(* Title: HOL/Library/Nonpos_Ints.thy
Author: Manuel Eberl, TU München
*)
section \<open>Non-negative, non-positive integers and reals\<close>
theory Nonpos_Ints
imports Complex_Main
begin
subsection\<open>Non-positive integers\<close>
text \<open>
The set of non-positive integers on a ring. (in analogy to the set of non-negative
integers @{term "\<nat>"}) This is useful e.g. for the Gamma function.
\<close>
definition nonpos_Ints ("\<int>\<^sub>\<le>\<^sub>0") where "\<int>\<^sub>\<le>\<^sub>0 = {of_int n |n. n \<le> 0}"
lemma zero_in_nonpos_Ints [simp,intro]: "0 \<in> \<int>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Ints_def by (auto intro!: exI[of _ "0::int"])
lemma neg_one_in_nonpos_Ints [simp,intro]: "-1 \<in> \<int>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Ints_def by (auto intro!: exI[of _ "-1::int"])
lemma neg_numeral_in_nonpos_Ints [simp,intro]: "-numeral n \<in> \<int>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Ints_def by (auto intro!: exI[of _ "-numeral n::int"])
lemma one_notin_nonpos_Ints [simp]: "(1 :: 'a :: ring_char_0) \<notin> \<int>\<^sub>\<le>\<^sub>0"
by (auto simp: nonpos_Ints_def)
lemma numeral_notin_nonpos_Ints [simp]: "(numeral n :: 'a :: ring_char_0) \<notin> \<int>\<^sub>\<le>\<^sub>0"
by (auto simp: nonpos_Ints_def)
lemma minus_of_nat_in_nonpos_Ints [simp, intro]: "- of_nat n \<in> \<int>\<^sub>\<le>\<^sub>0"
proof -
have "- of_nat n = of_int (-int n)" by simp
also have "-int n \<le> 0" by simp
hence "of_int (-int n) \<in> \<int>\<^sub>\<le>\<^sub>0" unfolding nonpos_Ints_def by blast
finally show ?thesis .
qed
lemma of_nat_in_nonpos_Ints_iff: "(of_nat n :: 'a :: {ring_1,ring_char_0}) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n = 0"
proof
assume "(of_nat n :: 'a) \<in> \<int>\<^sub>\<le>\<^sub>0"
then obtain m where "of_nat n = (of_int m :: 'a)" "m \<le> 0" by (auto simp: nonpos_Ints_def)
hence "(of_int m :: 'a) = of_nat n" by simp
also have "... = of_int (int n)" by simp
finally have "m = int n" by (subst (asm) of_int_eq_iff)
with \<open>m \<le> 0\<close> show "n = 0" by auto
qed simp
lemma nonpos_Ints_of_int: "n \<le> 0 \<Longrightarrow> of_int n \<in> \<int>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Ints_def by blast
lemma nonpos_IntsI:
"x \<in> \<int> \<Longrightarrow> x \<le> 0 \<Longrightarrow> (x :: 'a :: linordered_idom) \<in> \<int>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Ints_def Ints_def by auto
lemma nonpos_Ints_subset_Ints: "\<int>\<^sub>\<le>\<^sub>0 \<subseteq> \<int>"
unfolding nonpos_Ints_def Ints_def by blast
lemma nonpos_Ints_nonpos [dest]: "x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x \<le> (0 :: 'a :: linordered_idom)"
unfolding nonpos_Ints_def by auto
lemma nonpos_Ints_Int [dest]: "x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x \<in> \<int>"
unfolding nonpos_Ints_def Ints_def by blast
lemma nonpos_Ints_cases:
assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0"
obtains n where "x = of_int n" "n \<le> 0"
using assms unfolding nonpos_Ints_def by (auto elim!: Ints_cases)
