added lemma.
(* 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 [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 ==> !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 ==> (!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];
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";
(** 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];
(* A bounded set of natural numbers is finite *)
Goal "!N. (!i:N. i<(n::nat)) --> finite N";
by(induct_tac "n" 1);
by(Simp_tac 1);
by(asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
br allI 1;
by(case_tac "n : N" 1);
by(ftac insert_Diff 1);
by(etac subst 1);
br impI 1;
br (finite_insert RS iffD2) 1;
by(EVERY1[etac allE, etac mp]);
by(Blast_tac 1);
by(Blast_tac 1);
qed_spec_mp "bounded_nat_set_is_finite";
section "Finite cardinality -- 'card'";
(* Ugly proofs for the traditional definition
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 (case_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];
*)
val cardR_emptyE = cardR.mk_cases "({},n) : cardR";
AddSEs [cardR_emptyE];
val cardR_insertE = cardR.mk_cases "(insert a A,n) : cardR";
AddSIs cardR.intrs;
Goal "[| (A,n) : cardR |] ==> a : A --> (? m. n = Suc m)";
by (etac cardR.induct 1);
by (Blast_tac 1);
by (Blast_tac 1);
qed "cardR_SucD";
Goal "(A,m): cardR ==> (!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 ==> (!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);
by (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 "? 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 select_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(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 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 [le_SucI, subset_insert_iff]) 2);
by (force_tac (claset(),
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 |] ==> 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 ==> 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";
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 (simpset() addsimps [less_SucI])));
qed_spec_mp "psubset_card" ;
Goal "[| A <= B; card B <= card A; finite B |] ==> A = B";
by (case_tac "A < B" 1);
by (datac psubset_card 1 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [psubset_eq])));
qed "card_seteq";
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 (etac card_seteq 1);
by (dtac (card_image RS sym) 1);
by Auto_tac;
qed "endo_inj_surj";
(*** Cardinality of the Powerset ***)
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";
(*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";
(*** foldSet ***)
val 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);
by (case_tac "xa=xb" 1);
by (subgoal_tac "Aa = Ab" 1);
by (blast_tac (claset() addEs [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() addEs [equalityE]) 2);
by (Clarify_tac 1);
by (subgoal_tac "Aa = insert xb Ab - {xa}" 1);
by (blast_tac (claset() addEs [equalityE]) 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 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() 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 [normalize_thm [RSspec, RSmp] 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 select_equality 1);
by (auto_tac (claset() addIs [finite_imp_foldSet],
simpset() addcongs [conj_cong]
addsimps [symmetric fold_def,
fold_equality]));
qed "fold_insert";
(* Delete rules to do with foldSet relation: obsolete *)
Delsimps [foldSet_imp_finite];
Delrules [empty_foldSetE];
Delrules foldSet.intrs;
Close_locale "LC";
Open_locale "ACe";
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";
Delrules ([empty_foldSetE] @ foldSet.intrs);
Delsimps [foldSet_imp_finite];
(*** setsum ***)
Goalw [setsum_def] "setsum f {} = 0";
by (Simp_tac 1);
qed "setsum_empty";
Addsimps [setsum_empty];
Goalw [setsum_def]
"[| finite F; a ~: F |] ==> setsum f (insert a F) = f(a) + setsum f F";
by (asm_simp_tac (simpset() addsimps [export fold_insert]) 1);
qed "setsum_insert";
Addsimps [setsum_insert];
Goalw [setsum_def]
"[| finite A; finite B; A Int B = {} |] ==> \
\ setsum f (A Un B) = setsum f A + setsum f B";
by (asm_simp_tac (simpset() addsimps [export fold_Un_disjoint2]) 1);
qed_spec_mp "setsum_disj_Un";
Goal "[| finite F |] ==> \
\ setsum f (F-{a}) = (if a:F then setsum f F - f a else setsum f F)";
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";
(*** Basic theorem about "choose". By Florian Kammueller, tidied by LCP ***)
Goal "finite S ==> (card S = 0) = (S = {})";
by (auto_tac (claset() addDs [card_Suc_Diff1],
simpset()));
qed "card_0_empty_iff";
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_empty_iff]) 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 M; x ~: M |] \
\ ==> card{s. EX t. t <= M & card(t) = k & s = insert x t} = \
\ card {s. s <= M & card(s) = k}";
by (res_inst_tac [("f", "%s. s - {x}"), ("g","insert x")] card_bij_eq 1);
by (res_inst_tac [("B","Pow(insert x M)")] finite_subset 1);
by (res_inst_tac [("B","Pow(M)")] 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 {s. s <= A & card s = 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
(simp_tac (simpset() addcongs [conj_cong] addsimps [card_s_0_eq_empty])));
by (stac choose_deconstruct 1);
by (assume_tac 1);
by (assume_tac 1);
by (stac card_Un_disjoint 1);
by (Force_tac 3);
(** LEVEL 10 **)
(* use bijection *)
by (force_tac (claset(), simpset() addsimps [constr_bij]) 3);
(* finite goal *)
by (res_inst_tac [("B","Pow F")] finite_subset 1);
by (Blast_tac 1);
by (etac (finite_Pow_iff RS iffD2) 1);
(* finite goal *)
by (res_inst_tac [("B","Pow (insert x F)")] finite_subset 1);
by (Blast_tac 1);
by (blast_tac (claset() addIs [finite_Pow_iff RS iffD2]) 1);
qed "n_sub_lemma";
Goal "finite M ==> card {s. s <= M & card(s) = k} = ((card M) choose k)";
by (asm_simp_tac (simpset() addsimps [n_sub_lemma]) 1);
qed "n_subsets";