(* Title: HOL/Quadratic_Reciprocity/Finite2.thy
ID: $Id$
Authors: Jeremy Avigad, David Gray, and Adam Kramer
*)
header {*Finite Sets and Finite Sums*}
theory Finite2
imports IntFact
begin
text{*These are useful for combinatorial and number-theoretic counting
arguments.*}
text{*Note. This theory is being revised. See the web page
\url{http://www.andrew.cmu.edu/~avigad/isabelle}.*}
(******************************************************************)
(* *)
(* Useful properties of sums and products *)
(* *)
(******************************************************************)
subsection {* Useful properties of sums and products *}
lemma setsum_same_function_zcong:
assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
shows "[setsum f S = setsum g S] (mod m)"
proof cases
assume "finite S"
thus ?thesis using a by induct (simp_all add: zcong_zadd)
next
assume "infinite S" thus ?thesis by(simp add:setsum_def)
qed
lemma setprod_same_function_zcong:
assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
shows "[setprod f S = setprod g S] (mod m)"
proof cases
assume "finite S"
thus ?thesis using a by induct (simp_all add: zcong_zmult)
next
assume "infinite S" thus ?thesis by(simp add:setprod_def)
qed
lemma setsum_const: "finite X ==> setsum (%x. (c :: int)) X = c * int(card X)"
apply (induct set: Finites)
apply (auto simp add: left_distrib right_distrib int_eq_of_nat)
done
lemma setsum_const2: "finite X ==> int (setsum (%x. (c :: nat)) X) =
int(c) * int(card X)"
apply (induct set: Finites)
apply (auto simp add: zadd_zmult_distrib2)
done
lemma setsum_const_mult: "finite A ==> setsum (%x. c * ((f x)::int)) A =
c * setsum f A"
by (induct set: Finites) (auto simp add: zadd_zmult_distrib2)
(******************************************************************)
(* *)
(* Cardinality of some explicit finite sets *)
(* *)
(******************************************************************)
subsection {* Cardinality of explicit finite sets *}
lemma finite_surjI: "[| B \<subseteq> f ` A; finite A |] ==> finite B"
by (simp add: finite_subset finite_imageI)
lemma bdd_nat_set_l_finite: "finite {y::nat . y < x}"
by (rule bounded_nat_set_is_finite) blast
lemma bdd_nat_set_le_finite: "finite {y::nat . y \<le> x}"
proof -
have "{y::nat . y \<le> x} = {y::nat . y < Suc x}" by auto
then show ?thesis by (auto simp add: bdd_nat_set_l_finite)
qed
lemma bdd_int_set_l_finite: "finite {x::int. 0 \<le> x & x < n}"
apply (subgoal_tac " {(x :: int). 0 \<le> x & x < n} \<subseteq>
int ` {(x :: nat). x < nat n}")
apply (erule finite_surjI)
apply (auto simp add: bdd_nat_set_l_finite image_def)
apply (rule_tac x = "nat x" in exI, simp)
done
lemma bdd_int_set_le_finite: "finite {x::int. 0 \<le> x & x \<le> n}"
apply (subgoal_tac "{x. 0 \<le> x & x \<le> n} = {x. 0 \<le> x & x < n + 1}")
apply (erule ssubst)
apply (rule bdd_int_set_l_finite)
apply auto
done
lemma bdd_int_set_l_l_finite: "finite {x::int. 0 < x & x < n}"
proof -
have "{x::int. 0 < x & x < n} \<subseteq> {x::int. 0 \<le> x & x < n}"
by auto
then show ?thesis by (auto simp add: bdd_int_set_l_finite finite_subset)
qed
lemma bdd_int_set_l_le_finite: "finite {x::int. 0 < x & x \<le> n}"
proof -
have "{x::int. 0 < x & x \<le> n} \<subseteq> {x::int. 0 \<le> x & x \<le> n}"
by auto
then show ?thesis by (auto simp add: bdd_int_set_le_finite finite_subset)
qed
lemma card_bdd_nat_set_l: "card {y::nat . y < x} = x"
proof (induct x)
show "card {y::nat . y < 0} = 0" by simp
next
fix n::nat
assume "card {y. y < n} = n"
have "{y. y < Suc n} = insert n {y. y < n}"
by auto
then have "card {y. y < Suc n} = card (insert n {y. y < n})"
by auto
also have "... = Suc (card {y. y < n})"
by (rule card_insert_disjoint) (auto simp add: bdd_nat_set_l_finite)
finally show "card {y. y < Suc n} = Suc n"
by (simp add: prems)
qed
lemma card_bdd_nat_set_le: "card { y::nat. y \<le> x} = Suc x"
proof -
have "{y::nat. y \<le> x} = { y::nat. y < Suc x}"
by auto
then show ?thesis by (auto simp add: card_bdd_nat_set_l)
qed
lemma card_bdd_int_set_l: "0 \<le> (n::int) ==> card {y. 0 \<le> y & y < n} = nat n"
proof -
assume "0 \<le> n"
have "inj_on (%y. int y) {y. y < nat n}"
by (auto simp add: inj_on_def)
hence "card (int ` {y. y < nat n}) = card {y. y < nat n}"
by (rule card_image)
also from prems have "int ` {y. y < nat n} = {y. 0 \<le> y & y < n}"
apply (auto simp add: zless_nat_eq_int_zless image_def)
apply (rule_tac x = "nat x" in exI)
apply (auto simp add: nat_0_le)
done
also have "card {y. y < nat n} = nat n"
by (rule card_bdd_nat_set_l)
finally show "card {y. 0 \<le> y & y < n} = nat n" .
