clasohm@1465: (*  Title:      HOL/Finite.thy
clasohm@923:     ID:         $Id$
nipkow@1531:     Author:     Lawrence C Paulson & Tobias Nipkow
nipkow@1531:     Copyright   1995  University of Cambridge & TU Muenchen
clasohm@923: 
nipkow@1531: Finite sets and their cardinality
clasohm@923: *)
clasohm@923: 
clasohm@923: open Finite;
clasohm@923: 
nipkow@3413: section "finite";
nipkow@1531: 
nipkow@3413: (*
clasohm@923: goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";
clasohm@1465: by (rtac lfp_mono 1);
clasohm@923: by (REPEAT (ares_tac basic_monos 1));
clasohm@923: qed "Fin_mono";
clasohm@923: 
clasohm@923: goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)";
wenzelm@4089: by (blast_tac (claset() addSIs [lfp_lowerbound]) 1);
clasohm@923: qed "Fin_subset_Pow";
clasohm@923: 
clasohm@923: (* A : Fin(B) ==> A <= B *)
clasohm@923: val FinD = Fin_subset_Pow RS subsetD RS PowD;
nipkow@3413: *)
clasohm@923: 
clasohm@923: (*Discharging ~ x:y entails extra work*)
clasohm@923: val major::prems = goal Finite.thy 
nipkow@3413:     "[| finite F;  P({}); \
nipkow@3413: \       !!F x. [| finite F;  x ~: F;  P(F) |] ==> P(insert x F) \
clasohm@923: \    |] ==> P(F)";
nipkow@3413: by (rtac (major RS Finites.induct) 1);
nipkow@3413: by (excluded_middle_tac "a:A" 2);
clasohm@923: by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3);   (*backtracking!*)
clasohm@923: by (REPEAT (ares_tac prems 1));
nipkow@3413: qed "finite_induct";
nipkow@3413: 
nipkow@3413: val major::prems = goal Finite.thy 
nipkow@3413:     "[| finite F; \
nipkow@3413: \       P({}); \
nipkow@3413: \       !!F a. [| finite F; a:A; a ~: F;  P(F) |] ==> P(insert a F) \
nipkow@3413: \    |] ==> F <= A --> P(F)";
nipkow@3413: by (rtac (major RS finite_induct) 1);
wenzelm@4089: by (ALLGOALS (blast_tac (claset() addIs prems)));
nipkow@3413: val lemma = result();
clasohm@923: 
nipkow@3413: val prems = goal Finite.thy 
nipkow@3413:     "[| finite F;  F <= A; \
nipkow@3413: \       P({}); \
nipkow@3413: \       !!F a. [| finite F; a:A; a ~: F;  P(F) |] ==> P(insert a F) \
nipkow@3413: \    |] ==> P(F)";
paulson@3457: by (blast_tac (HOL_cs addIs ((lemma RS mp)::prems)) 1);
nipkow@3413: qed "finite_subset_induct";
nipkow@3413: 
nipkow@3413: Addsimps Finites.intrs;
nipkow@3413: AddSIs Finites.intrs;
clasohm@923: 
clasohm@923: (*The union of two finite sets is finite*)
clasohm@923: val major::prems = goal Finite.thy
nipkow@3413:     "[| finite F;  finite G |] ==> finite(F Un G)";
nipkow@3413: by (rtac (major RS finite_induct) 1);
wenzelm@4089: by (ALLGOALS (asm_simp_tac (simpset() addsimps prems)));
nipkow@3413: qed "finite_UnI";
clasohm@923: 
clasohm@923: (*Every subset of a finite set is finite*)
nipkow@3413: val [subs,fin] = goal Finite.thy "[| A<=B;  finite B |] ==> finite A";
nipkow@3413: by (EVERY1 [subgoal_tac "ALL C. C<=B --> finite C",
clasohm@1465:             rtac mp, etac spec,
clasohm@1465:             rtac subs]);
nipkow@3413: by (rtac (fin RS finite_induct) 1);
wenzelm@4089: by (simp_tac (simpset() addsimps [subset_Un_eq]) 1);
wenzelm@4089: by (safe_tac (claset() addSDs [subset_insert_iff RS iffD1]));
clasohm@923: by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2);
clasohm@1264: by (ALLGOALS Asm_simp_tac);
nipkow@3413: qed "finite_subset";
clasohm@923: 
nipkow@3413: goal Finite.thy "finite(F Un G) = (finite F & finite G)";
wenzelm@4089: by (blast_tac (claset() addIs [finite_UnI] addDs
