(*  Title:      HOL/Finite.thy

     ID:         $Id$

     Author:     Lawrence C Paulson & Tobias Nipkow

     Copyright   1995  University of Cambridge & TU Muenchen

 Finite sets and their cardinality

 *)

 open Finite;

 section "finite";

 (*

 goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";

 by (rtac lfp_mono 1);

 by (REPEAT (ares_tac basic_monos 1));

 qed "Fin_mono";

 goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)";

 by (blast_tac (claset() addSIs [lfp_lowerbound]) 1);

 qed "Fin_subset_Pow";

 (* A : Fin(B) ==> A <= B *)

 val FinD = Fin_subset_Pow RS subsetD RS PowD;

 *)

 (*Discharging ~ x:y entails extra work*)

 val major::prems = goal Finite.thy 

     "[| 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::prems = goal Finite.thy 

     "[| finite F; \

 \       P({}); \

 \       !!F a. [| finite F; a:A; a ~: F;  P(F) |] ==> P(insert a F) \

 \    |] ==> F <= A --> P(F)";

 by (rtac (major RS finite_induct) 1);

 by (ALLGOALS (blast_tac (claset() addIs prems)));

 val lemma = result();

 val prems = goal Finite.thy 

     "[| finite F;  F <= A; \

 \       P({}); \

 \       !!F a. [| finite F; a:A; a ~: F;  P(F) |] ==> P(insert a F) \

 \    |] ==> P(F)";

 by (blast_tac (HOL_cs addIs ((lemma RS mp)::prems)) 1);

 qed "finite_subset_induct";

 Addsimps Finites.intrs;

 AddSIs Finites.intrs;

 (*The union of two finite sets is finite*)

 val major::prems = goal Finite.thy

     "[| finite F;  finite G |] ==> finite(F Un G)";

 by (rtac (major RS finite_induct) 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps prems)));

 qed "finite_UnI";

 (*Every subset of a finite set is finite*)

 val [subs,fin] = goal Finite.thy "[| A<=B;  finite B |] ==> finite A";

 by (EVERY1 [subgoal_tac "ALL C. C<=B --> finite C",

             rtac mp, etac spec,

             rtac subs]);

 by (rtac (fin RS finite_induct) 1);

 by (simp_tac (simpset() addsimps [subset_Un_eq]) 1);

 by (safe_tac (claset() addSDs [subset_insert_iff RS iffD1]));

 by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2);

 by (ALLGOALS Asm_simp_tac);

 qed "finite_subset";

 goal Finite.thy "finite(F Un G) = (finite F & finite G)";

 by (blast_tac (claset() addIs [finite_UnI] addDs

                 [Un_upper1 RS finite_subset, Un_upper2 RS finite_subset]) 1);

 qed "finite_Un";

 AddIffs[finite_Un];

 goal Finite.thy "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.thy  "!!F. finite F ==> finite(h``F)";

 by (etac finite_induct 1);

 by (Simp_tac 1);

 by (Asm_simp_tac 1);

 qed "finite_imageI";

 val major::prems = goal Finite.thy 

     "[| 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.thy 

     "[| 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.thy "finite(A-{a}) = finite(A)";

 by (case_tac "a:A" 1);

 by (rtac (finite_insert RS sym RS trans) 1);

 by (stac insert_Diff 1);

 by (ALLGOALS Asm_simp_tac);

 qed "finite_Diff_singleton";

 AddIffs [finite_Diff_singleton];

 (*Lemma for proving finite_imageD*)

 goal Finite.thy "!!A. finite B ==> !A. f``A = B --> inj_onto 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_onto_def]) 1);

  by (Blast_tac 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_onto_image_set_diff])));

 val lemma = result();

 goal Finite.thy "!!A. [| finite(f``A);  inj_onto f A |] ==> finite A";

 by (dtac lemma 1);

 by (Blast_tac 1);

 qed "finite_imageD";

 (** The finite UNION of finite sets **)

 val [prem] = goal Finite.thy

  "finite A ==> (!a:A. finite(B a)) --> finite(UN a:A. B a)";

 by (rtac (prem RS finite_induct) 1);

 by (ALLGOALS Asm_simp_tac);

 bind_thm("finite_UnionI", ballI RSN (2, result() RS mp));

 Addsimps [finite_UnionI];

 (** Sigma of finite sets **)

 goalw Finite.thy [Sigma_def]

  "!!A. [| finite A; !a:A. finite(B a) |] ==> finite(SIGMA a:A. B a)";

 by (blast_tac (claset() addSIs [finite_UnionI]) 1);

 bind_thm("finite_SigmaI", ballI RSN (2,result()));

 Addsimps [finite_SigmaI];

