equal
deleted
inserted
replaced
2804 and f0: "Df 0 = f" |
2804 and f0: "Df 0 = f" |
2805 and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow> |
2805 and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow> |
2806 (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})" |
2806 (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})" |
2807 and ivl: "a \<le> b" |
2807 and ivl: "a \<le> b" |
2808 defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x" |
2808 defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x" |
2809 shows taylor_has_integral: |
2809 shows Taylor_has_integral: |
2810 "(i has_integral f b - (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}" |
2810 "(i has_integral f b - (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}" |
2811 and taylor_integral: |
2811 and Taylor_integral: |
2812 "f b = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i" |
2812 "f b = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i" |
2813 and taylor_integrable: |
2813 and Taylor_integrable: |
2814 "i integrable_on {a..b}" |
2814 "i integrable_on {a..b}" |
2815 proof goal_cases |
2815 proof goal_cases |
2816 case 1 |
2816 case 1 |
2817 interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a" |
2817 interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a" |
2818 by (rule bounded_bilinear_scaleR) |
2818 by (rule bounded_bilinear_scaleR) |