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(*
File: Circle_Area.thy
Author: Manuel Eberl <eberlm@in.tum.de>
An proof that the area of a circle with radius R is R²\<pi>.
The proof uses integration by substitution on interval integrals.
*)
theory Circle_Area
imports Complex_Main Interval_Integral
begin
section {* Auxiliary lemmas *}
lemma plus_emeasure':
assumes "A \<in> sets M" "B \<in> sets M" "A \<inter> B \<in> null_sets M"
shows "emeasure M A + emeasure M B = emeasure M (A \<union> B)"
proof-
let ?C = "A \<inter> B"
have "A \<union> B = A \<union> (B - ?C)" by blast
with assms have "emeasure M (A \<union> B) = emeasure M A + emeasure M (B - ?C)"
by (subst plus_emeasure) auto
also from assms(3,2) have "emeasure M (B - ?C) = emeasure M B"
by (rule emeasure_Diff_null_set)
finally show ?thesis ..
qed
lemma real_sqrt_square:
"x \<ge> 0 \<Longrightarrow> sqrt (x^2) = (x::real)" by simp
section {* Circle area theorems *}
lemma unit_circle_area_aux:
"LBINT x=-1..1. 2 * sqrt (1 - x^2) = pi"
proof-
have "LBINT x=-1..1. 2 * sqrt (1 - x^2) =
LBINT x=ereal (sin (-pi/2))..ereal (sin (pi/2)). 2 * sqrt (1 - x^2)"
by (simp_all add: one_ereal_def)
also have "... = LBINT x=-pi/2..pi/2. cos x *\<^sub>R (2 * sqrt (1 - (sin x)\<^sup>2))"
by (rule interval_integral_substitution_finite[symmetric])
(auto intro: DERIV_subset[OF DERIV_sin] intro!: continuous_intros)
also have "... = LBINT x=-pi/2..pi/2. 2 * cos x * sqrt ((cos x)^2)"
by (simp add: cos_squared_eq field_simps)
also {
fix x assume "x \<in> {-pi/2<..<pi/2}"
hence "cos x \<ge> 0" by (intro cos_ge_zero) simp_all
hence "sqrt ((cos x)^2) = cos x" by simp
} note A = this
have "LBINT x=-pi/2..pi/2. 2 * cos x * sqrt ((cos x)^2) = LBINT x=-pi/2..pi/2. 2 * (cos x)^2"
by (intro interval_integral_cong, subst A) (simp_all add: min_def max_def power2_eq_square)
also let ?F = "\<lambda>x. x + sin x * cos x"
{
fix x A
have "(?F has_real_derivative 1 - (sin x)^2 + (cos x)^2) (at x)"
by (auto simp: power2_eq_square intro!: derivative_eq_intros)
also have "1 - (sin x)^2 + (cos x)^2 = 2 * (cos x)^2" by (simp add: cos_squared_eq)
finally have "(?F has_real_derivative 2 * (cos x)^2) (at x within A)"
by (rule DERIV_subset) simp
}
hence "LBINT x=-pi/2..pi/2. 2 * (cos x)^2 = ?F (pi/2) - ?F (-pi/2)"
by (intro interval_integral_FTC_finite)
(auto simp: has_field_derivative_iff_has_vector_derivative intro!: continuous_intros)
also have "... = pi" by simp
finally show ?thesis .
