src/HOL/Finite.ML
author berghofe
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(*  Title:      HOL/Finite.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson & Tobias Nipkow
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    Copyright   1995  University of Cambridge & TU Muenchen
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Finite sets and their cardinality
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*)
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open Finite;
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section "The finite powerset operator -- Fin";
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goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)";
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by (rtac lfp_mono 1);
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by (REPEAT (ares_tac basic_monos 1));
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qed "Fin_mono";
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goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)";
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by (fast_tac (!claset addSIs [lfp_lowerbound]) 1);
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qed "Fin_subset_Pow";
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(* A : Fin(B) ==> A <= B *)
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val FinD = Fin_subset_Pow RS subsetD RS PowD;
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(*Discharging ~ x:y entails extra work*)
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val major::prems = goal Finite.thy 
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    "[| F:Fin(A);  P({}); \
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\       !!F x. [| x:A;  F:Fin(A);  x~:F;  P(F) |] ==> P(insert x F) \
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\    |] ==> P(F)";
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by (rtac (major RS Fin.induct) 1);
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by (excluded_middle_tac "a:b" 2);
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by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3);   (*backtracking!*)
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by (REPEAT (ares_tac prems 1));
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qed "Fin_induct";
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Addsimps Fin.intrs;
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(*The union of two finite sets is finite*)
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val major::prems = goal Finite.thy
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    "[| F: Fin(A);  G: Fin(A) |] ==> F Un G : Fin(A)";
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by (rtac (major RS Fin_induct) 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps (prems @ [Un_insert_left]))));
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qed "Fin_UnI";
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(*Every subset of a finite set is finite*)
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val [subs,fin] = goal Finite.thy "[| A<=B;  B: Fin(M) |] ==> A: Fin(M)";
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by (EVERY1 [subgoal_tac "ALL C. C<=B --> C: Fin(M)",
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            rtac mp, etac spec,
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            rtac subs]);
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by (rtac (fin RS Fin_induct) 1);
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by (simp_tac (!simpset addsimps [subset_Un_eq]) 1);
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by (safe_tac (!claset addSDs [subset_insert_iff RS iffD1]));
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by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2);
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by (ALLGOALS Asm_simp_tac);
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qed "Fin_subset";
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goal Finite.thy "(F Un G : Fin(A)) = (F: Fin(A) & G: Fin(A))";
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by (fast_tac (!claset addIs [Fin_UnI] addDs
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                [Un_upper1 RS Fin_subset, Un_upper2 RS Fin_subset]) 1);
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qed "subset_Fin";
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Addsimps[subset_Fin];
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goal Finite.thy "(insert a A : Fin M) = (a:M & A : Fin M)";
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by (stac insert_is_Un 1);
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by (Simp_tac 1);
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by (fast_tac (!claset addSIs Fin.intrs addDs [FinD]) 1);
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qed "insert_Fin";
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Addsimps[insert_Fin];
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(*The image of a finite set is finite*)
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val major::_ = goal Finite.thy
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    "F: Fin(A) ==> h``F : Fin(h``A)";
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by (rtac (major RS Fin_induct) 1);
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by (Simp_tac 1);
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by (asm_simp_tac
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    (!simpset addsimps [image_eqI RS Fin.insertI, image_insert]
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	      delsimps [insert_Fin]) 1);
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qed "Fin_imageI";
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val major::prems = goal Finite.thy 
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    "[| c: Fin(A);  b: Fin(A);                                  \
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\       P(b);                                                   \
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\       !!(x::'a) y. [| x:A; y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
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\    |] ==> c<=b --> P(b-c)";
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by (rtac (major RS Fin_induct) 1);
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by (rtac (Diff_insert RS ssubst) 2);
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by (ALLGOALS (asm_simp_tac
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                (!simpset addsimps (prems@[Diff_subset RS Fin_subset]))));
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val lemma = result();
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val prems = goal Finite.thy 
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    "[| b: Fin(A);                                              \
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\       P(b);                                                   \
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\       !!x y. [| x:A; y: Fin(A);  x:y;  P(y) |] ==> P(y-{x}) \
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\    |] ==> P({})";
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by (rtac (Diff_cancel RS subst) 1);
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by (rtac (lemma RS mp) 1);
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by (REPEAT (ares_tac (subset_refl::prems) 1));
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qed "Fin_empty_induct";
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section "The predicate 'finite'";
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val major::prems = goalw Finite.thy [finite_def]
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    "[| finite F;  P({}); \
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\       !!F x. [| finite F;  x~:F;  P(F) |] ==> P(insert x F) \
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\    |] ==> P(F)";
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by (rtac (major RS Fin_induct) 1);
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by (REPEAT (ares_tac prems 1));
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qed "finite_induct";
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goalw Finite.thy [finite_def] "finite {}";
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by (Simp_tac 1);
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qed "finite_emptyI";
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Addsimps [finite_emptyI];
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goalw Finite.thy [finite_def] "!!A. finite A ==> finite(insert a A)";
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by (Asm_simp_tac 1);
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qed "finite_insertI";
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(*The union of two finite sets is finite*)
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goalw Finite.thy [finite_def]
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    "!!F. [| finite F;  finite G |] ==> finite(F Un G)";
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by (Asm_simp_tac 1);
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qed "finite_UnI";
