src/HOL/NumberTheory/Finite2.thy
 author haftmann Wed Sep 26 20:27:55 2007 +0200 (2007-09-26) changeset 24728 e2b3a1065676 parent 22274 ce1459004c8d child 25592 e8ddaf6bf5df permissions -rw-r--r--
moved Finite_Set before Datatype
```     1 (*  Title:      HOL/Quadratic_Reciprocity/Finite2.thy
```
```     2     ID:         \$Id\$
```
```     3     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     4 *)
```
```     5
```
```     6 header {*Finite Sets and Finite Sums*}
```
```     7
```
```     8 theory Finite2
```
```     9 imports IntFact Infinite_Set
```
```    10 begin
```
```    11
```
```    12 text{*
```
```    13   These are useful for combinatorial and number-theoretic counting
```
```    14   arguments.
```
```    15 *}
```
```    16
```
```    17
```
```    18 subsection {* Useful properties of sums and products *}
```
```    19
```
```    20 lemma setsum_same_function_zcong:
```
```    21   assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
```
```    22   shows "[setsum f S = setsum g S] (mod m)"
```
```    23 proof cases
```
```    24   assume "finite S"
```
```    25   thus ?thesis using a by induct (simp_all add: zcong_zadd)
```
```    26 next
```
```    27   assume "infinite S" thus ?thesis by(simp add:setsum_def)
```
```    28 qed
```
```    29
```
```    30 lemma setprod_same_function_zcong:
```
```    31   assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
```
```    32   shows "[setprod f S = setprod g S] (mod m)"
```
```    33 proof cases
```
```    34   assume "finite S"
```
```    35   thus ?thesis using a by induct (simp_all add: zcong_zmult)
```
```    36 next
```
```    37   assume "infinite S" thus ?thesis by(simp add:setprod_def)
```
```    38 qed
```
```    39
```
```    40 lemma setsum_const: "finite X ==> setsum (%x. (c :: int)) X = c * int(card X)"
```
```    41   apply (induct set: finite)
```
```    42   apply (auto simp add: left_distrib right_distrib int_eq_of_nat)
```
```    43   done
```
```    44
```
```    45 lemma setsum_const2: "finite X ==> int (setsum (%x. (c :: nat)) X) =
```
```    46     int(c) * int(card X)"
```
```    47   apply (induct set: finite)
```
```    48   apply (auto simp add: zadd_zmult_distrib2)
```
```    49   done
```
```    50
```
```    51 lemma setsum_const_mult: "finite A ==> setsum (%x. c * ((f x)::int)) A =
```
```    52     c * setsum f A"
```
```    53   by (induct set: finite) (auto simp add: zadd_zmult_distrib2)
```
```    54
```
```    55
```
```    56 subsection {* Cardinality of explicit finite sets *}
```
```    57
```
```    58 lemma finite_surjI: "[| B \<subseteq> f ` A; finite A |] ==> finite B"
```
```    59   by (simp add: finite_subset finite_imageI)
```
```    60
```
```    61 lemma bdd_nat_set_l_finite: "finite {y::nat . y < x}"
```
```    62   by (rule bounded_nat_set_is_finite) blast
```
```    63
```
```    64 lemma bdd_nat_set_le_finite: "finite {y::nat . y \<le> x}"
```
```    65 proof -
```
```    66   have "{y::nat . y \<le> x} = {y::nat . y < Suc x}" by auto
```
```    67   then show ?thesis by (auto simp add: bdd_nat_set_l_finite)
```
```    68 qed
```
```    69
```
```    70 lemma  bdd_int_set_l_finite: "finite {x::int. 0 \<le> x & x < n}"
```
```    71   apply (subgoal_tac " {(x :: int). 0 \<le> x & x < n} \<subseteq>
```
```    72       int ` {(x :: nat). x < nat n}")
```
```    73    apply (erule finite_surjI)
```
```    74    apply (auto simp add: bdd_nat_set_l_finite image_def)
```
```    75   apply (rule_tac x = "nat x" in exI, simp)
```
```    76   done
```
```    77
```
```    78 lemma bdd_int_set_le_finite: "finite {x::int. 0 \<le> x & x \<le> n}"
```
```    79   apply (subgoal_tac "{x. 0 \<le> x & x \<le> n} = {x. 0 \<le> x & x < n + 1}")
```
```    80    apply (erule ssubst)
```
```    81    apply (rule bdd_int_set_l_finite)
```
```    82   apply auto
```
```    83   done
```
