--- a/src/HOL/Old_Number_Theory/Finite2.thy Tue Oct 18 07:04:08 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,184 +0,0 @@
-(* Title: HOL/Old_Number_Theory/Finite2.thy
- Authors: Jeremy Avigad, David Gray, and Adam Kramer
-*)
-
-section \<open>Finite Sets and Finite Sums\<close>
-
-theory Finite2
-imports IntFact "~~/src/HOL/Library/Infinite_Set"
-begin
-
-text\<open>
- These are useful for combinatorial and number-theoretic counting
- arguments.
-\<close>
-
-
-subsection \<open>Useful properties of sums and products\<close>
-
-lemma sum_same_function_zcong:
- assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
- shows "[sum f S = sum 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
-qed
-
-lemma prod_same_function_zcong:
- assumes a: "\<forall>x \<in> S. [f x = g x](mod m)"
- shows "[prod f S = prod 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
-qed
-
-lemma sum_const: "finite X ==> sum (%x. (c :: int)) X = c * int(card X)"
-by (simp add: of_nat_mult)
-
-lemma sum_const2: "finite X ==> int (sum (%x. (c :: nat)) X) =
- int(c) * int(card X)"
-by (simp add: of_nat_mult)
-
-lemma sum_const_mult: "finite A ==> sum (%x. c * ((f x)::int)) A =
- c * sum f A"
- by (induct set: finite) (auto simp add: distrib_left)
-
-
-subsection \<open>Cardinality of explicit finite sets\<close>
-
-lemma finite_surjI: "[| B \<subseteq> f ` A; finite A |] ==> finite B"
-by (simp add: finite_subset)
-
-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)
- case 0
- show "card {y::nat . y < 0} = 0" by simp
-next
- case (Suc 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"
- using \<open>card {y. y < n} = n\<close> by simp
-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 \<open>0 \<le> n\<close> 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 \<open>0 \<le> n\<close> 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)
- 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)
-
-
-end