(* 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 "The finite powerset operator -- Fin";
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 (fast_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
"[| F:Fin(A); P({}); \
\ !!F x. [| x:A; F:Fin(A); x~:F; P(F) |] ==> P(insert x F) \
\ |] ==> P(F)";
by (rtac (major RS Fin.induct) 1);
by (excluded_middle_tac "a:b" 2);
by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*)
by (REPEAT (ares_tac prems 1));
qed "Fin_induct";
Addsimps Fin.intrs;
(*The union of two finite sets is finite*)
val major::prems = goal Finite.thy
"[| F: Fin(A); G: Fin(A) |] ==> F Un G : Fin(A)";
by (rtac (major RS Fin_induct) 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps (prems @ [Un_insert_left]))));
qed "Fin_UnI";
(*Every subset of a finite set is finite*)
val [subs,fin] = goal Finite.thy "[| A<=B; B: Fin(M) |] ==> A: Fin(M)";
by (EVERY1 [subgoal_tac "ALL C. C<=B --> C: Fin(M)",
rtac mp, etac spec,
rtac subs]);
by (rtac (fin RS Fin_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 "Fin_subset";
goal Finite.thy "(F Un G : Fin(A)) = (F: Fin(A) & G: Fin(A))";
by (fast_tac (!claset addIs [Fin_UnI] addDs
[Un_upper1 RS Fin_subset, Un_upper2 RS Fin_subset]) 1);
qed "subset_Fin";
Addsimps[subset_Fin];
goal Finite.thy "(insert a A : Fin M) = (a:M & A : Fin M)";
by (stac insert_is_Un 1);
by (Simp_tac 1);
by (fast_tac (!claset addSIs Fin.intrs addDs [FinD]) 1);
qed "insert_Fin";
Addsimps[insert_Fin];
(*The image of a finite set is finite*)
val major::_ = goal Finite.thy
"F: Fin(A) ==> h``F : Fin(h``A)";
by (rtac (major RS Fin_induct) 1);
by (Simp_tac 1);
by (asm_simp_tac
(!simpset addsimps [image_eqI RS Fin.insertI, image_insert]
delsimps [insert_Fin]) 1);
qed "Fin_imageI";
val major::prems = goal Finite.thy
"[| c: Fin(A); b: Fin(A); \
\ P(b); \
\ !!(x::'a) y. [| x:A; y: Fin(A); x:y; P(y) |] ==> P(y-{x}) \
\ |] ==> c<=b --> P(b-c)";
by (rtac (major RS Fin_induct) 1);
by (stac Diff_insert 2);
by (ALLGOALS (asm_simp_tac
(!simpset addsimps (prems@[Diff_subset RS Fin_subset]))));
val lemma = result();
val prems = goal Finite.thy
"[| b: Fin(A); \
\ P(b); \
\ !!x y. [| x:A; y: Fin(A); x:y; P(y) |] ==> P(y-{x}) \
\ |] ==> P({})";
by (rtac (Diff_cancel RS subst) 1);
by (rtac (lemma RS mp) 1);
by (REPEAT (ares_tac (subset_refl::prems) 1));
qed "Fin_empty_induct";
section "The predicate 'finite'";
val major::prems = goalw Finite.thy [finite_def]
"[| finite F; P({}); \
\ !!F x. [| finite F; x~:F; P(F) |] ==> P(insert x F) \
\ |] ==> P(F)";
by (rtac (major RS Fin_induct) 1);
by (REPEAT (ares_tac prems 1));
qed "finite_induct";
goalw Finite.thy [finite_def] "finite {}";
by (Simp_tac 1);
qed "finite_emptyI";
Addsimps [finite_emptyI];
goalw Finite.thy [finite_def] "!!A. finite A ==> finite(insert a A)";
by (Asm_simp_tac 1);
qed "finite_insertI";
(*The union of two finite sets is finite*)
goalw Finite.thy [finite_def]
"!!F. [| finite F; finite G |] ==> finite(F Un G)";
by (Asm_simp_tac 1);
qed "finite_UnI";
goalw Finite.thy [finite_def] "!!A. [| A<=B; finite B |] ==> finite A";
by (etac Fin_subset 1);
by (assume_tac 1);
qed "finite_subset";
goalw Finite.thy [finite_def] "finite(F Un G) = (finite F & finite G)";
by (Simp_tac 1);
qed "subset_finite";
Addsimps[subset_finite];
goalw Finite.thy [finite_def] "finite(insert a A) = finite(A)";
by (Simp_tac 1);
qed "insert_finite";
Addsimps[insert_finite];
(* finite B ==> finite (B - Ba) *)
bind_thm ("finite_Diff", Diff_subset RS finite_subset);
Addsimps [finite_Diff];
(*The image of a finite set is finite*)
goal Finite.thy "!!F. finite F ==> finite(h``F)";
by (etac finite_induct 1);
by (ALLGOALS Asm_simp_tac);
qed "finite_imageI";
val major::prems = goalw Finite.thy [finite_def]
"[| finite A; \
\ P(A); \
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \
\ |] ==> P({})";
by (rtac (major RS Fin_empty_induct) 1);
by (REPEAT (ares_tac (subset_refl::prems) 1));
qed "finite_empty_induct";
section "Finite cardinality -- 'card'";
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}";
by (Fast_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 (Fast_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 (SELECT_GOAL(safe_tac (!claset))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 (Fast_tac 2);
by (subgoal_tac "? k. f k = x & k<m" 1);
by (best_tac (!claset) 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 (best_tac (!claset) 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 (Fast_tac 2);
by (subgoal_tac "? k. f k = x & k<m" 1);
by (best_tac (!claset) 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 (best_tac (!claset) 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 (Fast_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 (best_tac (!claset) 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 (best_tac (!claset) 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";
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 (!simpset addsimps [card_insert_disjoint]) 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";
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 (Fast_tac 1);
by (Fast_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 ==> !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 (dtac 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";