src/HOL/Probability/Distributions.thy
changeset 56371 fb9ae0727548
parent 53077 a1b3784f8129
child 56381 0556204bc230
equal deleted inserted replaced
56370:7c717ba55a0b 56371:fb9ae0727548
   178     proof (rule integral_FTC_atLeastAtMost)
   178     proof (rule integral_FTC_atLeastAtMost)
   179       fix x assume "0 \<le> x" "x \<le> b"
   179       fix x assume "0 \<le> x" "x \<le> b"
   180       show "DERIV (\<lambda>x. x * exp (-x)) x :> exp (-x) - x * exp (-x)"
   180       show "DERIV (\<lambda>x. x * exp (-x)) x :> exp (-x) - x * exp (-x)"
   181         by (auto intro!: DERIV_intros)
   181         by (auto intro!: DERIV_intros)
   182       show "isCont (\<lambda>x. exp (- x) - x * exp (- x)) x "
   182       show "isCont (\<lambda>x. exp (- x) - x * exp (- x)) x "
   183         by (intro isCont_intros isCont_exp')
   183         by (intro continuous_intros)
   184     qed fact
   184     qed fact
   185     also have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) = - exp (- b) - - exp (- 0)"
   185     also have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) = - exp (- b) - - exp (- 0)"
   186       by (rule integral_FTC_atLeastAtMost) (auto intro!: DERIV_intros)
   186       by (rule integral_FTC_atLeastAtMost) (auto intro!: DERIV_intros)
   187     finally show "1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)"
   187     finally show "1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)"
   188       by (auto simp: field_simps exp_minus inverse_eq_divide)
   188       by (auto simp: field_simps exp_minus inverse_eq_divide)