nipkow@3413:                 [Un_upper1 RS finite_subset, Un_upper2 RS finite_subset]) 1);
nipkow@3413: qed "finite_Un";
nipkow@3413: AddIffs[finite_Un];
nipkow@1531: 
nipkow@3413: goal Finite.thy "finite(insert a A) = finite A";
paulson@1553: by (stac insert_is_Un 1);
nipkow@3413: by (simp_tac (HOL_ss addsimps [finite_Un]) 1);
paulson@3427: by (Blast_tac 1);
nipkow@3413: qed "finite_insert";
nipkow@3413: Addsimps[finite_insert];
nipkow@1531: 
nipkow@3413: (*The image of a finite set is finite *)
nipkow@3413: goal Finite.thy  "!!F. finite F ==> finite(h``F)";
nipkow@3413: by (etac finite_induct 1);
clasohm@1264: by (Simp_tac 1);
nipkow@3413: by (Asm_simp_tac 1);
nipkow@3413: qed "finite_imageI";
clasohm@923: 
clasohm@923: val major::prems = goal Finite.thy 
nipkow@3413:     "[| finite c;  finite b;                                  \
clasohm@1465: \       P(b);                                                   \
nipkow@3413: \       !!x y. [| finite y;  x:y;  P(y) |] ==> P(y-{x}) \
clasohm@923: \    |] ==> c<=b --> P(b-c)";
nipkow@3413: by (rtac (major RS finite_induct) 1);
paulson@2031: by (stac Diff_insert 2);
clasohm@923: by (ALLGOALS (asm_simp_tac
wenzelm@4089:                 (simpset() addsimps (prems@[Diff_subset RS finite_subset]))));
nipkow@1531: val lemma = result();
clasohm@923: 
clasohm@923: val prems = goal Finite.thy 
nipkow@3413:     "[| finite A;                                       \
nipkow@3413: \       P(A);                                           \
nipkow@3413: \       !!a A. [| finite A;  a:A;  P(A) |] ==> P(A-{a}) \
clasohm@923: \    |] ==> P({})";
clasohm@923: by (rtac (Diff_cancel RS subst) 1);
nipkow@1531: by (rtac (lemma RS mp) 1);
clasohm@923: by (REPEAT (ares_tac (subset_refl::prems) 1));
nipkow@3413: qed "finite_empty_induct";
nipkow@1531: 
nipkow@1531: 
paulson@1618: (* finite B ==> finite (B - Ba) *)
paulson@1618: bind_thm ("finite_Diff", Diff_subset RS finite_subset);
nipkow@1531: Addsimps [finite_Diff];
nipkow@1531: 
paulson@3368: goal Finite.thy "finite(A-{a}) = finite(A)";
paulson@3368: by (case_tac "a:A" 1);
paulson@3457: by (rtac (finite_insert RS sym RS trans) 1);
paulson@3368: by (stac insert_Diff 1);
paulson@3368: by (ALLGOALS Asm_simp_tac);
paulson@3368: qed "finite_Diff_singleton";
paulson@3368: AddIffs [finite_Diff_singleton];
paulson@3368: 
paulson@4059: (*Lemma for proving finite_imageD*)
nipkow@3413: goal Finite.thy "!!A. finite B ==> !A. f``A = B --> inj_onto f A --> finite A";
paulson@1553: by (etac finite_induct 1);
nipkow@3413:  by (ALLGOALS Asm_simp_tac);
paulson@3708: by (Clarify_tac 1);
nipkow@3413: by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1);
paulson@3708:  by (Clarify_tac 1);
wenzelm@4089:  by (full_simp_tac (simpset() addsimps [inj_onto_def]) 1);
nipkow@3413:  by (Blast_tac 1);
paulson@3368: by (thin_tac "ALL A. ?PP(A)" 1);
nipkow@3413: by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1);
paulson@3708: by (Clarify_tac 1);
paulson@3368: by (res_inst_tac [("x","xa")] bexI 1);
paulson@4059: by (ALLGOALS 
wenzelm@4089:     (asm_full_simp_tac (simpset() addsimps [inj_onto_image_set_diff])));
paulson@3368: val lemma = result();
paulson@3368: 
paulson@3368: goal Finite.thy "!!A. [| finite(f``A);  inj_onto f A |] ==> finite A";
paulson@3457: by (dtac lemma 1);
paulson@3368: by (Blast_tac 1);
paulson@3368: qed "finite_imageD";
paulson@3368: 
nipkow@4014: (** The finite UNION of finite sets **)
nipkow@4014: 