 (** The powerset of a finite set **)

 goal Finite.thy "!!A. 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_onto_def] finite_imageD])));

 val lemma = result();

 goal Finite.thy "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.thy "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_onto_def] finite_imageD) 1);

   by (simp_tac (simpset() addsplits [expand_split]) 1);

  by (etac finite_imageI 1);

 by (simp_tac (simpset() addsimps [inverse_def,image_def]) 1);

 by (Auto_tac());

  by (rtac bexI 1);

  by (assume_tac 2);

  by (Simp_tac 1);

 by (split_all_tac 1);

 by (Asm_full_simp_tac 1);

 qed "finite_inverse";

 AddIffs [finite_inverse];

 section "Finite cardinality -- 'card'";

 goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}";

 by (Blast_tac 1);

 val Collect_conv_insert = result();

 goalw Finite.thy [card_def] "card {} = 0";

 by (rtac Least_equality 1);

 by (ALLGOALS Asm_full_simp_tac);

 qed "card_empty";

 Addsimps [card_empty];

 val [major] = goal Finite.thy

   "finite A ==> ? (n::nat) f. A = {f i |i. i<n}";

 by (rtac (major RS finite_induct) 1);

  by (res_inst_tac [("x","0")] exI 1);

  by (Simp_tac 1);

 by (etac exE 1);

 by (etac exE 1);

 by (hyp_subst_tac 1);

 by (res_inst_tac [("x","Suc n")] exI 1);

 by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1);

 by (asm_simp_tac (simpset() addsimps [Collect_conv_insert, less_Suc_eq]

                           addcongs [rev_conj_cong]) 1);

 qed "finite_has_card";

 goal Finite.thy

   "!!A.[| x ~: A; insert x A = {f i|i. i<n} |] ==> \

 \  ? m::nat. m<n & (? g. A = {g i|i. i<m})";

 by (res_inst_tac [("n","n")] natE 1);

  by (hyp_subst_tac 1);

  by (Asm_full_simp_tac 1);

 by (rename_tac "m" 1);

 by (hyp_subst_tac 1);

 by (case_tac "? a. a:A" 1);

  by (res_inst_tac [("x","0")] exI 2);

  by (Simp_tac 2);

  by (Blast_tac 2);

 by (etac exE 1);

 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);

 by (rtac exI 1);

 by (rtac (refl RS disjI2 RS conjI) 1);

 by (etac equalityE 1);

 by (asm_full_simp_tac

      (simpset() addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1);

 by Safe_tac;

   by (Asm_full_simp_tac 1);

   by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);

   by (SELECT_GOAL Safe_tac 1);

    by (subgoal_tac "x ~= f m" 1);

     by (Blast_tac 2);

    by (subgoal_tac "? k. f k = x & k<m" 1);

     by (Blast_tac 2);

    by (SELECT_GOAL Safe_tac 1);

    by (res_inst_tac [("x","k")] exI 1);

    by (Asm_simp_tac 1);

   by (simp_tac (simpset() addsplits [expand_if]) 1);

   by (Blast_tac 1);

  by (dtac sym 1);

  by (rotate_tac ~1 1);

  by (Asm_full_simp_tac 1);

  by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);

  by (SELECT_GOAL Safe_tac 1);

   by (subgoal_tac "x ~= f m" 1);

    by (Blast_tac 2);

   by (subgoal_tac "? k. f k = x & k<m" 1);

    by (Blast_tac 2);

   by (SELECT_GOAL Safe_tac 1);

   by (res_inst_tac [("x","k")] exI 1);

   by (Asm_simp_tac 1);

  by (simp_tac (simpset() addsplits [expand_if]) 1);

  by (Blast_tac 1);

 by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1);

 by (SELECT_GOAL Safe_tac 1);

  by (subgoal_tac "x ~= f i" 1);

   by (Blast_tac 2);

  by (case_tac "x = f m" 1);

   by (res_inst_tac [("x","i")] exI 1);

   by (Asm_simp_tac 1);

  by (subgoal_tac "? k. f k = x & k<m" 1);

   by (Blast_tac 2);

  by (SELECT_GOAL Safe_tac 1);

  by (res_inst_tac [("x","k")] exI 1);

  by (Asm_simp_tac 1);

 by (simp_tac (simpset() addsplits [expand_if]) 1);

 by (Blast_tac 1);

 val lemma = result();

 goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \

 \ (LEAST n. ? f. insert x A = {f i|i. i<n}) = Suc(LEAST n. ? f. A={f i|i. i<n})";

 by (rtac Least_equality 1);

  by (dtac finite_has_card 1);

  by (etac exE 1);

  by (dres_inst_tac [("P","%n.? f. A={f i|i. i<n}")] LeastI 1);

  by (etac exE 1);

  by (res_inst_tac

    [("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1);

  by (simp_tac

     (simpset() addsimps [Collect_conv_insert, less_Suc_eq] 

               addcongs [rev_conj_cong]) 1);

  by (etac subst 1);

  by (rtac refl 1);

 by (rtac notI 1);

 by (etac exE 1);

 by (dtac lemma 1);

  by (assume_tac 1);

 by (etac exE 1);

 by (etac conjE 1);

 by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1);

 by (dtac le_less_trans 1 THEN atac 1);