lemma nonpos_Ints_cases':
assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0"
obtains n where "x = -of_nat n"
proof -
from assms obtain m where "x = of_int m" and m: "m \<le> 0" by (auto elim!: nonpos_Ints_cases)
hence "x = - of_int (-m)" by auto
also from m have "(of_int (-m) :: 'a) = of_nat (nat (-m))" by simp_all
finally show ?thesis by (rule that)
qed
lemma of_real_in_nonpos_Ints_iff: "(of_real x :: 'a :: real_algebra_1) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> x \<in> \<int>\<^sub>\<le>\<^sub>0"
proof
assume "of_real x \<in> (\<int>\<^sub>\<le>\<^sub>0 :: 'a set)"
then obtain n where "(of_real x :: 'a) = of_int n" "n \<le> 0" by (erule nonpos_Ints_cases)
note \<open>of_real x = of_int n\<close>
also have "of_int n = of_real (of_int n)" by (rule of_real_of_int_eq [symmetric])
finally have "x = of_int n" by (subst (asm) of_real_eq_iff)
with \<open>n \<le> 0\<close> show "x \<in> \<int>\<^sub>\<le>\<^sub>0" by (simp add: nonpos_Ints_of_int)
qed (auto elim!: nonpos_Ints_cases intro!: nonpos_Ints_of_int)
lemma nonpos_Ints_altdef: "\<int>\<^sub>\<le>\<^sub>0 = {n \<in> \<int>. (n :: 'a :: linordered_idom) \<le> 0}"
by (auto intro!: nonpos_IntsI elim!: nonpos_Ints_cases)
lemma uminus_in_Nats_iff: "-x \<in> \<nat> \<longleftrightarrow> x \<in> \<int>\<^sub>\<le>\<^sub>0"
proof
assume "-x \<in> \<nat>"
then obtain n where "n \<ge> 0" "-x = of_int n" by (auto simp: Nats_altdef1)
hence "-n \<le> 0" "x = of_int (-n)" by (simp_all add: eq_commute minus_equation_iff[of x])
thus "x \<in> \<int>\<^sub>\<le>\<^sub>0" unfolding nonpos_Ints_def by blast
next
assume "x \<in> \<int>\<^sub>\<le>\<^sub>0"
then obtain n where "n \<le> 0" "x = of_int n" by (auto simp: nonpos_Ints_def)
hence "-n \<ge> 0" "-x = of_int (-n)" by (simp_all add: eq_commute minus_equation_iff[of x])
thus "-x \<in> \<nat>" unfolding Nats_altdef1 by blast
qed
lemma uminus_in_nonpos_Ints_iff: "-x \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> x \<in> \<nat>"
using uminus_in_Nats_iff[of "-x"] by simp
lemma nonpos_Ints_mult: "x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> y \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x * y \<in> \<nat>"
using Nats_mult[of "-x" "-y"] by (simp add: uminus_in_Nats_iff)
lemma Nats_mult_nonpos_Ints: "x \<in> \<nat> \<Longrightarrow> y \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x * y \<in> \<int>\<^sub>\<le>\<^sub>0"
using Nats_mult[of x "-y"] by (simp add: uminus_in_Nats_iff)
lemma nonpos_Ints_mult_Nats:
"x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> y \<in> \<nat> \<Longrightarrow> x * y \<in> \<int>\<^sub>\<le>\<^sub>0"
using Nats_mult[of "-x" y] by (simp add: uminus_in_Nats_iff)
lemma nonpos_Ints_add:
"x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> y \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x + y \<in> \<int>\<^sub>\<le>\<^sub>0"
using Nats_add[of "-x" "-y"] uminus_in_Nats_iff[of "y+x", simplified minus_add]
by (simp add: uminus_in_Nats_iff add.commute)
lemma nonpos_Ints_diff_Nats:
"x \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> y \<in> \<nat> \<Longrightarrow> x - y \<in> \<int>\<^sub>\<le>\<^sub>0"
using Nats_add[of "-x" "y"] uminus_in_Nats_iff[of "x-y", simplified minus_add]
by (simp add: uminus_in_Nats_iff add.commute)
lemma Nats_diff_nonpos_Ints:
"x \<in> \<nat> \<Longrightarrow> y \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x - y \<in> \<nat>"
using Nats_add[of "x" "-y"] by (simp add: uminus_in_Nats_iff add.commute)
lemma plus_of_nat_eq_0_imp: "z + of_nat n = 0 \<Longrightarrow> z \<in> \<int>\<^sub>\<le>\<^sub>0"
proof -
assume "z + of_nat n = 0"
hence A: "z = - of_nat n" by (simp add: eq_neg_iff_add_eq_0)
show "z \<in> \<int>\<^sub>\<le>\<^sub>0" by (subst A) simp
qed
subsection\<open>Non-negative reals\<close>
definition nonneg_Reals :: "'a::real_algebra_1 set" ("\<real>\<^sub>\<ge>\<^sub>0")
where "\<real>\<^sub>\<ge>\<^sub>0 = {of_real r | r. r \<ge> 0}"
lemma nonneg_Reals_of_real_iff [simp]: "of_real r \<in> \<real>\<^sub>\<ge>\<^sub>0 \<longleftrightarrow> r \<ge> 0"
by (force simp add: nonneg_Reals_def)
lemma nonneg_Reals_subset_Reals: "\<real>\<^sub>\<ge>\<^sub>0 \<subseteq> \<real>"
unfolding nonneg_Reals_def Reals_def by blast
lemma nonneg_Reals_Real [dest]: "x \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> x \<in> \<real>"
unfolding nonneg_Reals_def Reals_def by blast
lemma nonneg_Reals_of_nat_I [simp]: "of_nat n \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (metis nonneg_Reals_of_real_iff of_nat_0_le_iff of_real_of_nat_eq)
lemma nonneg_Reals_cases:
assumes "x \<in> \<real>\<^sub>\<ge>\<^sub>0"
obtains r where "x = of_real r" "r \<ge> 0"
using assms unfolding nonneg_Reals_def by (auto elim!: Reals_cases)
lemma nonneg_Reals_zero_I [simp]: "0 \<in> \<real>\<^sub>\<ge>\<^sub>0"
unfolding nonneg_Reals_def by auto
lemma nonneg_Reals_one_I [simp]: "1 \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (metis (mono_tags, lifting) nonneg_Reals_of_nat_I of_nat_1)
lemma nonneg_Reals_minus_one_I [simp]: "-1 \<notin> \<real>\<^sub>\<ge>\<^sub>0"
by (metis nonneg_Reals_of_real_iff le_minus_one_simps(3) of_real_1 of_real_def real_vector.scale_minus_left)
lemma nonneg_Reals_numeral_I [simp]: "numeral w \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (metis (no_types) nonneg_Reals_of_nat_I of_nat_numeral)
lemma nonneg_Reals_minus_numeral_I [simp]: "- numeral w \<notin> \<real>\<^sub>\<ge>\<^sub>0"
using nonneg_Reals_of_real_iff not_zero_le_neg_numeral by fastforce
lemma nonneg_Reals_add_I [simp]: "\<lbrakk>a \<in> \<real>\<^sub>\<ge>\<^sub>0; b \<in> \<real>\<^sub>\<ge>\<^sub>0\<rbrakk> \<Longrightarrow> a + b \<in> \<real>\<^sub>\<ge>\<^sub>0"
apply (simp add: nonneg_Reals_def)
apply clarify
apply (rename_tac r s)
apply (rule_tac x="r+s" in exI, auto)
done
lemma nonneg_Reals_mult_I [simp]: "\<lbrakk>a \<in> \<real>\<^sub>\<ge>\<^sub>0; b \<in> \<real>\<^sub>\<ge>\<^sub>0\<rbrakk> \<Longrightarrow> a * b \<in> \<real>\<^sub>\<ge>\<^sub>0"
unfolding nonneg_Reals_def by (auto simp: of_real_def)
lemma nonneg_Reals_inverse_I [simp]:
fixes a :: "'a::real_div_algebra"