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goalw Finite.thy [finite_def] "!!A. [| A<=B;  finite B |] ==> finite A";
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by (etac Fin_subset 1);
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by (assume_tac 1);
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qed "finite_subset";
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goalw Finite.thy [finite_def] "finite(F Un G) = (finite F & finite G)";
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by (Simp_tac 1);
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qed "subset_finite";
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Addsimps[subset_finite];
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goalw Finite.thy [finite_def] "finite(insert a A) = finite(A)";
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by (Simp_tac 1);
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qed "insert_finite";
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Addsimps[insert_finite];
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(* finite B ==> finite (B - Ba) *)
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bind_thm ("finite_Diff", Diff_subset RS finite_subset);
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Addsimps [finite_Diff];
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(*The image of a finite set is finite*)
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goal Finite.thy "!!F. finite F ==> finite(h``F)";
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by (etac finite_induct 1);
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by (ALLGOALS Asm_simp_tac);
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qed "finite_imageI";
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   152
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   153
val major::prems = goalw Finite.thy [finite_def]
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   154
    "[| finite A;                                       \
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   155
\       P(A);                                           \
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   156
\       !!a A. [| finite A;  a:A;  P(A) |] ==> P(A-{a}) \
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   157
\    |] ==> P({})";
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   158
by (rtac (major RS Fin_empty_induct) 1);
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   159
by (REPEAT (ares_tac (subset_refl::prems) 1));
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   160
qed "finite_empty_induct";
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   161
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   162
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   163
section "Finite cardinality -- 'card'";
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   164
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   165
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}";
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   166
by (Fast_tac 1);
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   167
val Collect_conv_insert = result();
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   168
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   169
goalw Finite.thy [card_def] "card {} = 0";
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   170
by (rtac Least_equality 1);
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   171
by (ALLGOALS Asm_full_simp_tac);
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   172
qed "card_empty";
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   173
Addsimps [card_empty];
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   174
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   175
val [major] = goal Finite.thy
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   176
  "finite A ==> ? (n::nat) f. A = {f i |i. i<n}";
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   177
by (rtac (major RS finite_induct) 1);
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   178
 by (res_inst_tac [("x","0")] exI 1);
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   179
 by (Simp_tac 1);
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   180
by (etac exE 1);
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   181
by (etac exE 1);
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   182
by (hyp_subst_tac 1);
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   183
by (res_inst_tac [("x","Suc n")] exI 1);
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   184
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1);
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   185
by (asm_simp_tac (!simpset addsimps [Collect_conv_insert, less_Suc_eq]
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                          addcongs [rev_conj_cong]) 1);
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qed "finite_has_card";
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   189
goal Finite.thy
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  "!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \
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   191
\  ? m::nat. m<n & (? g. A = {g i|i.i<m})";
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   192
by (res_inst_tac [("n","n")] natE 1);
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   193
 by (hyp_subst_tac 1);
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   194
 by (Asm_full_simp_tac 1);
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   195
by (rename_tac "m" 1);
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   196
by (hyp_subst_tac 1);
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   197
by (case_tac "? a. a:A" 1);
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   198
 by (res_inst_tac [("x","0")] exI 2);
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   199
 by (Simp_tac 2);
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   200
 by (Fast_tac 2);
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   201
by (etac exE 1);
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   202
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);
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   203
by (rtac exI 1);
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   204
by (rtac (refl RS disjI2 RS conjI) 1);
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   205
by (etac equalityE 1);
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   206
by (asm_full_simp_tac
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   207
     (!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1);
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   208
by (SELECT_GOAL(safe_tac (!claset))1);
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   209
  by (Asm_full_simp_tac 1);
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   210
  by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
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   211
  by (SELECT_GOAL(safe_tac (!claset))1);
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   212
   by (subgoal_tac "x ~= f m" 1);
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berghofe
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   213
    by (Fast_tac 2);
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   214
   by (subgoal_tac "? k. f k = x & k<m" 1);
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   215
    by (best_tac (!claset) 2);
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berghofe
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   216
   by (SELECT_GOAL(safe_tac (!claset))1);
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   217
   by (res_inst_tac [("x","k")] exI 1);
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   218
   by (Asm_simp_tac 1);
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   219
  by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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   220
  by (best_tac (!claset) 1);
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   221
 bd sym 1;
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   222
 by (rotate_tac ~1 1);
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   223
 by (Asm_full_simp_tac 1);
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   224
 by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
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   225
 by (SELECT_GOAL(safe_tac (!claset))1);
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   226
  by (subgoal_tac "x ~= f m" 1);
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   227
   by (Fast_tac 2);
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   228
  by (subgoal_tac "? k. f k = x & k<m" 1);
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diff changeset
   229
   by (best_tac (!claset) 2);
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berghofe
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diff changeset
   230
  by (SELECT_GOAL(safe_tac (!claset))1);
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diff changeset
   231
  by (res_inst_tac [("x","k")] exI 1);
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   232
  by (Asm_simp_tac 1);
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   233