 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);

 by (etac disjE 1);

 by (etac less_asym 1 THEN atac 1);

 by (hyp_subst_tac 1);

 by (Asm_full_simp_tac 1);

 val lemma = result();

 goalw Finite.thy [card_def]

   "!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)";

 by (etac lemma 1);

 by (assume_tac 1);

 qed "card_insert_disjoint";

 Addsimps [card_insert_disjoint];

 goal Finite.thy  "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)";

 by (etac finite_induct 1);

 by (Simp_tac 1);

 by (Clarify_tac 1);

 by (case_tac "x:B" 1);

  by (dres_inst_tac [("A","B")] mk_disjoint_insert 1);

  by (SELECT_GOAL Safe_tac 1);

  by (rotate_tac ~1 1);

  by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);

 by (rotate_tac ~1 1);

 by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);

 qed_spec_mp "card_mono";

 goal Finite.thy "!!A B. [| finite A; finite B |]\

 \                       ==> A Int B = {} --> card(A Un B) = card A + card B";

 by (etac finite_induct 1);

 by (ALLGOALS 

     (asm_simp_tac (simpset() addsimps [Int_insert_left]

 	                    addsplits [expand_if])));

 qed_spec_mp "card_Un_disjoint";

 goal Finite.thy "!!A. [| 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.thy "!!A. [| 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_Diff";

 goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A";

 by (rtac Suc_less_SucD 1);

 by (asm_simp_tac (simpset() addsimps [card_Suc_Diff]) 1);

 qed "card_Diff";

 (*** Cardinality of the Powerset ***)

 val [major] = goal Finite.thy

   "finite A ==> card(insert x A) = Suc(card(A-{x}))";

 by (case_tac "x:A" 1);

 by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1);

 by (dtac mk_disjoint_insert 1);

 by (etac exE 1);

 by (Asm_simp_tac 1);

 by (rtac card_insert_disjoint 1);

 by (rtac (major RSN (2,finite_subset)) 1);

 by (Blast_tac 1);

 by (Blast_tac 1);

 by (asm_simp_tac (simpset() addsimps [major RS card_insert_disjoint]) 1);

 qed "card_insert";

 Addsimps [card_insert];

 goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A";

 by (etac finite_induct 1);

 by (ALLGOALS Asm_simp_tac);

 by Safe_tac;

 by (rewtac inj_onto_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 thy "!!A. finite A ==> card (Pow A) = 2 ^ card A";

 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_onto (insert x) (Pow F)" 1);

 by (asm_simp_tac (simpset() addsimps [card_image, Pow_insert]) 1);

 by (rewtac inj_onto_def);

 by (blast_tac (claset() addSEs [equalityE]) 1);

 qed "card_Pow";

 Addsimps [card_Pow];

 (*Proper subsets*)

 goalw Finite.thy [psubset_def]

 "!!B. finite B ==> !A. A < B --> card(A) < card(B)";

 by (etac finite_induct 1);

 by (Simp_tac 1);

 by (Clarify_tac 1);

 by (case_tac "x:A" 1);

 (*1*)

 by (dres_inst_tac [("A","A")]mk_disjoint_insert 1);

 by (etac exE 1);

 by (etac conjE 1);

 by (hyp_subst_tac 1);

 by (rotate_tac ~1 1);

 by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);

 by (Blast_tac 1);

 (*2*)

 by (rotate_tac ~1 1);

 by (eres_inst_tac [("P","?a<?b")] notE 1);

 by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff,finite_subset]) 1);

 by (case_tac "A=F" 1);

 by (ALLGOALS Asm_simp_tac);

 qed_spec_mp "psubset_card" ;

 (*Relates to equivalence classes.   Based on a theorem of F. Kammueller's.

   The "finite C" premise is redundant*)

 goal thy "!!C. finite C ==> finite (Union C) --> \

 \          (! c : C. k dvd card c) -->  \

 \          (! c1: C. ! 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 "!c:F. ?PP(c)" 1);

 by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1);

 by (Clarify_tac 1);

 by (ball_tac 1);

 by (Blast_tac 1);

 qed_spec_mp "dvd_partition";