```    84
```
```    85 lemma bdd_int_set_l_l_finite: "finite {x::int. 0 < x & x < n}"
```
```    86 proof -
```
```    87   have "{x::int. 0 < x & x < n} \<subseteq> {x::int. 0 \<le> x & x < n}"
```
```    88     by auto
```
```    89   then show ?thesis by (auto simp add: bdd_int_set_l_finite finite_subset)
```
```    90 qed
```
```    91
```
```    92 lemma bdd_int_set_l_le_finite: "finite {x::int. 0 < x & x \<le> n}"
```
```    93 proof -
```
```    94   have "{x::int. 0 < x & x \<le> n} \<subseteq> {x::int. 0 \<le> x & x \<le> n}"
```
```    95     by auto
```
```    96   then show ?thesis by (auto simp add: bdd_int_set_le_finite finite_subset)
```
```    97 qed
```
```    98
```
```    99 lemma card_bdd_nat_set_l: "card {y::nat . y < x} = x"
```
```   100 proof (induct x)
```
```   101   case 0
```
```   102   show "card {y::nat . y < 0} = 0" by simp
```
```   103 next
```
```   104   case (Suc n)
```
```   105   have "{y. y < Suc n} = insert n {y. y < n}"
```
```   106     by auto
```
```   107   then have "card {y. y < Suc n} = card (insert n {y. y < n})"
```
```   108     by auto
```
```   109   also have "... = Suc (card {y. y < n})"
```
```   110     by (rule card_insert_disjoint) (auto simp add: bdd_nat_set_l_finite)
```
```   111   finally show "card {y. y < Suc n} = Suc n"
```
```   112     using `card {y. y < n} = n` by simp
```
```   113 qed
```
```   114
```
```   115 lemma card_bdd_nat_set_le: "card { y::nat. y \<le> x} = Suc x"
```
```   116 proof -
```
```   117   have "{y::nat. y \<le> x} = { y::nat. y < Suc x}"
```
```   118     by auto
```
```   119   then show ?thesis by (auto simp add: card_bdd_nat_set_l)
```
```   120 qed
```
```   121
```
```   122 lemma card_bdd_int_set_l: "0 \<le> (n::int) ==> card {y. 0 \<le> y & y < n} = nat n"
```
```   123 proof -
```
```   124   assume "0 \<le> n"
```
```   125   have "inj_on (%y. int y) {y. y < nat n}"
```
```   126     by (auto simp add: inj_on_def)
```
```   127   hence "card (int ` {y. y < nat n}) = card {y. y < nat n}"
```
```   128     by (rule card_image)
```
```   129   also from `0 \<le> n` have "int ` {y. y < nat n} = {y. 0 \<le> y & y < n}"
```
```   130     apply (auto simp add: zless_nat_eq_int_zless image_def)
```
```   131     apply (rule_tac x = "nat x" in exI)
```
```   132     apply (auto simp add: nat_0_le)
```
```   133     done
```
```   134   also have "card {y. y < nat n} = nat n"
```
```   135     by (rule card_bdd_nat_set_l)
```
```   136   finally show "card {y. 0 \<le> y & y < n} = nat n" .
```
```   137 qed
```
```   138
```
```   139 lemma card_bdd_int_set_le: "0 \<le> (n::int) ==> card {y. 0 \<le> y & y \<le> n} =
```
```   140   nat n + 1"
```
```   141 proof -
```
```   142   assume "0 \<le> n"
```
```   143   moreover have "{y. 0 \<le> y & y \<le> n} = {y. 0 \<le> y & y < n+1}" by auto
```
```   144   ultimately show ?thesis
```
```   145     using card_bdd_int_set_l [of "n + 1"]
```
```   146     by (auto simp add: nat_add_distrib)
```
```   147 qed
```
```   148
```
```   149 lemma card_bdd_int_set_l_le: "0 \<le> (n::int) ==>
```
```   150     card {x. 0 < x & x \<le> n} = nat n"
```
```   151 proof -
```
```   152   assume "0 \<le> n"
```
```   153   have "inj_on (%x. x+1) {x. 0 \<le> x & x < n}"
```
```   154     by (auto simp add: inj_on_def)
```
```   155   hence "card ((%x. x+1) ` {x. 0 \<le> x & x < n}) =
```
```   156      card {x. 0 \<le> x & x < n}"
```
```   157     by (rule card_image)
```
```   158   also from `0 \<le> n` have "... = nat n"
```
```   159     by (rule card_bdd_int_set_l)
```
```   160   also have "(%x. x + 1) ` {x. 0 \<le> x & x < n} = {x. 0 < x & x<= n}"
```
```   161     apply (auto simp add: image_def)
```
```   162     apply (rule_tac x = "x - 1" in exI)
```
```   163     apply arith
```
```   164     done
```
```   165   finally show "card {x. 0 < x & x \<le> n} = nat n" .