nipkow@4014: val [prem] = goal Finite.thy
nipkow@4014:  "finite A ==> (!a:A. finite(B a)) --> finite(UN a:A. B a)";
paulson@4153: by (rtac (prem RS finite_induct) 1);
paulson@4153: by (ALLGOALS Asm_simp_tac);
nipkow@4014: bind_thm("finite_UnionI", ballI RSN (2, result() RS mp));
nipkow@4014: Addsimps [finite_UnionI];
nipkow@4014: 
nipkow@4014: (** Sigma of finite sets **)
nipkow@4014: 
nipkow@4014: goalw Finite.thy [Sigma_def]
nipkow@4014:  "!!A. [| finite A; !a:A. finite(B a) |] ==> finite(SIGMA a:A. B a)";
paulson@4153: by (blast_tac (claset() addSIs [finite_UnionI]) 1);
nipkow@4014: bind_thm("finite_SigmaI", ballI RSN (2,result()));
nipkow@4014: Addsimps [finite_SigmaI];
paulson@3368: 
paulson@3368: (** The powerset of a finite set **)
paulson@3368: 
paulson@3368: goal Finite.thy "!!A. finite(Pow A) ==> finite A";
paulson@3368: by (subgoal_tac "finite ((%x.{x})``A)" 1);
paulson@3457: by (rtac finite_subset 2);
paulson@3457: by (assume_tac 3);
paulson@3368: by (ALLGOALS
wenzelm@4089:     (fast_tac (claset() addSDs [rewrite_rule [inj_onto_def] finite_imageD])));
paulson@3368: val lemma = result();
paulson@3368: 
paulson@3368: goal Finite.thy "finite(Pow A) = finite A";
paulson@3457: by (rtac iffI 1);
paulson@3457: by (etac lemma 1);
paulson@3368: (*Opposite inclusion: finite A ==> finite (Pow A) *)
paulson@3340: by (etac finite_induct 1);
paulson@3340: by (ALLGOALS 
paulson@3340:     (asm_simp_tac
wenzelm@4089:      (simpset() addsimps [finite_UnI, finite_imageI, Pow_insert])));
paulson@3368: qed "finite_Pow_iff";
paulson@3368: AddIffs [finite_Pow_iff];
paulson@3340: 
nipkow@3439: goal Finite.thy "finite(r^-1) = finite r";
paulson@3457: by (subgoal_tac "r^-1 = (%(x,y).(y,x))``r" 1);
paulson@3457:  by (Asm_simp_tac 1);
paulson@3457:  by (rtac iffI 1);
paulson@3457:   by (etac (rewrite_rule [inj_onto_def] finite_imageD) 1);
wenzelm@4089:   by (simp_tac (simpset() addsplits [expand_split]) 1);
paulson@3457:  by (etac finite_imageI 1);
wenzelm@4089: by (simp_tac (simpset() addsimps [inverse_def,image_def]) 1);
paulson@3457: by (Auto_tac());
paulson@3457:  by (rtac bexI 1);
paulson@3457:  by (assume_tac 2);
paulson@3457:  by (Simp_tac 1);
paulson@3457: by (split_all_tac 1);
paulson@3457: by (Asm_full_simp_tac 1);
nipkow@3439: qed "finite_inverse";
nipkow@3439: AddIffs [finite_inverse];
nipkow@1531: 
nipkow@1548: section "Finite cardinality -- 'card'";
nipkow@1531: 
nipkow@1531: goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}";
paulson@2922: by (Blast_tac 1);
nipkow@1531: val Collect_conv_insert = result();
nipkow@1531: 
nipkow@1531: goalw Finite.thy [card_def] "card {} = 0";
paulson@1553: by (rtac Least_equality 1);
paulson@1553: by (ALLGOALS Asm_full_simp_tac);
nipkow@1531: qed "card_empty";
nipkow@1531: Addsimps [card_empty];
nipkow@1531: 
nipkow@1531: val [major] = goal Finite.thy
nipkow@1531:   "finite A ==> ? (n::nat) f. A = {f i |i. i<n}";
paulson@1553: by (rtac (major RS finite_induct) 1);
paulson@1553:  by (res_inst_tac [("x","0")] exI 1);
paulson@1553:  by (Simp_tac 1);
paulson@1553: by (etac exE 1);
paulson@1553: by (etac exE 1);
paulson@1553: by (hyp_subst_tac 1);
paulson@1553: by (res_inst_tac [("x","Suc n")] exI 1);
paulson@1553: by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [Collect_conv_insert, less_Suc_eq]
nipkow@1548:                           addcongs [rev_conj_cong]) 1);
nipkow@1531: qed "finite_has_card";
nipkow@1531: 
