(* Title : HOL/RealPow.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
*)
header {* Natural powers theory *}
theory RealPow
imports RealDef RComplete
begin
(* FIXME: declare this in Rings.thy or not at all *)
declare abs_mult_self [simp]
(* used by Import/HOL/real.imp *)
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
by simp
lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
apply (induct "n")
apply (auto simp add: real_of_nat_Suc)
apply (subst mult_2)
apply (erule add_less_le_mono)
apply (rule two_realpow_ge_one)
done
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma realpow_Suc_le_self:
fixes r :: "'a::linordered_semidom"
shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
by (insert power_decreasing [of 1 "Suc n" r], simp)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma realpow_minus_mult:
fixes x :: "'a::monoid_mult"
shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
by (simp add: power_commutes split add: nat_diff_split)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma realpow_two_diff:
fixes x :: "'a::comm_ring_1"
shows "x^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
by (simp add: algebra_simps)
(* TODO: move elsewhere *)
lemma add_eq_0_iff:
fixes x y :: "'a::group_add"
shows "x + y = 0 \<longleftrightarrow> y = - x"
by (auto dest: minus_unique)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma realpow_two_disj:
fixes x :: "'a::idom"
shows "(x^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
using realpow_two_diff [of x y]
by (auto simp add: add_eq_0_iff)
subsection{* Squares of Reals *}
(* FIXME: declare this [simp] for all types, or not at all *)
lemma real_two_squares_add_zero_iff [simp]:
"(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
by (rule sum_squares_eq_zero_iff)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma real_squared_diff_one_factored:
fixes x :: "'a::ring_1"
shows "x * x - 1 = (x + 1) * (x - 1)"
by (simp add: algebra_simps)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma real_mult_is_one [simp]:
fixes x :: "'a::ring_1_no_zero_divisors"
shows "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
proof -
have "x * x = 1 \<longleftrightarrow> (x + 1) * (x - 1) = 0"
by (simp add: algebra_simps)
also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1"
by (auto simp add: add_eq_0_iff minus_equation_iff [of _ 1])
finally show ?thesis .
qed
(* FIXME: declare this [simp] for all types, or not at all *)
lemma realpow_two_sum_zero_iff [simp]:
"(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
by (rule sum_power2_eq_zero_iff)
(* FIXME: declare this [simp] for all types, or not at all *)
lemma realpow_two_le_add_order [simp]: "(0::real) \<le> u ^ 2 + v ^ 2"
by (rule sum_power2_ge_zero)
(* FIXME: declare this [simp] for all types, or not at all *)
lemma realpow_two_le_add_order2 [simp]: "(0::real) \<le> u ^ 2 + v ^ 2 + w ^ 2"
by (intro add_nonneg_nonneg zero_le_power2)
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
by (rule_tac j = 0 in real_le_trans, auto)
lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
by (auto simp add: power2_eq_square)
(* The following theorem is by Benjamin Porter *)
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma real_sq_order:
fixes x :: "'a::linordered_semidom"
assumes xgt0: "0 \<le> x" and ygt0: "0 \<le> y" and sq: "x^2 \<le> y^2"
shows "x \<le> y"
proof -
from sq have "x ^ Suc (Suc 0) \<le> y ^ Suc (Suc 0)"
by (simp only: numeral_2_eq_2)
thus "x \<le> y" using ygt0
by (rule power_le_imp_le_base)
qed
subsection {*Floor*}
lemma floor_power:
assumes "x = real (floor x)"
shows "floor (x ^ n) = floor x ^ n"
proof -
have *: "x ^ n = real (floor x ^ n)"
using assms by (induct n arbitrary: x) simp_all
show ?thesis unfolding real_of_int_inject[symmetric]
unfolding * floor_real_of_int ..
qed
lemma natfloor_power:
assumes "x = real (natfloor x)"
shows "natfloor (x ^ n) = natfloor x ^ n"
proof -
from assms have "0 \<le> floor x" by auto
note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
from floor_power[OF this]
show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
by simp
qed
subsection {*Various Other Theorems*}
lemma real_le_add_half_cancel: "(x + y/2 \<le> (y::real)) = (x \<le> y /2)"
by auto
lemma real_mult_inverse_cancel:
"[|(0::real) < x; 0 < x1; x1 * y < x * u |]
==> inverse x * y < inverse x1 * u"
apply (rule_tac c=x in mult_less_imp_less_left)
apply (auto simp add: real_mult_assoc [symmetric])
apply (simp (no_asm) add: mult_ac)
apply (rule_tac c=x1 in mult_less_imp_less_right)
apply (auto simp add: mult_ac)
done
lemma real_mult_inverse_cancel2:
"[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
done
(* TODO: no longer real-specific; rename and move elsewhere *)
lemma realpow_num_eq_if:
fixes m :: "'a::power"
shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
by (cases n, auto)
end