(* Title: HOL/Finite.thy
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 (Simp_tac 1);
by (safe_tac (claset() addSDs [subset_insert_iff RS iffD1]));
by (eres_inst_tac [("t","A")] (insert_Diff RS subst) 2);
by (ALLGOALS Asm_simp_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 [read_instantiate [("B", "?X Un ?Y")] finite_subset,
finite_UnI]) 1);
qed "finite_Un";
AddIffs[finite_Un];
Goal "finite F ==> finite(F Int G)";
by (blast_tac (claset() addIs [finite_subset]) 1);
qed "finite_Int1";
Goal "finite G ==> finite(F Int G)";
by (blast_tac (claset() addIs [finite_subset]) 1);
qed "finite_Int2";
Addsimps[finite_Int1, finite_Int2];
AddIs[finite_Int1, finite_Int2];
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";
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-{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 B ==> !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 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 ==> (!a:A. finite(B a)) --> finite(UN a:A. B a)";
by (etac 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 [Sigma_def]
"[| 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(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];
section "Finite cardinality -- 'card'";
Goal "{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 [card_def] "card {} = 0";
by (rtac Least_equality 1);
by (ALLGOALS Asm_full_simp_tac);
qed "card_empty";
Addsimps [card_empty];
Goal "finite A ==> ? (n::nat) f. A = {f i |i. i<n}";
by (etac 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 "[| x ~: A; insert x A = {f i|i. i<n} |] \
\ ==> ? m::nat. m<n & (? g. A = {g i|i. i<m})";
by (exhaust_tac "n" 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 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 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 1);
by (Blast_tac 1);
val lemma = result();
Goal "[| 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 [card_def] "[| 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 A ==> card A <= card (insert x A)";
by (case_tac "x: A" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [insert_absorb])));
qed "card_insert_le";
Goal "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 (asm_full_simp_tac (simpset() addsimps [subset_insert_iff]) 2);
by (fast_tac (claset() addss
(simpset() addsimps [subset_insert_iff, finite_subset]
delsimps [insert_subset])) 1);
qed_spec_mp "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 |] ==> 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 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";
Goal "finite A ==> card(A-{x}) <= card A";
by (case_tac "x: A" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [card_Diff, less_imp_le])));
qed "card_Diff_le";
(*** Cardinality of the Powerset ***)
Goal "finite A ==> card(insert x A) = Suc(card(A-{x}))";
by (case_tac "x:A" 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [card_Suc_Diff, insert_absorb])));
qed "card_insert";
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 ==> 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_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";
Addsimps [card_Pow];
(*Proper subsets*)
Goalw [psubset_def] "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 (Clarify_tac 1);
by (rotate_tac ~3 1);
by (asm_full_simp_tac (simpset() addsimps [finite_subset]) 1);
by (Blast_tac 1);
(*2*)
by (eres_inst_tac [("P","?a<?b")] notE 1);
by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff]) 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 "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";