src/HOL/Finite.ML
author paulson
Tue, 27 May 1997 13:03:41 +0200
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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 (blast_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 (blast_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 (blast_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 (stac Diff_insert 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 "finite_Un_eq";
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Addsimps[finite_Un_eq];
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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 "finite_insert";
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Addsimps[finite_insert];
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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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(*The powerset of a finite set is finite*)
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   154
goal Finite.thy "!!A. finite A ==> finite(Pow A)";
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   155
by (etac finite_induct 1);
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   156
by (ALLGOALS 
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   157
    (asm_simp_tac
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   158
     (!simpset addsimps [finite_UnI, finite_imageI, Pow_insert])));
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   159
qed "finite_PowI";
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   160
AddSIs [finite_PowI];
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   161
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   162
val major::prems = goalw Finite.thy [finite_def]
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   163
    "[| finite A;                                       \
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   164
\       P(A);                                           \
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   165
\       !!a A. [| finite A;  a:A;  P(A) |] ==> P(A-{a}) \
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   166
\    |] ==> P({})";
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   167
by (rtac (major RS Fin_empty_induct) 1);
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diff changeset
   168
by (REPEAT (ares_tac (subset_refl::prems) 1));
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   169
qed "finite_empty_induct";
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   170
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   171
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   172
section "Finite cardinality -- 'card'";
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   173
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   174
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}";
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   175
by (Blast_tac 1);
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   176
val Collect_conv_insert = result();
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   177
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   178
goalw Finite.thy [card_def] "card {} = 0";
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   179
by (rtac Least_equality 1);
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   180
by (ALLGOALS Asm_full_simp_tac);
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   181
qed "card_empty";
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   182
Addsimps [card_empty];
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   183
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   184
val [major] = goal Finite.thy
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   185
  "finite A ==> ? (n::nat) f. A = {f i |i. i<n}";
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diff changeset
   186
by (rtac (major RS finite_induct) 1);
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   187
 by (res_inst_tac [("x","0")] exI 1);
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diff changeset
   188
 by (Simp_tac 1);
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paulson
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diff changeset
   189
by (etac exE 1);
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paulson
parents: 1548
diff changeset
   190
by (etac exE 1);
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paulson
parents: 1548
diff changeset
   191
by (hyp_subst_tac 1);
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diff changeset
   192
by (res_inst_tac [("x","Suc n")] exI 1);
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diff changeset
   193
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1);
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diff changeset
   194
by (asm_simp_tac (!simpset addsimps [Collect_conv_insert, less_Suc_eq]
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   195
                          addcongs [rev_conj_cong]) 1);
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   196
qed "finite_has_card";
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   197
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   198
goal Finite.thy
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   199
  "!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \
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   200
\  ? m::nat. m<n & (? g. A = {g i|i.i<m})";
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diff changeset
   201
by (res_inst_tac [("n","n")] natE 1);
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diff changeset
   202
 by (hyp_subst_tac 1);
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diff changeset
   203
 by (Asm_full_simp_tac 1);
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diff changeset
   204
by (rename_tac "m" 1);
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parents: 1548
diff changeset
   205
by (hyp_subst_tac 1);
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diff changeset
   206
by (case_tac "? a. a:A" 1);
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diff changeset
   207
 by (res_inst_tac [("x","0")] exI 2);
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diff changeset
   208
 by (Simp_tac 2);
2922
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diff changeset
   209
 by (Blast_tac 2);
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diff changeset
   210
by (etac exE 1);
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diff changeset
   211
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);
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diff changeset
   212
by (rtac exI 1);
1782
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paulson
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diff changeset
   213
by (rtac (refl RS disjI2 RS conjI) 1);
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diff changeset
   214
by (etac equalityE 1);
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diff changeset
   215
by (asm_full_simp_tac
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   216
     (!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1);
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diff changeset
   217
by (safe_tac (!claset));
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diff changeset
   218
  by (Asm_full_simp_tac 1);
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diff changeset
   219
  by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
1786
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berghofe
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diff changeset
   220
  by (SELECT_GOAL(safe_tac (!claset))1);
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diff changeset
   221
   by (subgoal_tac "x ~= f m" 1);
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diff changeset
   222
    by (Blast_tac 2);
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diff changeset
   223
   by (subgoal_tac "? k. f k = x & k<m" 1);
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diff changeset
   224