qed
lemma unit_circle_area:
"emeasure lborel {z::real\<times>real. norm z \<le> 1} = pi" (is "emeasure _ ?A = _")
proof-
let ?A1 = "{(x,y)\<in>?A. y \<ge> 0}" and ?A2 = "{(x,y)\<in>?A. y \<le> 0}"
have [measurable]: "(\<lambda>x. snd (x :: real \<times> real)) \<in> measurable borel borel"
by (simp add: borel_prod[symmetric])
have "?A1 = ?A \<inter> {x\<in>space lborel. snd x \<ge> 0}" by auto
also have "?A \<inter> {x\<in>space lborel. snd x \<ge> 0} \<in> sets borel"
by (intro sets.Int pred_Collect_borel) simp_all
finally have A1_in_sets: "?A1 \<in> sets lborel" by (subst sets_lborel)
have "?A2 = ?A \<inter> {x\<in>space lborel. snd x \<le> 0}" by auto
also have "... \<in> sets borel"
by (intro sets.Int pred_Collect_borel) simp_all
finally have A2_in_sets: "?A2 \<in> sets lborel" by (subst sets_lborel)
have A12: "?A = ?A1 \<union> ?A2" by auto
have sq_le_1_iff: "\<And>x. x\<^sup>2 \<le> 1 \<longleftrightarrow> abs (x::real) \<le> 1" using real_abs_le_square_iff[of _ 1] by simp
have "?A1 \<inter> ?A2 = {x. abs x \<le> 1} \<times> {0}" by (auto simp: norm_Pair field_simps sq_le_1_iff)
also have "... \<in> null_sets lborel"
by (subst lborel_prod[symmetric]) (auto simp: lborel.emeasure_pair_measure_Times)
finally have "emeasure lborel ?A = emeasure lborel ?A1 + emeasure lborel ?A2"
by (subst A12, rule plus_emeasure'[OF A1_in_sets A2_in_sets, symmetric])
also have "emeasure lborel ?A1 = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel \<partial>lborel"
by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure)
(simp_all only: lborel_prod A1_in_sets)
also have "emeasure lborel ?A2 = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A2 (x,y) \<partial>lborel \<partial>lborel"
by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure)
(simp_all only: lborel_prod A2_in_sets)
also have "distr lborel lborel uminus = (lborel :: real measure)"
by (subst (3) lborel_real_affine[of "-1" 0])
(simp_all add: one_ereal_def[symmetric] density_1 cong: distr_cong)
hence "(\<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A2 (x,y) \<partial>lborel \<partial>lborel) =
\<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A2 (x,y) \<partial>distr lborel lborel uminus \<partial>lborel" by simp
also have "... = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A2 (x,-y) \<partial>lborel \<partial>lborel"
apply (intro nn_integral_cong nn_integral_distr, simp)
apply (intro measurable_compose[OF _ borel_measurable_indicator[OF A2_in_sets]], simp)
done
also have "... = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel \<partial>lborel"
by (intro nn_integral_cong) (auto split: split_indicator simp: norm_Pair)
also have "... + ... = (1+1) * ..." by (subst ereal_left_distrib) simp_all
also have "... = \<integral>\<^sup>+x. 2 * \<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel \<partial>lborel"
by (subst nn_integral_cmult) simp_all
also {
fix x y :: real assume "x \<notin> {-1..1}"
hence "abs x > 1" by auto
also have "norm (x,y) \<ge> abs x" by (simp add: norm_Pair)
finally have "(x,y) \<notin> ?A1" by auto
}
hence "... = \<integral>\<^sup>+x. 2 * (\<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel) * indicator {-1..1} x \<partial>lborel"
by (intro nn_integral_cong) (auto split: split_indicator)
also {
fix x :: real assume "x \<in> {-1..1}"
hence x: "1 - x\<^sup>2 \<ge> 0" by (simp add: field_simps sq_le_1_iff abs_real_def)
have "\<And>y. (y::real) \<ge> 0 \<Longrightarrow> norm (x,y) \<le> 1 \<longleftrightarrow> y \<le> sqrt (1-x\<^sup>2)"
by (subst (5) real_sqrt_square[symmetric], simp, subst real_sqrt_le_iff)
(simp_all add: norm_Pair field_simps)
hence "(\<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel) = (\<integral>\<^sup>+y. indicator {0..sqrt (1-x\<^sup>2)} y \<partial>lborel)"
by (intro nn_integral_cong) (auto split: split_indicator)
also from x have "... = sqrt (1-x\<^sup>2)" using x by simp
finally have "(\<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel) = sqrt (1-x\<^sup>2)" .