nipkow@1531: goal Finite.thy
wenzelm@3842:   "!!A.[| x ~: A; insert x A = {f i|i. i<n} |] ==> \
wenzelm@3842: \  ? m::nat. m<n & (? g. A = {g i|i. i<m})";
paulson@1553: by (res_inst_tac [("n","n")] natE 1);
paulson@1553:  by (hyp_subst_tac 1);
paulson@1553:  by (Asm_full_simp_tac 1);
paulson@1553: by (rename_tac "m" 1);
paulson@1553: by (hyp_subst_tac 1);
paulson@1553: by (case_tac "? a. a:A" 1);
paulson@1553:  by (res_inst_tac [("x","0")] exI 2);
paulson@1553:  by (Simp_tac 2);
paulson@2922:  by (Blast_tac 2);
paulson@1553: by (etac exE 1);
wenzelm@4089: by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
paulson@1553: by (rtac exI 1);
paulson@1782: by (rtac (refl RS disjI2 RS conjI) 1);
paulson@1553: by (etac equalityE 1);
paulson@1553: by (asm_full_simp_tac
wenzelm@4089:      (simpset() addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1);
paulson@4153: by Safe_tac;
paulson@1553:   by (Asm_full_simp_tac 1);
paulson@1553:   by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
paulson@4153:   by (SELECT_GOAL Safe_tac 1);
paulson@1553:    by (subgoal_tac "x ~= f m" 1);
paulson@2922:     by (Blast_tac 2);
paulson@1553:    by (subgoal_tac "? k. f k = x & k<m" 1);
paulson@2922:     by (Blast_tac 2);
paulson@4153:    by (SELECT_GOAL Safe_tac 1);
paulson@1553:    by (res_inst_tac [("x","k")] exI 1);
paulson@1553:    by (Asm_simp_tac 1);
wenzelm@4089:   by (simp_tac (simpset() addsplits [expand_if]) 1);
paulson@2922:   by (Blast_tac 1);
paulson@3457:  by (dtac sym 1);
paulson@1553:  by (rotate_tac ~1 1);
paulson@1553:  by (Asm_full_simp_tac 1);
paulson@1553:  by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
paulson@4153:  by (SELECT_GOAL Safe_tac 1);
paulson@1553:   by (subgoal_tac "x ~= f m" 1);
paulson@2922:    by (Blast_tac 2);
paulson@1553:   by (subgoal_tac "? k. f k = x & k<m" 1);
paulson@2922:    by (Blast_tac 2);
paulson@4153:   by (SELECT_GOAL Safe_tac 1);
paulson@1553:   by (res_inst_tac [("x","k")] exI 1);
paulson@1553:   by (Asm_simp_tac 1);
wenzelm@4089:  by (simp_tac (simpset() addsplits [expand_if]) 1);
paulson@2922:  by (Blast_tac 1);
paulson@1553: by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1);
paulson@4153: by (SELECT_GOAL Safe_tac 1);
paulson@1553:  by (subgoal_tac "x ~= f i" 1);
paulson@2922:   by (Blast_tac 2);
paulson@1553:  by (case_tac "x = f m" 1);
paulson@1553:   by (res_inst_tac [("x","i")] exI 1);
paulson@1553:   by (Asm_simp_tac 1);
paulson@1553:  by (subgoal_tac "? k. f k = x & k<m" 1);
paulson@2922:   by (Blast_tac 2);
paulson@4153:  by (SELECT_GOAL Safe_tac 1);
paulson@1553:  by (res_inst_tac [("x","k")] exI 1);
paulson@1553:  by (Asm_simp_tac 1);
wenzelm@4089: by (simp_tac (simpset() addsplits [expand_if]) 1);
paulson@2922: by (Blast_tac 1);
nipkow@1531: val lemma = result();
nipkow@1531: 
nipkow@1531: goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \
wenzelm@3842: \ (LEAST n. ? f. insert x A = {f i|i. i<n}) = Suc(LEAST n. ? f. A={f i|i. i<n})";
paulson@1553: by (rtac Least_equality 1);
paulson@3457:  by (dtac finite_has_card 1);
paulson@3457:  by (etac exE 1);
wenzelm@3842:  by (dres_inst_tac [("P","%n.? f. A={f i|i. i<n}")] LeastI 1);
paulson@3457:  by (etac exE 1);
paulson@1553:  by (res_inst_tac
nipkow@1531:    [("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1);
paulson@1553:  by (simp_tac
wenzelm@4089:     (simpset() addsimps [Collect_conv_insert, less_Suc_eq] 
paulson@2031:               addcongs [rev_conj_cong]) 1);