shows "a \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> inverse a \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (simp add: nonneg_Reals_def image_iff) (metis inverse_nonnegative_iff_nonnegative of_real_inverse)
lemma nonneg_Reals_divide_I [simp]:
fixes a :: "'a::real_div_algebra"
shows "\<lbrakk>a \<in> \<real>\<^sub>\<ge>\<^sub>0; b \<in> \<real>\<^sub>\<ge>\<^sub>0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (simp add: divide_inverse)
lemma nonneg_Reals_pow_I [simp]: "a \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> a^n \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (induction n) auto
lemma complex_nonneg_Reals_iff: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<longleftrightarrow> Re z \<ge> 0 \<and> Im z = 0"
by (auto simp: nonneg_Reals_def) (metis complex_of_real_def complex_surj)
lemma ii_not_nonneg_Reals [iff]: "\<i> \<notin> \<real>\<^sub>\<ge>\<^sub>0"
by (simp add: complex_nonneg_Reals_iff)
subsection\<open>Non-positive reals\<close>
definition nonpos_Reals :: "'a::real_algebra_1 set" ("\<real>\<^sub>\<le>\<^sub>0")
where "\<real>\<^sub>\<le>\<^sub>0 = {of_real r | r. r \<le> 0}"
lemma nonpos_Reals_of_real_iff [simp]: "of_real r \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> r \<le> 0"
by (force simp add: nonpos_Reals_def)
lemma nonpos_Reals_subset_Reals: "\<real>\<^sub>\<le>\<^sub>0 \<subseteq> \<real>"
unfolding nonpos_Reals_def Reals_def by blast
lemma nonpos_Ints_subset_nonpos_Reals: "\<int>\<^sub>\<le>\<^sub>0 \<subseteq> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonpos_Ints_cases nonpos_Ints_nonpos nonpos_Ints_of_int
nonpos_Reals_of_real_iff of_real_of_int_eq subsetI)
lemma nonpos_Reals_of_nat_iff [simp]: "of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n=0"
by (metis nonpos_Reals_of_real_iff of_nat_le_0_iff of_real_of_nat_eq)
lemma nonpos_Reals_Real [dest]: "x \<in> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x \<in> \<real>"
unfolding nonpos_Reals_def Reals_def by blast
lemma nonpos_Reals_cases:
assumes "x \<in> \<real>\<^sub>\<le>\<^sub>0"
obtains r where "x = of_real r" "r \<le> 0"
using assms unfolding nonpos_Reals_def by (auto elim!: Reals_cases)
lemma uminus_nonneg_Reals_iff [simp]: "-x \<in> \<real>\<^sub>\<ge>\<^sub>0 \<longleftrightarrow> x \<in> \<real>\<^sub>\<le>\<^sub>0"
apply (auto simp: nonpos_Reals_def nonneg_Reals_def)
apply (metis nonpos_Reals_of_real_iff minus_minus neg_le_0_iff_le of_real_minus)
done
lemma uminus_nonpos_Reals_iff [simp]: "-x \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> x \<in> \<real>\<^sub>\<ge>\<^sub>0"
by (metis (no_types) minus_minus uminus_nonneg_Reals_iff)
lemma nonpos_Reals_zero_I [simp]: "0 \<in> \<real>\<^sub>\<le>\<^sub>0"
unfolding nonpos_Reals_def by force
lemma nonpos_Reals_one_I [simp]: "1 \<notin> \<real>\<^sub>\<le>\<^sub>0"
using nonneg_Reals_minus_one_I uminus_nonneg_Reals_iff by blast
lemma nonpos_Reals_numeral_I [simp]: "numeral w \<notin> \<real>\<^sub>\<le>\<^sub>0"
using nonneg_Reals_minus_numeral_I uminus_nonneg_Reals_iff by blast
