(* Title: HOL/Analysis/ex/Circle_Area.thy Author: Manuel Eberl, TU Muenchen A proof that the area of a circle with radius R is R⇧^{2}π. *) section ‹The area of a circle› theory Circle_Area imports Complex_Main "HOL-Analysis.Interval_Integral" begin lemma plus_emeasure': assumes "A ∈ sets M" "B ∈ sets M" "A ∩ B ∈ null_sets M" shows "emeasure M A + emeasure M B = emeasure M (A ∪ B)" proof- let ?C = "A ∩ B" have "A ∪ B = A ∪ (B - ?C)" by blast with assms have "emeasure M (A ∪ 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 ≥ 0 ⟹ sqrt (x^2) = (x::real)" by simp 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 *⇩_{R}(2 * sqrt (1 - (sin x)⇧^{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 ∈ {-pi/2<..<pi/2}" hence "cos x ≥ 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 = "λx. 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×real. norm z ≤ 1} = pi" (is "emeasure _ ?A = _") proof- let ?A1 = "{(x,y)∈?A. y ≥ 0}" and ?A2 = "{(x,y)∈?A. y ≤ 0}" have [measurable]: "(λx. snd (x :: real × real)) ∈ measurable borel borel" by (simp add: borel_prod[symmetric]) have "?A1 = ?A ∩ {x∈space lborel. snd x ≥ 0}" by auto also have "?A ∩ {x∈space lborel. snd x ≥ 0} ∈ sets borel" by (intro sets.Int pred_Collect_borel) simp_all finally have A1_in_sets: "?A1 ∈ sets lborel" by (subst sets_lborel) have "?A2 = ?A ∩ {x∈space lborel. snd x ≤ 0}" by auto also have "... ∈ sets borel" by (intro sets.Int pred_Collect_borel) simp_all finally have A2_in_sets: "?A2 ∈ sets lborel" by (subst sets_lborel) have A12: "?A = ?A1 ∪ ?A2" by auto have sq_le_1_iff: "⋀x. x⇧^{2}≤ 1 ⟷ abs (x::real) ≤ 1" by (simp add: abs_square_le_1) have "?A1 ∩ ?A2 = {x. abs x ≤ 1} × {0}" by (auto simp: norm_Pair field_simps sq_le_1_iff) also have "... ∈ 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 = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel ∂lborel" by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure) (simp_all only: lborel_prod A1_in_sets) also have "emeasure lborel ?A2 = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A2 (x,y) ∂lborel ∂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 "(∫⇧^{+}x. ∫⇧^{+}y. indicator ?A2 (x,y) ∂lborel ∂lborel) = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A2 (x,y) ∂distr lborel lborel uminus ∂lborel" by simp also have "... = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A2 (x,-y) ∂lborel ∂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 "... = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel ∂lborel" by (intro nn_integral_cong) (auto split: split_indicator simp: norm_Pair) also have "... + ... = (1+1) * ..." by (subst ring_distribs) simp_all also have "... = ∫⇧^{+}x. 2 * ∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel ∂lborel" by (subst nn_integral_cmult) simp_all also { fix x y :: real assume "x ∉ {-1..1}" hence "abs x > 1" by auto also have "norm (x,y) ≥ abs x" by (simp add: norm_Pair) finally have "(x,y) ∉ ?A1" by auto } hence "... = ∫⇧^{+}x. 2 * (∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel) * indicator {-1..1} x ∂lborel" by (intro nn_integral_cong) (auto split: split_indicator) also { fix x :: real assume "x ∈ {-1..1}" hence x: "1 - x⇧^{2}≥ 0" by (simp add: field_simps sq_le_1_iff abs_real_def) have "⋀y. (y::real) ≥ 0 ⟹ norm (x,y) ≤ 1 ⟷ y ≤ sqrt (1-x⇧^{2})" by (subst (5) real_sqrt_square[symmetric], simp, subst real_sqrt_le_iff) (simp_all add: norm_Pair field_simps) hence "(∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel) = (∫⇧^{+}y. indicator {0..sqrt (1-x⇧^{2})} y ∂lborel)" by (intro nn_integral_cong) (auto split: split_indicator) also from x have "... = sqrt (1-x⇧^{2})" using x by simp finally have "(∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel) = sqrt (1-x⇧^{2})" . } hence "(∫⇧^{+}x. 2 * (∫⇧^{+}y. indicator ?A1 (x,y) ∂lborel) * indicator {-1..1} x ∂lborel) = ∫⇧^{+}x. 2 * sqrt (1-x⇧^{2}) * indicator {-1..1} x ∂lborel" by (intro nn_integral_cong) (simp split: split_indicator add: ennreal_mult') also have A: "⋀x. -1 ≤ x ⟹ x ≤ 1 ⟹ ¬x^2 > (1::real)" by (subst not_less, subst sq_le_1_iff) (simp add: abs_real_def) have "integrable lborel (λx. 2 * sqrt (1-x⇧^{2}) * indicator {-1..1::real} x)" by (intro borel_integrable_atLeastAtMost continuous_intros) hence "(∫⇧^{+}x. 2 * sqrt (1-x⇧^{2}) * indicator {-1..1} x ∂lborel) = ennreal (∫x. 2 * sqrt (1-x⇧^{2}) * indicator {-1..1} x ∂lborel)" by (intro nn_integral_eq_integral AE_I2) (auto split: split_indicator simp: field_simps sq_le_1_iff) also have "(∫x. 2 * sqrt (1-x⇧^{2}) * indicator {-1..1} x ∂lborel) = LBINT x:{-1..1}. 2 * sqrt (1-x⇧^{2})" by (simp add: field_simps) also have "... = LBINT x=-1..1. 2 * sqrt (1-x⇧^{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_area: assumes "R ≥ 0" shows "emeasure lborel {z::real×real. norm z ≤ R} = R^2 * pi" (is "emeasure _ ?A = _") proof (cases "R = 0") assume "R ≠ 0" with assms have R: "R > 0" by simp let ?A' = "{z::real×real. norm z ≤ 1}" have "emeasure lborel ?A = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A (x,y) ∂lborel ∂lborel" by (subst lborel_prod[symmetric], subst lborel.emeasure_pair_measure, subst lborel_prod) simp_all also have "... = ∫⇧^{+}x. R * ∫⇧^{+}y. indicator ?A (x,R*y) ∂lborel ∂lborel" proof (rule nn_integral_cong) fix x from R show "(∫⇧^{+}y. indicator ?A (x,y) ∂lborel) = R * ∫⇧^{+}y. indicator ?A (x,R*y) ∂lborel" by (subst nn_integral_real_affine[OF _ ‹R ≠ 0›, of _ 0]) simp_all qed also have "... = R * ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A (x,R*y) ∂lborel ∂lborel" using R by (intro nn_integral_cmult) simp_all also from R have "(∫⇧^{+}x. ∫⇧^{+}y. indicator ?A (x,R*y) ∂lborel ∂lborel) = R * ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A (R*x,R*y) ∂lborel ∂lborel" by (subst nn_integral_real_affine[OF _ ‹R ≠ 0›, of _ 0]) simp_all also { fix x y have A: "(R*x, R*y) = R *⇩_{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) ∈ ?A ⟷ (x, y) ∈ ?A'" by (auto simp: field_simps) } hence "(∫⇧^{+}x. ∫⇧^{+}y. indicator ?A (R*x,R*y) ∂lborel ∂lborel) = ∫⇧^{+}x. ∫⇧^{+}y. indicator ?A' (x,y) ∂lborel ∂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 using assms by (simp add: power2_eq_square ennreal_mult mult_ac) qed simp end