    by (Blast_tac 2);
1786
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berghofe
parents: 1782
diff changeset
   225
   by (SELECT_GOAL(safe_tac (!claset))1);
1553
4eb4a9c7d736 Ran expandshort
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parents: 1548
diff changeset
   226
   by (res_inst_tac [("x","k")] exI 1);
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paulson
parents: 1548
diff changeset
   227
   by (Asm_simp_tac 1);
4eb4a9c7d736 Ran expandshort
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diff changeset
   228
  by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
2922
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diff changeset
   229
  by (Blast_tac 1);
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   230
 bd sym 1;
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diff changeset
   231
 by (rotate_tac ~1 1);
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parents: 1548
diff changeset
   232
 by (Asm_full_simp_tac 1);
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diff changeset
   233
 by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1);
1786
8a31d85d27b8 best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
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diff changeset
   234
 by (SELECT_GOAL(safe_tac (!claset))1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   235
  by (subgoal_tac "x ~= f m" 1);
2922
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diff changeset
   236
   by (Blast_tac 2);
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4eb4a9c7d736 Ran expandshort
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parents: 1548
diff changeset
   237
  by (subgoal_tac "? k. f k = x & k<m" 1);
2922
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paulson
parents: 2031
diff changeset
   238
   by (Blast_tac 2);
1786
8a31d85d27b8 best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents: 1782
diff changeset
   239
  by (SELECT_GOAL(safe_tac (!claset))1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   240
  by (res_inst_tac [("x","k")] exI 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   241
  by (Asm_simp_tac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   242
 by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
2922
580647a879cf Using Blast_tac
paulson
parents: 2031
diff changeset
   243
 by (Blast_tac 1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   244
by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1);
1786
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berghofe
parents: 1782
diff changeset
   245
by (SELECT_GOAL(safe_tac (!claset))1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   246
 by (subgoal_tac "x ~= f i" 1);
2922
580647a879cf Using Blast_tac
paulson
parents: 2031
diff changeset
   247
  by (Blast_tac 2);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   248
 by (case_tac "x = f m" 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   249
  by (res_inst_tac [("x","i")] exI 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   250
  by (Asm_simp_tac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   251
 by (subgoal_tac "? k. f k = x & k<m" 1);
2922
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paulson
parents: 2031
diff changeset
   252
  by (Blast_tac 2);
1786
8a31d85d27b8 best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents: 1782
diff changeset
   253
 by (SELECT_GOAL(safe_tac (!claset))1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   254
 by (res_inst_tac [("x","k")] exI 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   255
 by (Asm_simp_tac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   256
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
2922
580647a879cf Using Blast_tac
paulson
parents: 2031
diff changeset
   257
by (Blast_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   258
val lemma = result();
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diff changeset
   259
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   260
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   261
\ (LEAST n. ? f. insert x A = {f i|i.i<n}) = Suc(LEAST n. ? f. A={f i|i.i<n})";
1553
4eb4a9c7d736 Ran expandshort
paulson
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diff changeset
   262
by (rtac Least_equality 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   263
 bd finite_has_card 1;
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diff changeset
   264
 be exE 1;
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   265
 by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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parents: 1465
diff changeset
   266
 be exE 1;
1553
4eb4a9c7d736 Ran expandshort
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diff changeset
   267
 by (res_inst_tac
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e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
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diff changeset
   268
   [("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1);
1553
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paulson
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diff changeset
   269
 by (simp_tac
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parents: 1618
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   270
    (!simpset addsimps [Collect_conv_insert, less_Suc_eq] 
2031
03a843f0f447 Ran expandshort
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diff changeset
   271
              addcongs [rev_conj_cong]) 1);
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diff changeset
   272
 be subst 1;
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diff changeset
   273
 br refl 1;
1553
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paulson
parents: 1548
diff changeset
   274
by (rtac notI 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   275
by (etac exE 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   276
by (dtac lemma 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   277
 ba 1;
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   278
by (etac exE 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   279
by (etac conjE 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   280
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   281
by (dtac le_less_trans 1 THEN atac 1);
1660
8cb42cd97579 *** empty log message ***
oheimb
parents: 1618
diff changeset
   282
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   283
by (etac disjE 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   284
by (etac less_asym 1 THEN atac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   285
by (hyp_subst_tac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   286
by (Asm_full_simp_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   287
val lemma = result();
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diff changeset
   288
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   289
goalw Finite.thy [card_def]
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diff changeset
   290
  "!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)";
1553
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paulson
parents: 1548
diff changeset
   291
by (etac lemma 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   292
by (assume_tac 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
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diff changeset
   293
qed "card_insert_disjoint";
3352
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   294
Addsimps [card_insert_disjoint];
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   295
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   296
goal Finite.thy  "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)";
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   297
by (etac finite_induct 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   298
by (Simp_tac 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   299
by (strip_tac 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   300
