(* 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 @ [Un_insert_left]))));
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 (!claset addSIs Finites.intrs) 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);
br (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];
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 (Step_tac 1);
by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1);
by (Step_tac 1);
bw inj_onto_def;
by (Blast_tac 1);
by (thin_tac "ALL A. ?PP(A)" 1);
by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1);
by (Step_tac 1);
by (res_inst_tac [("x","xa")] bexI 1);
by (ALLGOALS Asm_simp_tac);
be equalityE 1;
br equalityI 1;
by (Blast_tac 2);
by (Best_tac 1);
val lemma = result();
goal Finite.thy "!!A. [| finite(f``A); inj_onto f A |] ==> finite A";
bd lemma 1;
by (Blast_tac 1);
qed "finite_imageD";
(** The powerset of a finite set **)
goal Finite.thy "!!A. finite(Pow A) ==> finite A";
by (subgoal_tac "finite ((%x.{x})``A)" 1);
br finite_subset 2;
ba 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";
br iffI 1;
be 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(converse r) = finite r";
by(subgoal_tac "converse r = (%(x,y).(y,x))``r" 1);
by(Asm_simp_tac 1);
br iffI 1;
be (rewrite_rule [inj_onto_def] finite_imageD) 1;
by(simp_tac (!simpset setloop (split_tac[expand_split])) 1);
be finite_imageI 1;
by(simp_tac (!simpset addsimps [converse_def,image_def]) 1);
by(Auto_tac());
br bexI 1;
ba 2;
by(Simp_tac 1);
by(split_all_tac 1);
by(Asm_full_simp_tac 1);
qed "finite_converse";
AddIffs [finite_converse];
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 (!claset));
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 (!claset))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 (!claset))1);
by (res_inst_tac [("x","k")] exI 1);
by (Asm_simp_tac 1);
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
by (Blast_tac 1);
bd 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 (!claset))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 (!claset))1);
by (res_inst_tac [("x","k")] exI 1);
by (Asm_simp_tac 1);
by (simp_tac (!simpset setloop (split_tac [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 (!claset))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 (!claset))1);
by (res_inst_tac [("x","k")] exI 1);
by (Asm_simp_tac 1);
by (simp_tac (!simpset setloop (split_tac [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);
bd finite_has_card 1;
be exE 1;
by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1);
be 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);
be subst 1;
br refl 1;
by (rtac notI 1);
by (etac exE 1);
by (dtac lemma 1);
ba 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 (strip_tac 1);
by (case_tac "x:B" 1);
by (dres_inst_tac [("A","B")] mk_disjoint_insert 1);
by (SELECT_GOAL(safe_tac (!claset))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 [Un_insert_left, Int_insert_left]
setloop split_tac [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);
br (add_right_cancel RS iffD1) 1;
br (card_Un_disjoint RS subst) 1;
be 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 (Step_tac 1);
bw 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);
bw 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 (Blast_tac 1);
by (strip_tac 1);
by (etac conjE 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 (dtac insert_lim 1);
by (Asm_full_simp_tac 1);
(*2*)
by (rotate_tac ~1 1);
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
by (case_tac "A=F" 1);
by (Asm_simp_tac 1);
by (Asm_simp_tac 1);
qed_spec_mp "psubset_card" ;
(*Relates to equivalence classes. Based on a theorem of F. Kammüller'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 (strip_tac 1);
by (REPEAT (etac conjE 1));
by (stac card_Un_disjoint 1);
by (ALLGOALS
(asm_full_simp_tac (!simpset
addsimps [dvd_add, disjoint_eq_subset_Compl])));
by (thin_tac "?PP-->?QQ" 1);
by (thin_tac "!c:F. ?PP(c)" 1);
by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1);
by (Step_tac 1);
by (ball_tac 1);
by (Blast_tac 1);
qed_spec_mp "dvd_partition";