```
```   166 qed
```
```   167
```
```   168 lemma card_bdd_int_set_l_l: "0 < (n::int) ==>
```
```   169   card {x. 0 < x & x < n} = nat n - 1"
```
```   170 proof -
```
```   171   assume "0 < n"
```
```   172   moreover have "{x. 0 < x & x < n} = {x. 0 < x & x \<le> n - 1}"
```
```   173     by simp
```
```   174   ultimately show ?thesis
```
```   175     using insert card_bdd_int_set_l_le [of "n - 1"]
```
```   176     by (auto simp add: nat_diff_distrib)
```
```   177 qed
```
```   178
```
```   179 lemma int_card_bdd_int_set_l_l: "0 < n ==>
```
```   180     int(card {x. 0 < x & x < n}) = n - 1"
```
```   181   apply (auto simp add: card_bdd_int_set_l_l)
```
```   182   done
```
```   183
```
```   184 lemma int_card_bdd_int_set_l_le: "0 \<le> n ==>
```
```   185     int(card {x. 0 < x & x \<le> n}) = n"
```
```   186   by (auto simp add: card_bdd_int_set_l_le)
```
```   187
```
```   188
```
```   189 subsection {* Cardinality of finite cartesian products *}
```
```   190
```
```   191 (* FIXME could be useful in general but not needed here
```
```   192 lemma insert_Sigma [simp]: "(insert x A) <*> B = ({ x } <*> B) \<union> (A <*> B)"
```
```   193   by blast
```
```   194  *)
```
```   195
```
```   196 text {* Lemmas for counting arguments. *}
```
```   197
```
```   198 lemma setsum_bij_eq: "[| finite A; finite B; f ` A \<subseteq> B; inj_on f A;
```
```   199     g ` B \<subseteq> A; inj_on g B |] ==> setsum g B = setsum (g \<circ> f) A"
```
```   200   apply (frule_tac h = g and f = f in setsum_reindex)
```
```   201   apply (subgoal_tac "setsum g B = setsum g (f ` A)")
```
```   202    apply (simp add: inj_on_def)
```
```   203   apply (subgoal_tac "card A = card B")
```
```   204    apply (drule_tac A = "f ` A" and B = B in card_seteq)
```
```   205      apply (auto simp add: card_image)
```
```   206   apply (frule_tac A = A and B = B and f = f in card_inj_on_le, auto)
```
```   207   apply (frule_tac A = B and B = A and f = g in card_inj_on_le)
```
```   208     apply auto
```
```   209   done
```
```   210
```
```   211 lemma setprod_bij_eq: "[| finite A; finite B; f ` A \<subseteq> B; inj_on f A;
```
```   212     g ` B \<subseteq> A; inj_on g B |] ==> setprod g B = setprod (g \<circ> f) A"
```
```   213   apply (frule_tac h = g and f = f in setprod_reindex)
```
```   214   apply (subgoal_tac "setprod g B = setprod g (f ` A)")
```
```   215    apply (simp add: inj_on_def)
```
```   216   apply (subgoal_tac "card A = card B")
```
```   217    apply (drule_tac A = "f ` A" and B = B in card_seteq)
```
```   218      apply (auto simp add: card_image)
```
```   219   apply (frule_tac A = A and B = B and f = f in card_inj_on_le, auto)
```
```   220   apply (frule_tac A = B and B = A and f = g in card_inj_on_le, auto)
```
```   221   done
```
```   222
```
```   223 end
```