qed
lemma card_bdd_int_set_le: "0 \<le> (n::int) ==> card {y. 0 \<le> y & y \<le> n} =
nat n + 1"
proof -
assume "0 \<le> n"
moreover have "{y. 0 \<le> y & y \<le> n} = {y. 0 \<le> y & y < n+1}" by auto
ultimately show ?thesis
using card_bdd_int_set_l [of "n + 1"]
by (auto simp add: nat_add_distrib)
qed
lemma card_bdd_int_set_l_le: "0 \<le> (n::int) ==>
card {x. 0 < x & x \<le> n} = nat n"
proof -
assume "0 \<le> n"
have "inj_on (%x. x+1) {x. 0 \<le> x & x < n}"
by (auto simp add: inj_on_def)
hence "card ((%x. x+1) ` {x. 0 \<le> x & x < n}) =
card {x. 0 \<le> x & x < n}"
by (rule card_image)
also from `0 \<le> n` have "... = nat n"
by (rule card_bdd_int_set_l)
also have "(%x. x + 1) ` {x. 0 \<le> x & x < n} = {x. 0 < x & x<= n}"
apply (auto simp add: image_def)
apply (rule_tac x = "x - 1" in exI)
apply arith
done
finally show "card {x. 0 < x & x \<le> n} = nat n" .
qed
lemma card_bdd_int_set_l_l: "0 < (n::int) ==>
card {x. 0 < x & x < n} = nat n - 1"
proof -
assume "0 < n"
moreover have "{x. 0 < x & x < n} = {x. 0 < x & x \<le> n - 1}"
by simp
ultimately show ?thesis
using insert card_bdd_int_set_l_le [of "n - 1"]
by (auto simp add: nat_diff_distrib)
qed
lemma int_card_bdd_int_set_l_l: "0 < n ==>
int(card {x. 0 < x & x < n}) = n - 1"
apply (auto simp add: card_bdd_int_set_l_l)
apply (subgoal_tac "Suc 0 \<le> nat n")
apply (auto simp add: zdiff_int [symmetric])
apply (subgoal_tac "0 < nat n", arith)
apply (simp add: zero_less_nat_eq)
done
lemma int_card_bdd_int_set_l_le: "0 \<le> n ==>
int(card {x. 0 < x & x \<le> n}) = n"
by (auto simp add: card_bdd_int_set_l_le)
(******************************************************************)
(* *)
(* Cartesian products of finite sets *)
(* *)
(******************************************************************)
subsection {* Cardinality of finite cartesian products *}
(* FIXME could be useful in general but not needed here
lemma insert_Sigma [simp]: "(insert x A) <*> B = ({ x } <*> B) \<union> (A <*> B)"
by blast
*)
(******************************************************************)
(* *)
(* Sums and products over finite sets *)
(* *)
(******************************************************************)
subsection {* Lemmas for counting arguments *}
lemma setsum_bij_eq: "[| finite A; finite B; f ` A \<subseteq> B; inj_on f A;
g ` B \<subseteq> A; inj_on g B |] ==> setsum g B = setsum (g \<circ> f) A"
apply (frule_tac h = g and f = f in setsum_reindex)
apply (subgoal_tac "setsum g B = setsum g (f ` A)")
apply (simp add: inj_on_def)
apply (subgoal_tac "card A = card B")
apply (drule_tac A = "f ` A" and B = B in card_seteq)
apply (auto simp add: card_image)
apply (frule_tac A = A and B = B and f = f in card_inj_on_le, auto)
apply (frule_tac A = B and B = A and f = g in card_inj_on_le)
apply auto
done
lemma setprod_bij_eq: "[| finite A; finite B; f ` A \<subseteq> B; inj_on f A;
g ` B \<subseteq> A; inj_on g B |] ==> setprod g B = setprod (g \<circ> f) A"
apply (frule_tac h = g and f = f in setprod_reindex)
apply (subgoal_tac "setprod g B = setprod g (f ` A)")
apply (simp add: inj_on_def)
apply (subgoal_tac "card A = card B")
apply (drule_tac A = "f ` A" and B = B in card_seteq)
apply (auto simp add: card_image)
apply (frule_tac A = A and B = B and f = f in card_inj_on_le, auto)
apply (frule_tac A = B and B = A and f = g in card_inj_on_le, auto)
done
end