}
hence "(\<integral>\<^sup>+x. 2 * (\<integral>\<^sup>+y. indicator ?A1 (x,y) \<partial>lborel) * indicator {-1..1} x \<partial>lborel) =
\<integral>\<^sup>+x. 2 * sqrt (1-x\<^sup>2) * indicator {-1..1} x \<partial>lborel"
by (intro nn_integral_cong) (simp split: split_indicator)
also have A: "\<And>x. -1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<not>x^2 > (1::real)"
by (subst not_less, subst sq_le_1_iff) (simp add: abs_real_def)
have "integrable lborel (\<lambda>x. 2 * sqrt (1-x\<^sup>2) * indicator {-1..1::real} x)"
apply (rule integrable_bound[OF integrable_indicator[of "{-1..1}"], of _ _ "2::real"],
simp, simp, simp)
apply (rule AE_I2, auto simp: abs_real_def power2_eq_square field_simps
dest: A split: split_indicator)
done
hence "(\<integral>\<^sup>+x. 2 * sqrt (1-x\<^sup>2) * indicator {-1..1} x \<partial>lborel) =
ereal (\<integral>x. 2 * sqrt (1-x\<^sup>2) * indicator {-1..1} x \<partial>lborel)"
by (intro nn_integral_eq_integral AE_I2)
(auto split: split_indicator simp: field_simps sq_le_1_iff)
also have "(\<integral>x. 2 * sqrt (1-x\<^sup>2) * indicator {-1..1} x \<partial>lborel) =
LBINT x:{-1..1}. 2 * sqrt (1-x\<^sup>2)" by (simp add: field_simps)
also have "... = LBINT x=-1..1. 2 * sqrt (1-x\<^sup>2)"
by (subst interval_integral_Icc[symmetric]) (simp_all add: one_ereal_def)
also have "... = pi" by (rule unit_circle_area_aux)
finally show ?thesis .
qed
lemma circle_sets_borel:
"{z::real\<times>real. norm z \<le> R} \<in> sets borel" by simp
lemma circle_area:
assumes "R \<ge> 0"
shows "emeasure lborel {z::real\<times>real. norm z \<le> R} = R^2 * pi" (is "emeasure _ ?A = _")
proof (cases "R = 0")
assume "R \<noteq> 0"
with assms have R: "R > 0" by simp
let ?A' = "{z::real\<times>real. norm z \<le> 1}"
have "emeasure lborel ?A = \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A (x,y) \<partial>lborel \<partial>lborel"
by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure, subst lborel_prod)
simp_all
also have "... = \<integral>\<^sup>+x. R * \<integral>\<^sup>+y. indicator ?A (x,R*y) \<partial>lborel \<partial>lborel"
proof (rule nn_integral_cong)
fix x from R show "(\<integral>\<^sup>+y. indicator ?A (x,y) \<partial>lborel) = R * \<integral>\<^sup>+y. indicator ?A (x,R*y) \<partial>lborel"
by (subst nn_integral_real_affine[OF _ `R \<noteq> 0`, of _ 0]) simp_all
qed
also have "... = R * \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A (x,R*y) \<partial>lborel \<partial>lborel"
using R by (intro nn_integral_cmult) simp_all
also from R have "(\<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A (x,R*y) \<partial>lborel \<partial>lborel) =
R * \<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A (R*x,R*y) \<partial>lborel \<partial>lborel"
by (subst nn_integral_real_affine[OF _ `R \<noteq> 0`, of _ 0]) simp_all
also {
fix x y
have A: "(R*x, R*y) = R *\<^sub>R (x,y)" by simp
from R have "norm (R*x, R*y) = R * norm (x,y)" by (subst A, subst norm_scaleR) simp_all
with R have "(R*x, R*y) \<in> ?A \<longleftrightarrow> (x, y) \<in> ?A'" by (auto simp: field_simps)
}
hence "(\<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A (R*x,R*y) \<partial>lborel \<partial>lborel) =
\<integral>\<^sup>+x. \<integral>\<^sup>+y. indicator ?A' (x,y) \<partial>lborel \<partial>lborel"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "... = emeasure lborel ?A'"
by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure, subst lborel_prod)
simp_all
also have "... = pi" by (rule unit_circle_area)
finally show ?thesis by (simp add: power2_eq_square)
qed simp
end