by (case_tac "x:B" 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   301
 by (dtac mk_disjoint_insert 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   302
 by (SELECT_GOAL(safe_tac (!claset))1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   303
 by (rotate_tac ~1 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   304
 by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   305
by (rotate_tac ~1 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   306
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   307
qed_spec_mp "card_mono";
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   308
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   309
goal Finite.thy "!!A B. [| finite A; finite B |]\
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   310
\                       ==> A Int B = {} --> card(A Un B) = card A + card B";
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   311
by (etac finite_induct 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   312
by (ALLGOALS 
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   313
    (asm_simp_tac (!simpset addsimps [Un_insert_left, Int_insert_left]
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   314
		            setloop split_tac [expand_if])));
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   315
qed_spec_mp "card_Un_disjoint";
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   316
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   317
goal Finite.thy "!!A. [| finite A; B<=A |] ==> card A - card B = card (A - B)";
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   318
by (subgoal_tac "(A-B) Un B = A" 1);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   319
by (Blast_tac 2);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   320
br (add_right_cancel RS iffD1) 1;
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   321
br (card_Un_disjoint RS subst) 1;
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   322
be ssubst 4;
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   323
by (Blast_tac 3);
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   324
by (ALLGOALS 
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   325
    (asm_simp_tac
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   326
     (!simpset addsimps [add_commute, not_less_iff_le, 
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   327
			 add_diff_inverse, card_mono, finite_subset])));
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   328
qed "card_Diff_subset";
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   329
1618
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   330
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A";
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   331
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1);
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   332
by (assume_tac 1);
3352
04502e5431fb New theorems suggested by Florian Kammueller
paulson
parents: 3340
diff changeset
   333
by (Asm_simp_tac 1);
1618
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   334
qed "card_Suc_Diff";
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   335
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   336
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A";
2031
03a843f0f447 Ran expandshort
paulson
parents: 1786
diff changeset
   337
by (rtac Suc_less_SucD 1);
1618
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   338
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1);
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   339
qed "card_Diff";
372880456b5b Library changes for mutilated checkerboard
paulson
parents: 1553
diff changeset
   340
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   341
val [major] = goal Finite.thy
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   342
  "finite A ==> card(insert x A) = Suc(card(A-{x}))";
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   343
by (case_tac "x:A" 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   344
by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   345
by (dtac mk_disjoint_insert 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   346
by (etac exE 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   347
by (Asm_simp_tac 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   348
by (rtac card_insert_disjoint 1);
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   349
by (rtac (major RSN (2,finite_subset)) 1);
2922
580647a879cf Using Blast_tac
paulson
parents: 2031
diff changeset
   350
by (Blast_tac 1);
580647a879cf Using Blast_tac
paulson
parents: 2031
diff changeset
   351
by (Blast_tac 1);
1553
4eb4a9c7d736 Ran expandshort
paulson
parents: 1548
diff changeset
   352
by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1);
1531
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   353
qed "card_insert";
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   354
Addsimps [card_insert];
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   355
e5eb247ad13c Added a constant UNIV == {x.True}
nipkow
parents: 1465
diff changeset
   356
3340
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   357
goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A";
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   358
by (etac finite_induct 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   359
by (ALLGOALS Asm_simp_tac);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   360
by (Step_tac 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   361
bw inj_onto_def;
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   362
by (Blast_tac 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   363
by (stac card_insert_disjoint 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   364
by (etac finite_imageI 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   365
by (Blast_tac 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   366
by (Blast_tac 1);
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   367
qed_spec_mp "card_image";
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   368
a886795c9dce Two results suggested by Florian Kammueller
paulson
parents: 3222
diff changeset
   369
3222
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   370
goalw Finite.thy [psubset_def]
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   371
"!!B. finite B ==> !A. A < B --> card(A) < card(B)";
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   372
by (etac finite_induct 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   373
by (Simp_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   374
by (Blast_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   375
by (strip_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   376
by (etac conjE 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   377
by (case_tac "x:A" 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   378
(*1*)
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   379
by (dtac mk_disjoint_insert 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   380
by (etac exE 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   381
by (etac conjE 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   382
by (hyp_subst_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   383
by (rotate_tac ~1 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   384
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   385
by (dtac insert_lim 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   386
by (Asm_full_simp_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   387
(*2*)
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   388
by (rotate_tac ~1 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   389
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   390
by (case_tac "A=F" 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   391
by (Asm_simp_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   392
by (Asm_simp_tac 1);
726a9b069947 Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents: 2922
diff changeset
   393
qed_spec_mp "psubset_card" ;