paulson@3457:  by (etac subst 1);
paulson@3457:  by (rtac refl 1);
paulson@1553: by (rtac notI 1);
paulson@1553: by (etac exE 1);
paulson@1553: by (dtac lemma 1);
paulson@3457:  by (assume_tac 1);
paulson@1553: by (etac exE 1);
paulson@1553: by (etac conjE 1);
paulson@1553: by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1);
paulson@1553: by (dtac le_less_trans 1 THEN atac 1);
wenzelm@4089: by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
paulson@1553: by (etac disjE 1);
paulson@1553: by (etac less_asym 1 THEN atac 1);
paulson@1553: by (hyp_subst_tac 1);
paulson@1553: by (Asm_full_simp_tac 1);
nipkow@1531: val lemma = result();
nipkow@1531: 
nipkow@1531: goalw Finite.thy [card_def]
nipkow@1531:   "!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)";
paulson@1553: by (etac lemma 1);
paulson@1553: by (assume_tac 1);
nipkow@1531: qed "card_insert_disjoint";
paulson@3352: Addsimps [card_insert_disjoint];
paulson@3352: 
paulson@3352: goal Finite.thy  "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)";
paulson@3352: by (etac finite_induct 1);
paulson@3352: by (Simp_tac 1);
paulson@3708: by (Clarify_tac 1);
paulson@3352: by (case_tac "x:B" 1);
nipkow@3413:  by (dres_inst_tac [("A","B")] mk_disjoint_insert 1);
paulson@4153:  by (SELECT_GOAL Safe_tac 1);
paulson@3352:  by (rotate_tac ~1 1);
wenzelm@4089:  by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);
paulson@3352: by (rotate_tac ~1 1);
wenzelm@4089: by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);
paulson@3352: qed_spec_mp "card_mono";
paulson@3352: 
paulson@3352: goal Finite.thy "!!A B. [| finite A; finite B |]\
paulson@3352: \                       ==> A Int B = {} --> card(A Un B) = card A + card B";
paulson@3352: by (etac finite_induct 1);
paulson@3352: by (ALLGOALS 
wenzelm@4089:     (asm_simp_tac (simpset() addsimps [Int_insert_left]
nipkow@3919: 	                    addsplits [expand_if])));
paulson@3352: qed_spec_mp "card_Un_disjoint";
paulson@3352: 
paulson@3352: goal Finite.thy "!!A. [| finite A; B<=A |] ==> card A - card B = card (A - B)";
paulson@3352: by (subgoal_tac "(A-B) Un B = A" 1);
paulson@3352: by (Blast_tac 2);
paulson@3457: by (rtac (add_right_cancel RS iffD1) 1);
paulson@3457: by (rtac (card_Un_disjoint RS subst) 1);
paulson@3457: by (etac ssubst 4);
paulson@3352: by (Blast_tac 3);
paulson@3352: by (ALLGOALS 
paulson@3352:     (asm_simp_tac
wenzelm@4089:      (simpset() addsimps [add_commute, not_less_iff_le, 
paulson@3352: 			 add_diff_inverse, card_mono, finite_subset])));
paulson@3352: qed "card_Diff_subset";
nipkow@1531: 
paulson@1618: goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A";
paulson@1618: by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1);
paulson@1618: by (assume_tac 1);
paulson@3352: by (Asm_simp_tac 1);
paulson@1618: qed "card_Suc_Diff";
paulson@1618: 
paulson@1618: goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A";
paulson@2031: by (rtac Suc_less_SucD 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [card_Suc_Diff]) 1);
paulson@1618: qed "card_Diff";
paulson@1618: 
paulson@3389: 
paulson@3389: (*** Cardinality of the Powerset ***)
paulson@3389: 
nipkow@1531: val [major] = goal Finite.thy
nipkow@1531:   "finite A ==> card(insert x A) = Suc(card(A-{x}))";
paulson@1553: by (case_tac "x:A" 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1);
paulson@1553: by (dtac mk_disjoint_insert 1);
