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I am trying to prove the following lemma using the theorem

lemma lm22:
fixes f :: "real ⇒ 'a::banach"
assumes "a ≤ b"
and "∀x∈{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b}
shows "(f' has_integral (f b - f a)) {a .. b}"

Isabelle output states that the goal can be solved directly with Integration.fundamental_theorem_of_calculus. However adding:

using fundamental_theorem_of_calculus [of a b f f'] by auto

does not work. I have tried defining a and b, and also replacing the variables by values but no success.

How can I apply the lemma to prove the goal?

A K
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1 Answers1

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Isabelle's solve-direct feature considers the assumptions from the assumes clauses, but such assumptions are not available to proof methods like by auto unless you explicitly chain them in. Write using assms before the rest of your proof, and it should go through.

Brian Huffman
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