ b ==> P (b - c)" + proof induct + case empty + from P1 show ?case by simp + next + case (insert F x) + have "P (b - F - {x})" + proof (rule P2) + from _ b show "finite (b - F)" by (rule finite_subset) blast + from insert show "x \ b - F" by simp + from insert show "P (b - F)" by simp + qed + also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) + finally show ?case . + qed + next + show "A \ A" .. + qed + thus "P {}" by simp +qed + +lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)" + by (rule Diff_subset [THEN finite_subset]) + +lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" + apply (subst Diff_insert) + apply (case_tac "a : A - B") + apply (rule finite_insert [symmetric, THEN trans]) + apply (subst insert_Diff) + apply simp_all + done + + +lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" +proof - + have aux: "!!A. finite (A - {}) = finite A" by simp + fix B :: "'a set" + assume "finite B" + thus "!!A. f`A = B ==> inj_on f A ==> finite A" + apply induct + apply simp + apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") + apply clarify + apply (simp (no_asm_use) add: inj_on_def) + apply (blast dest!: aux [THEN iffD1]) + apply atomize + apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) + apply (frule subsetD [OF equalityD2 insertI1]) + apply clarify + apply (rule_tac x = xa in bexI) + apply (simp_all add: inj_on_image_set_diff) + done +qed (rule refl) + + +subsubsection {* The finite UNION of finite sets *} + +lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" + by (induct set: Finites) simp_all + +text {* + Strengthen RHS to + @{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x ~= {}})"}? + + We'd need to prove + @{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x ~= {}}"} + by induction. *} + +lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" + by (blast intro: finite_UN_I finite_subset) + + +subsubsection {* Sigma of finite sets *} + +lemma finite_SigmaI [simp]: + "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" + by (unfold Sigma_def) (blast intro!: finite_UN_I) + +lemma finite_Prod_UNIV: + "finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" + apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") + apply (erule ssubst) + apply (erule finite_SigmaI) + apply auto + done + +instance unit :: finite +proof + have "finite {()}" by simp + also have "{()} = UNIV" by auto + finally show "finite (UNIV :: unit set)" . +qed + +instance * :: (finite, finite) finite +proof + show "finite (UNIV :: ('a \ 'b) set)" + proof (rule finite_Prod_UNIV) + show "finite (UNIV :: 'a set)" by (rule finite) + show "finite (UNIV :: 'b set)" by (rule finite) + qed +qed + + +subsubsection {* The powerset of a finite set *} + +lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" +proof + assume "finite (Pow A)" + with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast + thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp +next + assume "finite A" + thus "finite (Pow A)" + by induct (simp_all add: finite_UnI finite_imageI Pow_insert) +qed + +lemma finite_converse [iff]: "finite (r^-1) = finite r" + apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") + apply simp + apply (rule iffI) + apply (erule finite_imageD [unfolded inj_on_def]) + apply (simp split add: split_split) + apply (erule finite_imageI) + apply (simp add: converse_def image_def) + apply auto + apply (rule bexI) + prefer 2 apply assumption + apply simp + done + +lemma finite_lessThan [iff]: (fixes k :: nat) "finite {..k(}" + by (induct k) (simp_all add: lessThan_Suc) + +lemma finite_atMost [iff]: (fixes k :: nat) "finite {..k}" + by (induct k) (simp_all add: atMost_Suc) + +lemma bounded_nat_set_is_finite: + "(ALL i:N. i < (n::nat)) ==> finite N" + -- {* A bounded set of natural numbers is finite. *} + apply (rule finite_subset) + apply (rule_tac [2] finite_lessThan) + apply auto + done + + +subsubsection {* Finiteness of transitive closure *} + +text {* (Thanks to Sidi Ehmety) *} + +lemma finite_Field: "finite r ==> finite (Field r)" + -- {* A finite relation has a finite field (@{text "= domain \ range"}. *} + apply (induct set: Finites) + apply (auto simp add: Field_def Domain_insert Range_insert) + done + +lemma trancl_subset_Field2: "r^+ <= Field r \ Field r" + apply clarify + apply (erule trancl_induct) + apply (auto simp add: Field_def) + done + +lemma finite_trancl: "finite (r^+) = finite r" + apply auto + prefer 2 + apply (rule trancl_subset_Field2 [THEN finite_subset]) + apply (rule finite_SigmaI) + prefer 3 + apply (blast intro: r_into_trancl finite_subset) + apply (auto simp add: finite_Field) + done + + +subsection {* Finite cardinality *} + +text {* + This definition, although traditional, is ugly to work with: @{text + "card A == LEAST n. EX f. A = {f i | i. i < n}"}. Therefore we have + switched to an inductive one: +*} + +consts cardR :: "('a set \ nat) set" + +inductive cardR + intros + EmptyI: "({}, 0) : cardR" + InsertI: "(A, n) : cardR ==> a \ A ==> (insert a A, Suc n) : cardR" + +constdefs + card :: "'a set => nat" + "card A == THE n. (A, n) : cardR" + +inductive_cases cardR_emptyE: "({}, n) : cardR" +inductive_cases cardR_insertE: "(insert a A,n) : cardR" + +lemma cardR_SucD: "(A, n) : cardR ==> a : A --> (EX m. n = Suc m)" + by (induct set: cardR) simp_all + +lemma cardR_determ_aux1: + "(A, m): cardR ==> (!!n a. m = Suc n ==> a:A ==> (A - {a}, n) : cardR)" + apply (induct set: cardR) + apply auto + apply (simp add: insert_Diff_if) + apply auto + apply (drule cardR_SucD) + apply (blast intro!: cardR.intros) + done + +lemma cardR_determ_aux2: "(insert a A, Suc m) : cardR ==> a \ A ==> (A, m) : cardR" + by (drule cardR_determ_aux1) auto + +lemma cardR_determ: "(A, m): cardR ==> (!!n. (A, n) : cardR ==> n = m)" + apply (induct set: cardR) + apply (safe elim!: cardR_emptyE cardR_insertE) + apply (rename_tac B b m) + apply (case_tac "a = b") + apply (subgoal_tac "A = B") + prefer 2 apply (blast elim: equalityE) + apply blast + apply (subgoal_tac "EX C. A = insert b C & B = insert a C") + prefer 2 + apply (rule_tac x = "A Int B" in exI) + apply (blast elim: equalityE) + apply (frule_tac A = B in cardR_SucD) + apply (blast intro!: cardR.intros dest!: cardR_determ_aux2) + done + +lemma cardR_imp_finite: "(A, n) : cardR ==> finite A" + by (induct set: cardR) simp_all + +lemma finite_imp_cardR: "finite A ==> EX n. (A, n) : cardR" + by (induct set: Finites) (auto intro!: cardR.intros) + +lemma card_equality: "(A,n) : cardR ==> card A = n" + by (unfold card_def) (blast intro: cardR_determ) + +lemma card_empty [simp]: "card {} = 0" + by (unfold card_def) (blast intro!: cardR.intros elim!: cardR_emptyE) + +lemma card_insert_disjoint [simp]: + "finite A ==> x \ A ==> card (insert x A) = Suc(card A)" +proof - + assume x: "x \ A" + hence aux: "!!n. ((insert x A, n) : cardR) = (EX m. (A, m) : cardR & n = Suc m)" + apply (auto intro!: cardR.intros) + apply (rule_tac A1 = A in finite_imp_cardR [THEN exE]) + apply (force dest: cardR_imp_finite) + apply (blast intro!: cardR.intros intro: cardR_determ) + done + assume "finite A" + thus ?thesis + apply (simp add: card_def aux) + apply (rule the_equality) + apply (auto