paulson@1553: by (etac exE 1);
paulson@1553: by (Asm_simp_tac 1);
paulson@1553: by (rtac card_insert_disjoint 1);
paulson@1553: by (rtac (major RSN (2,finite_subset)) 1);
paulson@2922: by (Blast_tac 1);
paulson@2922: by (Blast_tac 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [major RS card_insert_disjoint]) 1);
nipkow@1531: qed "card_insert";
nipkow@1531: Addsimps [card_insert];
nipkow@1531: 
paulson@3340: goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A";
paulson@3340: by (etac finite_induct 1);
paulson@3340: by (ALLGOALS Asm_simp_tac);
paulson@3724: by Safe_tac;
paulson@3457: by (rewtac inj_onto_def);
paulson@3340: by (Blast_tac 1);
paulson@3340: by (stac card_insert_disjoint 1);
paulson@3340: by (etac finite_imageI 1);
paulson@3340: by (Blast_tac 1);
paulson@3340: by (Blast_tac 1);
paulson@3340: qed_spec_mp "card_image";
paulson@3340: 
paulson@3389: goal thy "!!A. finite A ==> card (Pow A) = 2 ^ card A";
paulson@3389: by (etac finite_induct 1);
wenzelm@4089: by (ALLGOALS (asm_simp_tac (simpset() addsimps [Pow_insert])));
paulson@3389: by (stac card_Un_disjoint 1);
wenzelm@4089: by (EVERY (map (blast_tac (claset() addIs [finite_imageI])) [3,2,1]));
paulson@3389: by (subgoal_tac "inj_onto (insert x) (Pow F)" 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [card_image, Pow_insert]) 1);
paulson@3457: by (rewtac inj_onto_def);
wenzelm@4089: by (blast_tac (claset() addSEs [equalityE]) 1);
paulson@3389: qed "card_Pow";
paulson@3389: Addsimps [card_Pow];
paulson@3340: 
paulson@3389: 
paulson@3389: (*Proper subsets*)
nipkow@3222: goalw Finite.thy [psubset_def]
nipkow@3222: "!!B. finite B ==> !A. A < B --> card(A) < card(B)";
nipkow@3222: by (etac finite_induct 1);
nipkow@3222: by (Simp_tac 1);
paulson@3708: by (Clarify_tac 1);
nipkow@3222: by (case_tac "x:A" 1);
nipkow@3222: (*1*)
nipkow@3413: by (dres_inst_tac [("A","A")]mk_disjoint_insert 1);
nipkow@3222: by (etac exE 1);
nipkow@3222: by (etac conjE 1);
nipkow@3222: by (hyp_subst_tac 1);
nipkow@3222: by (rotate_tac ~1 1);
wenzelm@4089: by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);
paulson@3708: by (Blast_tac 1);
nipkow@3222: (*2*)
nipkow@3222: by (rotate_tac ~1 1);
paulson@3708: by (eres_inst_tac [("P","?a<?b")] notE 1);
wenzelm@4089: by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);
nipkow@3222: by (case_tac "A=F" 1);
paulson@3708: by (ALLGOALS Asm_simp_tac);
nipkow@3222: qed_spec_mp "psubset_card" ;
paulson@3368: 
paulson@3368: 
wenzelm@3430: (*Relates to equivalence classes.   Based on a theorem of F. Kammueller's.
paulson@3368:   The "finite C" premise is redundant*)
paulson@3368: goal thy "!!C. finite C ==> finite (Union C) --> \
paulson@3368: \          (! c : C. k dvd card c) -->  \
paulson@3368: \          (! c1: C. ! c2: C. c1 ~= c2 --> c1 Int c2 = {}) \
paulson@3368: \          --> k dvd card(Union C)";
paulson@3368: by (etac finite_induct 1);
paulson@3368: by (ALLGOALS Asm_simp_tac);
paulson@3708: by (Clarify_tac 1);
paulson@3368: by (stac card_Un_disjoint 1);
paulson@3368: by (ALLGOALS
wenzelm@4089:     (asm_full_simp_tac (simpset()
paulson@3368: 			 addsimps [dvd_add, disjoint_eq_subset_Compl])));
paulson@3368: by (thin_tac "!c:F. ?PP(c)" 1);
paulson@3368: by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1);
paulson@3708: by (Clarify_tac 1);
paulson@3368: by (ball_tac 1);
paulson@3368: by (Blast_tac 1);
paulson@3368: qed_spec_mp "dvd_partition";
paulson@3368: