src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
changeset 70802 160eaf566bcb
parent 70760 ffbe7784cc85
child 70817 dd675800469d
equal deleted inserted replaced
70801:5352449209b1 70802:160eaf566bcb
  3583                (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow>
  3583                (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow>
  3584            (f has_integral I) (cbox a b)" (is "?lhs = ?rhs")
  3584            (f has_integral I) (cbox a b)" (is "?lhs = ?rhs")
  3585 proof
  3585 proof
  3586   assume ?lhs
  3586   assume ?lhs
  3587   from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close>
  3587   from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close>
  3588     show ?rhs by (simp add: field_simps vector_add_divide_simps)
  3588     show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps)
  3589 next
  3589 next
  3590   assume ?rhs
  3590   assume ?rhs
  3591   from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close>
  3591   from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close>
  3592   show ?lhs by simp
  3592   show ?lhs by simp
  3593 qed
  3593 qed
  4251         show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3"
  4251         show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3"
  4252           by (metis SUMEQ d1_fin norm_minus_cancel)
  4252           by (metis SUMEQ d1_fin norm_minus_cancel)
  4253         show "norm (?SUM p - integral {a..t} f) < e/3"
  4253         show "norm (?SUM p - integral {a..t} f) < e/3"
  4254           using d2_fin by blast
  4254           using d2_fin by blast
  4255         show "norm ((c - t) *\<^sub>R f c) < e/3"
  4255         show "norm ((c - t) *\<^sub>R f c) < e/3"
  4256           using w cwt \<open>t < c\<close> by (auto simp add: field_simps)
  4256           using w cwt \<open>t < c\<close> by simp (simp add: field_simps)
  4257       qed
  4257       qed
  4258       then show ?thesis by simp
  4258       then show ?thesis by simp
  4259     qed
  4259     qed
  4260   qed
  4260   qed
  4261 qed
  4261 qed