intro: finite_imp_cardR + cong: conj_cong simp: card_def [symmetric] card_equality) + done +qed + +lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})" + apply auto + apply (drule_tac a = x in mk_disjoint_insert) + apply clarify + apply (rotate_tac -1) + apply auto + done + +lemma card_insert_if: + "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))" + by (simp add: insert_absorb) + +lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A" + apply (rule_tac t = A in insert_Diff [THEN subst]) + apply assumption + apply simp + done + +lemma card_Diff_singleton: + "finite A ==> x: A ==> card (A - {x}) = card A - 1" + by (simp add: card_Suc_Diff1 [symmetric]) + +lemma card_Diff_singleton_if: + "finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)" + by (simp add: card_Diff_singleton) + +lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))" + by (simp add: card_insert_if card_Suc_Diff1) + +lemma card_insert_le: "finite A ==> card A <= card (insert x A)" + by (simp add: card_insert_if) + +lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)" + apply (induct set: Finites) + apply simp + apply clarify + apply (subgoal_tac "finite A & A - {x} <= F") + prefer 2 apply (blast intro: finite_subset) + apply atomize + apply (drule_tac x = "A - {x}" in spec) + apply (simp add: card_Diff_singleton_if split add: split_if_asm) + apply (case_tac "card A") + apply auto + done + +lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B" + apply (simp add: psubset_def linorder_not_le [symmetric]) + apply (blast dest: card_seteq) + done + +lemma card_mono: "finite B ==> A <= B ==> card A <= card B" + apply (case_tac "A = B") + apply simp + apply (simp add: linorder_not_less [symmetric]) + apply (blast dest: card_seteq intro: order_less_imp_le) + done + +lemma card_Un_Int: "finite A ==> finite B + ==> card A + card B = card (A Un B) + card (A Int B)" + apply (induct set: Finites) + apply simp + apply (simp add: insert_absorb Int_insert_left) + done + +lemma card_Un_disjoint: "finite A ==> finite B + ==> A Int B = {} ==> card (A Un B) = card A + card B" + by (simp add: card_Un_Int) + +lemma card_Diff_subset: + "finite A ==> B <= A ==> card A - card B = card (A - B)" + apply (subgoal_tac "(A - B) Un B = A") + prefer 2 apply blast + apply (rule add_right_cancel [THEN iffD1]) + apply (rule card_Un_disjoint [THEN subst]) + apply (erule_tac [4] ssubst) + prefer 3 apply blast + apply (simp_all add: add_commute not_less_iff_le + add_diff_inverse card_mono finite_subset) + done + +lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A" + apply (rule Suc_less_SucD) + apply (simp add: card_Suc_Diff1) + done + +lemma card_Diff2_less: + "finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A" + apply (case_tac "x = y") + apply (simp add: card_Diff1_less) + apply (rule less_trans) + prefer 2 apply (auto intro!: card_Diff1_less) + done + +lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A" + apply (case_tac "x : A") + apply (simp_all add: card_Diff1_less less_imp_le) + done + +lemma card_psubset: "finite B ==> A \ B ==> card A < card B ==> A < B" + apply (erule psubsetI) + apply blast + done + + +subsubsection {* Cardinality of image *} + +lemma card_image_le: "finite A ==> card (f ` A) <= card A" + apply (induct set: Finites) + apply simp + apply (simp add: le_SucI finite_imageI card_insert_if) + done + +lemma card_image: "finite A ==> inj_on f A ==> card (f ` A) = card A" + apply (induct set: Finites) + apply simp_all + apply atomize + apply safe + apply (unfold inj_on_def) + apply blast + apply (subst card_insert_disjoint) + apply (erule finite_imageI) + apply blast + apply blast + done + +lemma endo_inj_surj: "finite A ==> f ` A \ A ==> inj_on f A ==> f ` A = A" + by (simp add: card_seteq card_image) + + +subsubsection {* Cardinality of the Powerset *} + +lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *) + apply (induct set: Finites) + apply (simp_all add: Pow_insert) + apply (subst card_Un_disjoint) + apply blast + apply (blast intro: finite_imageI) + apply blast + apply (subgoal_tac "inj_on (insert x) (Pow F)") + apply (simp add: card_image Pow_insert) + apply (unfold inj_on_def) + apply (blast elim!: equalityE) + done + +text {* + \medskip Relates to equivalence classes. Based on a theorem of + F. Kammüller's. The @{prop "finite C"} premise is redundant. +*} + +lemma dvd_partition: + "finite C ==> finite (Union C) ==> + ALL c : C. k dvd card c ==> + (ALL c1: C. ALL c2: C. c1 ~= c2 --> c1 Int c2 = {}) ==> + k dvd card (Union C)" + apply (induct set: Finites) + apply simp_all + apply clarify + apply (subst card_Un_disjoint) + apply (auto simp add: dvd_add disjoint_eq_subset_Compl) + done + + +subsection {* A fold functional for finite sets *} + +text {* + For @{text n} non-negative we have @{text "fold f e {x1, ..., xn} = + f x1 (... (f xn e))"} where @{text f} is at least left-commutative. +*} + +consts + foldSet :: "('b => 'a => 'a) => 'a => ('b set \ 'a) set" + +inductive "foldSet f e" + intros + emptyI [intro]: "({}, e) : foldSet f e" + insertI [intro]: "x \ A ==> (A, y) : foldSet f e ==> (insert x A, f x y) : foldSet f e" + +inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f e" + +constdefs + fold :: "('b => 'a => 'a) => 'a => 'b set => 'a" + "fold f e A == THE x. (A, x) : foldSet f e" + +lemma Diff1_foldSet: "(A - {x}, y) : foldSet f e ==> x: A ==> (A, f x y) : foldSet f e" + apply (erule insert_Diff [THEN subst], rule foldSet.intros) + apply auto + done + +lemma foldSet_imp_finite [simp]: "(A, x) : foldSet f e ==> finite A" + by (induct set: foldSet) auto + +lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f e" + by (induct set: Finites) auto + + +subsubsection {* Left-commutative operations *} + +locale LC = + fixes f :: "'b => 'a => 'a" (infixl "\" 70) + assumes left_commute: "x \ (y \ z) = y \ (x \ z)" + +lemma (in LC) foldSet_determ_aux: + "ALL A x. card A < n --> (A, x) : foldSet f e --> + (ALL y. (A, y) : foldSet f e --> y = x)" + apply (induct n) + apply (auto simp add: less_Suc_eq) + apply (erule foldSet.cases) + apply blast + apply (erule foldSet.cases) + apply blast + apply clarify + txt {* force simplification of @{text "card A < card (insert ...)"}. *} + apply (erule rev_mp) + apply (simp add: less_Suc_eq_le) + apply (rule impI) + apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb") + apply (subgoal_tac "Aa = Ab") + prefer 2 apply (blast elim!: equalityE) + apply blast + txt {* case @{prop "xa \ xb"}. *} + apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb : Aa & xa : Ab") + prefer 2 apply (blast elim!: equalityE) + apply clarify + apply (subgoal_tac "Aa = insert xb Ab - {xa}") + prefer 2 apply blast + apply (subgoal_tac "card Aa <= card Ab") + prefer 2 + apply (rule Suc_le_mono [THEN subst]) + apply (simp add: card_Suc_Diff1) + apply (rule_tac A1 = "Aa - {xb}" and f1 = f in finite_imp_foldSet [THEN exE]) + apply (blast intro: foldSet_imp_finite finite_Diff) + apply (frule (1) Diff1_foldSet) + apply (subgoal_tac "ya = f xb x") + prefer 2 apply (blast del: equalityCE) + apply (subgoal_tac "(Ab - {xa}, x) : foldSet f e") + prefer 2 apply simp + apply (subgoal_tac "yb = f xa x") + prefer 2 apply (blast del: equalityCE dest: Diff1_foldSet) + apply (simp (no_asm_simp) add: left_commute) + done + +lemma (in LC) foldSet_determ: "(A, x) : foldSet f e ==> (A, y) : foldSet f e ==> y = x" + by (blast intro: foldSet_determ_aux [rule_format]) + +lemma (in LC) fold_equality: "(A, y) : foldSet f e ==> fold f e A = y" + by (unfold fold_def) (blast intro: foldSet_determ) + +lemma fold_empty [simp]: "fold f e {} = e" + by (unfold fold_def) blast + +lemma (in LC) fold_insert_aux: "x \ A ==> + ((insert x A, v) : foldSet f e) = + (EX y. (A, y) : foldSet f e & v = f x y)" + apply auto + apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE]) + apply (fastsimp dest: foldSet_imp_finite) + apply (blast intro: foldSet_determ) + done + +lemma (in LC) fold_insert: + "finite A ==> x \ A ==> fold f e (insert x A) = f x (fold f e A)" + apply (unfold fold_def) + apply (simp add: fold_insert_aux) + apply (rule the_equality) + apply (auto intro: finite_imp_foldSet + cong add: conj_cong simp add: fold_def [symmetric] fold_equality) + done + +lemma (in LC) fold_commute: "finite A ==> (!!e. f x (fold f e A) = fold f (f x e) A)" + apply (induct set: Finites) + apply simp + apply (simp add: left_commute fold_insert) + done + +lemma (in LC) fold_nest_Un_Int: + "finite A ==> finite B + ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)" + apply (induct set: Finites) + apply simp + apply (simp add: fold_insert fold_commute Int_insert_left insert_absorb) + done + +lemma (in LC) fold_nest_Un_disjoint: + "finite A ==> finite B ==> A Int B = {} + ==> fold f e (A Un B) = fold f (fold f e B) A" + by (simp add: fold_nest_Un_Int) + +declare foldSet_imp_finite [simp del] + empty_foldSetE [rule del] foldSet.intros [rule del] + -- {* Delete rules to do with @{text foldSet} relation. *} + + + +subsubsection {* Commutative monoids *} + +text {* + We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"} + instead of @{text "'b => 'a => 'a"}. +*} + +locale ACe = + fixes f :: "'a => 'a => 'a" (infixl "\" 70) + and e :: 'a + assumes ident [simp]: "x \ e = x" + and commute: "x \ y = y \ x" + and assoc: "(x \ y) \ z = x \ (y \ z)" + +lemma (in ACe) left_commute: "x \ (y \ z) = y \ (x \ z)" +proof - + have "x \ (y \ z) = (y \ z) \ x" by (simp only: commute) + also have "... = y \ (z \ x)" by (simp only: assoc) + also have "z \ x = x \ z" by (simp only: commute) + finally show ?thesis . +qed + +lemma (in ACe) + AC: "(x \ y) \ z = x \ (y \ z)" "x \ y = y \ x" "x \ (y \ z) = y \ (x \ z)" + by (rule assoc, rule commute, rule left_commute) (* FIXME localize "lemmas" (!??) *) + +lemma (in ACe [simp]) left_ident: "e \ x = x" +proof - + have "x \ e = x" by (rule ident) + thus ?thesis by (subst commute) +qed + +lemma (in ACe) fold_Un_Int: + "finite A ==> finite B ==> + fold f e A \ fold f e B = fold f e (A Un B) \ fold f e (A Int B)" + apply (induct set: Finites) + apply simp + apply (simp add: AC fold_insert insert_absorb Int_insert_left) + done + +lemma (in ACe) fold_Un_disjoint: + "finite A ==> finite B ==> A Int B = {} ==> + fold f e (A Un B) = fold f e A \ fold f e B" + by (simp add: fold_Un_Int) + +lemma (in ACe) fold_Un_disjoint2: + "finite A ==> finite B ==> A Int B = {} ==> + fold (f o g) e (A Un B) = fold (f o g) e A \ fold (f o g) e B" +proof - + assume b: "finite B" + assume "finite A" + thus "A Int B = {} ==> + fold (f o g) e (A Un B) = fold (f o g) e A \ fold (f o g) e B" + proof induct + case empty + thus ?case by simp + next + case (insert F x) + have "fold (f \ g) e (insert x F \ B) = fold (f \ g) e (insert x (F \ B))" + by simp + also have "... = (f \ g) x (fold (f \ g) e (F \ B))" + by (rule fold_insert) (insert b insert, auto simp add: left_commute) (* FIXME import of fold_insert (!?) *) + also from insert have "fold (f \ g) e (F \ B) = + fold (f \ g) e F \ fold (f \ g) e B" by blast + also have "(f \ g) x ... = (f \ g) x (fold (f \ g) e F) \ fold (f \ g) e B" + by (simp add: AC) + also have "(f \ g) x (fold (f \ g) e F) = fold (f \ g) e (insert x F)" + by (rule fold_insert [symmetric]) (insert b insert, auto simp add: left_commute) + finally show ?case . + qed +qed + + +subsection {* Generalized summation over a set *} + +constdefs + setsum :: "('a => 'b) => 'a set => 'b::plus_ac0" + "setsum f A == if finite A then fold (op + o f) 0 A else 0" + +syntax + "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\_:_. _" [0, 51, 10] 10) +syntax (xsymbols) + "_setsum" :: "idt => 'a set => 'b => 'b::plus_ac0" ("\_\_. _" [0, 51, 10] 10) +translations + "\i:A. b" == "setsum (%i. b) A" -- {* Beware of argument permutation! *} + + +lemma setsum_empty [simp]: "setsum f {} = 0" + by (simp add: setsum_def) + +lemma setsum_insert [simp]: + "finite F ==> a \ F ==> setsum f (insert a F) = f a + setsum f F" + by (simp add: setsum_def fold_insert plus_ac0_left_commute) + +lemma setsum_0: "setsum (\i. 0) A = 0" + apply (case_tac "finite A") + prefer 2 apply (simp add: setsum_def) + apply (erule finite_induct) + apply auto + done + +lemma setsum_eq_0_iff [simp]: + "finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))" + by (induct set: Finites) auto + +lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a" + apply (case_tac "finite A") + prefer 2 apply (simp add: setsum_def) + apply (erule rev_mp) + apply (erule finite_induct) + apply auto + done + +lemma card_eq_setsum: "finite A ==> card A = setsum (\x. 1) A" + -- {* Could allow many @{text "card"} proofs to be simplified. *} + by (induct set: Finites) auto + +lemma setsum_Un_Int: "finite A ==> finite B + ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" + -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} + apply (induct set: Finites) + apply simp + apply (simp add: plus_ac0 Int_insert_left insert_absorb) + done + +lemma setsum_Un_disjoint: "finite A ==> finite B + ==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" + apply (subst setsum_Un_Int [symmetric]) + apply auto + done + +lemma setsum_UN_disjoint: (fixes f :: "'a => 'b::plus_ac0") + "finite I ==> (ALL i:I. finite (A i)) ==> + (ALL i:I. ALL j:I. i \ j --> A i Int A j = {}) ==> + setsum f (UNION I A) = setsum (\i. setsum f (A i)) I" + apply (induct set: Finites) + apply simp + apply atomize + apply (subgoal_tac "ALL i:F. x \ i") + prefer 2 apply blast + apply (subgoal_tac "A x Int UNION F A = {}") + prefer 2 apply blast + apply (simp add: setsum_Un_disjoint) + done + +lemma setsum_addf: "setsum (\x. f x + g x) A = (setsum f A + setsum g A)" + apply (case_tac "finite A") + prefer 2 apply (simp add: setsum_def) + apply (erule finite_induct) + apply auto + apply (simp add: plus_ac0) + done + +lemma setsum_Un: "finite A ==> finite B ==> + (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)" + -- {* For the natural numbers, we have subtraction. *} + apply (subst setsum_Un_Int [symmetric]) + apply auto + done + +lemma setsum_diff1: "(setsum f (A - {a}) :: nat) = + (if a:A then setsum f A - f a else setsum f A)" + apply (case_tac "finite A") + prefer 2 apply (simp add: setsum_def) + apply (erule finite_induct) + apply (auto simp add: insert_Diff_if) + apply (drule_tac a = a in mk_disjoint_insert) + apply auto + done + +lemma setsum_cong: + "A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" + apply (case_tac "finite B") + prefer 2 apply (simp add: setsum_def) + apply simp + apply (subgoal_tac "ALL C. C <= B --> (ALL x:C. f x = g x) --> setsum f C = setsum g C") + apply simp + apply (erule finite_induct) + apply simp + apply (simp add: subset_insert_iff) + apply clarify + apply (subgoal_tac "finite C") + prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) + apply (subgoal_tac "C = insert x (C - {x})") + prefer 2 apply blast + apply (erule ssubst) + apply (drule spec) + apply (erule (1) notE impE) + apply (simp add: Ball_def) + done + + +text {* + \medskip Basic theorem about @{text "choose"}. By Florian + Kammüller, tidied by LCP. +*} + +lemma card_s_0_eq_empty: + "finite A ==> card {B. B \ A & card B = 0} = 1" + apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) + apply (simp cong add: rev_conj_cong) + done + +lemma choose_deconstruct: "finite M ==> x \ M + ==> {s. s <= insert x M & card(s) = Suc k} + = {s. s <= M & card(s) = Suc k} Un + {s. EX t. t <= M & card(t) = k & s = insert x t}" + apply safe + apply (auto intro: finite_subset [THEN card_insert_disjoint]) + apply (drule_tac x = "xa - {x}" in spec) + apply (subgoal_tac "x ~: xa") + apply auto + apply (erule rev_mp, subst card_Diff_singleton) + apply (auto intro: finite_subset) + done + +lemma card_inj_on_le: + "finite A ==> finite B ==> f ` A \ B ==> inj_on f A ==> card A <= card B" + by (auto intro: card_mono simp add: card_image [symmetric]) + +lemma card_bij_eq: "finite A ==> finite B ==> + f ` A \ B ==> inj_on f A ==> g ` B \ A ==> inj_on g B ==> card A = card B" + by (auto intro: le_anti_sym card_inj_on_le) + +lemma constr_bij: "finite A ==> x \ A ==> + card {B. EX C. C <= A & card(C) = k & B = insert x C} = + card {B. B <= A & card(B) = k}" + apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq) + apply (rule_tac B = "Pow (insert x A) " in finite_subset) + apply (rule_tac [3] B = "Pow (A) " in finite_subset) + apply fast+ + txt {* arity *} + apply (auto elim!: equalityE simp add: inj_on_def) + apply (subst Diff_insert0) + apply auto + done + +text {* + Main theorem: combinatorial statement about number of subsets of a set. +*} + +lemma n_sub_lemma: + "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)" + apply (induct k) + apply (simp add: card_s_0_eq_empty) + apply atomize + apply (rotate_tac -1, erule finite_induct) + apply (simp_all (no_asm_simp) cong add: conj_cong add: card_s_0_eq_empty choose_deconstruct) + apply (subst card_Un_disjoint) + prefer 4 apply (force simp add: constr_bij) + prefer 3 apply force + prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] + finite_subset [of _ "Pow (insert x F)", standard]) + apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) + done + +theorem n_subsets: "finite A ==> card {B. B <= A & card(B) = k} = (card A choose k)" + by (simp add: n_sub_lemma) + +end diff -r d6913de7655f -r 2298d5b8e530 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Thu Dec 06 00:37:59 2001 +0100 +++ b/src/HOL/IsaMakefile Thu Dec 06 00:38:55 2001 +0100 @@ -78,7 +78,7 @@ $(SRC)/TFL/rules.ML $(SRC)/TFL/tfl.ML $(SRC)/TFL/thms.ML $(SRC)/TFL/thry.ML \ $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML \ Datatype.thy Datatype_Universe.ML Datatype_Universe.thy Divides.ML \ - Divides.thy Finite.ML Finite.thy Fun.ML Fun.thy Gfp.ML Gfp.thy \ + Divides.thy Finite_Set.ML Finite_Set.thy Fun.ML Fun.thy Gfp.ML Gfp.thy \ Hilbert_Choice.thy Hilbert_Choice_lemmas.ML HOL.ML \ HOL.thy HOL_lemmas.ML Inductive.thy Integ/Bin.ML Integ/Bin.thy \ Integ/Equiv.ML Integ/Equiv.thy Integ/Int.ML Integ/Int.thy \