 by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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   234
 by (best_tac (!claset) 1);
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   235
by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1);
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berghofe
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diff changeset
   236
by (SELECT_GOAL(safe_tac (!claset))1);
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   237
 by (subgoal_tac "x ~= f i" 1);
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berghofe
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diff changeset
   238
  by (Fast_tac 2);
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   239
 by (case_tac "x = f m" 1);
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diff changeset
   240
  by (res_inst_tac [("x","i")] exI 1);
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paulson
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diff changeset
   241
  by (Asm_simp_tac 1);
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   242
 by (subgoal_tac "? k. f k = x & k<m" 1);
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diff changeset
   243
  by (best_tac (!claset) 2);
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berghofe
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   244
 by (SELECT_GOAL(safe_tac (!claset))1);
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   245
 by (res_inst_tac [("x","k")] exI 1);
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paulson
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diff changeset
   246
 by (Asm_simp_tac 1);
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diff changeset
   247
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
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diff changeset
   248
by (best_tac (!claset) 1);
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   249
val lemma = result();
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   250
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   251
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \
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   252
\ (LEAST n. ? f. insert x A = {f i|i.i<n}) = Suc(LEAST n. ? f. A={f i|i.i<n})";
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diff changeset
   253
by (rtac Least_equality 1);
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   254
 bd finite_has_card 1;
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   255
 be exE 1;
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diff changeset
   256
 by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1);
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   257
 be exE 1;
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   258
 by (res_inst_tac
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   259
   [("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1);
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   260
 by (simp_tac
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   261
    (!simpset addsimps [Collect_conv_insert, less_Suc_eq] 
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   262
	      addcongs [rev_conj_cong]) 1);
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   263
 be subst 1;
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   264
 br refl 1;
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   265
by (rtac notI 1);
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diff changeset
   266
by (etac exE 1);
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   267
by (dtac lemma 1);
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   268
 ba 1;
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diff changeset
   269
by (etac exE 1);
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diff changeset
   270
by (etac conjE 1);
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diff changeset
   271
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1);
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paulson
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diff changeset
   272
by (dtac le_less_trans 1 THEN atac 1);
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diff changeset
   273
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
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paulson
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diff changeset
   274
by (etac disjE 1);
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paulson
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diff changeset
   275
by (etac less_asym 1 THEN atac 1);
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paulson
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diff changeset
   276
by (hyp_subst_tac 1);
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   277
by (Asm_full_simp_tac 1);
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   278
val lemma = result();
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   279
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   280
goalw Finite.thy [card_def]
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   281
  "!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)";
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diff changeset
   282
by (etac lemma 1);
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diff changeset
   283
by (assume_tac 1);
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   284
qed "card_insert_disjoint";
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   285
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diff changeset
   286
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A";
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paulson
parents: 1553
diff changeset
   287
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1);
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paulson
parents: 1553
diff changeset
   288
by (assume_tac 1);
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   289
by (asm_simp_tac (!simpset addsimps [card_insert_disjoint]) 1);
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paulson
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   290
qed "card_Suc_Diff";
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paulson
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diff changeset
   291
372880456b5b Library changes for mutilated checkerboard
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diff changeset
   292
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A";
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   293
by (resolve_tac [Suc_less_SucD] 1);
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   294
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1);
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   295
qed "card_Diff";
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diff changeset
   296
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val [major] = goal Finite.thy
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  "finite A ==> card(insert x A) = Suc(card(A-{x}))";
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by (case_tac "x:A" 1);
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by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1);
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by (dtac mk_disjoint_insert 1);
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by (etac exE 1);
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by (Asm_simp_tac 1);
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by (rtac card_insert_disjoint 1);
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by (rtac (major RSN (2,finite_subset)) 1);
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6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
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by (Fast_tac 1);
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by (Fast_tac 1);
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by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1);
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qed "card_insert";
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Addsimps [card_insert];
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goal Finite.thy  "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)";
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by (etac finite_induct 1);
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by (Simp_tac 1);
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by (strip_tac 1);
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by (case_tac "x:B" 1);
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 by (dtac mk_disjoint_insert 1);
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8a31d85d27b8 best_tac, deepen_tac and safe_tac now also use default claset.
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 by (SELECT_GOAL(safe_tac (!claset))1);
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 by (rotate_tac ~1 1);
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 by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
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by (rotate_tac ~1 1);
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by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
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qed_spec_mp "card_mono";