lemma nonpos_Reals_add_I [simp]: "\<lbrakk>a \<in> \<real>\<^sub>\<le>\<^sub>0; b \<in> \<real>\<^sub>\<le>\<^sub>0\<rbrakk> \<Longrightarrow> a + b \<in> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonneg_Reals_add_I add_uminus_conv_diff minus_diff_eq minus_minus uminus_nonpos_Reals_iff)
lemma nonpos_Reals_mult_I1: "\<lbrakk>a \<in> \<real>\<^sub>\<ge>\<^sub>0; b \<in> \<real>\<^sub>\<le>\<^sub>0\<rbrakk> \<Longrightarrow> a * b \<in> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonneg_Reals_mult_I mult_minus_right uminus_nonneg_Reals_iff)
lemma nonpos_Reals_mult_I2: "\<lbrakk>a \<in> \<real>\<^sub>\<le>\<^sub>0; b \<in> \<real>\<^sub>\<ge>\<^sub>0\<rbrakk> \<Longrightarrow> a * b \<in> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonneg_Reals_mult_I mult_minus_left uminus_nonneg_Reals_iff)
lemma nonpos_Reals_mult_of_nat_iff:
fixes a:: "'a :: real_div_algebra" shows "a * of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> a \<in> \<real>\<^sub>\<le>\<^sub>0 \<or> n=0"
apply (auto intro: nonpos_Reals_mult_I2)
apply (auto simp: nonpos_Reals_def)
apply (rule_tac x="r/n" in exI)
apply (auto simp: divide_simps)
done
lemma nonpos_Reals_inverse_I:
fixes a :: "'a::real_div_algebra"
shows "a \<in> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> inverse a \<in> \<real>\<^sub>\<le>\<^sub>0"
using nonneg_Reals_inverse_I uminus_nonneg_Reals_iff by fastforce
lemma nonpos_Reals_divide_I1:
fixes a :: "'a::real_div_algebra"
shows "\<lbrakk>a \<in> \<real>\<^sub>\<ge>\<^sub>0; b \<in> \<real>\<^sub>\<le>\<^sub>0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>\<^sub>\<le>\<^sub>0"
by (simp add: nonpos_Reals_inverse_I nonpos_Reals_mult_I1 divide_inverse)
lemma nonpos_Reals_divide_I2:
fixes a :: "'a::real_div_algebra"
shows "\<lbrakk>a \<in> \<real>\<^sub>\<le>\<^sub>0; b \<in> \<real>\<^sub>\<ge>\<^sub>0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonneg_Reals_divide_I minus_divide_left uminus_nonneg_Reals_iff)
lemma nonpos_Reals_divide_of_nat_iff:
fixes a:: "'a :: real_div_algebra" shows "a / of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> a \<in> \<real>\<^sub>\<le>\<^sub>0 \<or> n=0"
apply (auto intro: nonpos_Reals_divide_I2)
apply (auto simp: nonpos_Reals_def)
apply (rule_tac x="r*n" in exI)
apply (auto simp: divide_simps mult_le_0_iff)
done
lemma nonpos_Reals_inverse_iff [simp]:
fixes a :: "'a::real_div_algebra"
shows "inverse a \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> a \<in> \<real>\<^sub>\<le>\<^sub>0"
using nonpos_Reals_inverse_I by fastforce
lemma nonpos_Reals_pow_I: "\<lbrakk>a \<in> \<real>\<^sub>\<le>\<^sub>0; odd n\<rbrakk> \<Longrightarrow> a^n \<in> \<real>\<^sub>\<le>\<^sub>0"
by (metis nonneg_Reals_pow_I power_minus_odd uminus_nonneg_Reals_iff)
lemma complex_nonpos_Reals_iff: "z \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> Re z \<le> 0 \<and> Im z = 0"
using complex_is_Real_iff by (force simp add: nonpos_Reals_def)
lemma ii_not_nonpos_Reals [iff]: "\<i> \<notin> \<real>\<^sub>\<le>\<^sub>0"
by (simp add: complex_nonpos_Reals_iff)
end