equal
deleted
inserted
replaced
3386 with True show ?thesis |
3386 with True show ?thesis |
3387 by (auto simp: absolutely_integrable_on_def integrable_negligible integral_negligible) |
3387 by (auto simp: absolutely_integrable_on_def integrable_negligible integral_negligible) |
3388 next |
3388 next |
3389 case False |
3389 case False |
3390 then obtain h where h: "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x" "linear h" |
3390 then obtain h where h: "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x" "linear h" |
3391 using assms det_nz_iff_inj linear_injective_isomorphism by blast |
3391 using assms det_nz_iff_inj linear_injective_isomorphism by metis |
3392 show ?thesis |
3392 show ?thesis |
3393 proof (rule has_absolute_integral_change_of_variables_invertible) |
3393 proof (rule has_absolute_integral_change_of_variables_invertible) |
3394 show "(g has_derivative g) (at x within S)" for x |
3394 show "(g has_derivative g) (at x within S)" for x |
3395 by (simp add: assms linear_imp_has_derivative) |
3395 by (simp add: assms linear_imp_has_derivative) |
3396 show "continuous_on (g ` S) h" |
3396 show "continuous_on (g ` S) h" |