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23 May 2024: a - b)x = a (ax2 b) a x b b x" by (simp add:have "(kn. ... norm(f x - g x) < e)" proof - { fix b :: 'b assume "bhave "pp (x S.

23 May 2024: k K open k" unfolding eq by auto from ex_countable_basis obtain B :: "'awhere B: "b. ... fix X::"'a set" assume "open X" from open_countable_basisE[OF this] obtain B' where B': "B' B" "XB'".

23 May 2024: thesis unfolding mset_less_eq_Bag0 by auto qed declare inter_mset_def [code] declare union_mset_def [code] declare mset.simps [code] fun fold_impl :: "('a nat 'b 'b) 'b ... ab" where "fold_impl fn e ((a,n) # ms) = (fold_impl fn ((fn a n) e) ms)" |

23 May 2024: auto simp add: set_plus_def) lemma set_plus_elim: assumes "x A B" obtains a b where "x = a b" and "a A" and "b B" using assms unfolding ... b" and "a A" and "b B" using assms unfolding set_times_def by fast lemma set_times_intro2 [intro!]: "b C a b a o C"

23 May 2024: nm#k))" by (auto intro: less_imp_succ_add) lemma add_lt_elim2: "⟦a # d = b # c; a < b; bcd⟧ c < d" by (drulelemma add_le_elim2: "⟦a # d = b # c; ... add:lemma raw_nat_diff_split: "⟦ab⟧ (P(a #- b)) ((a < b P(dnat.

23 May 2024: g = "gcd (a b' b a') (b b')" have gz: "?gusing False by simp show? ... g = "gcd (a a') (b b')" have gz: "?gusing neq by simp fromwhere?

23 May 2024: bool: "C) = C(C(1)" by (unfoldIntroduction rules for the injections ) lemma InlI [intro!,simp,TC]: "a A Inl(a) AB" by (unfoldlemma InrI [intro!,simp,TC]: "b B Inr(b) ... b): AB b B" by blast lemma sum_iff: "u ABx.

23 May 2024: h xa.b} g (h x) = x" assumes range: "{a.bh" shows "f) lborel h) {a.bλx. ... where B=B])(auto simp add: f_bounded) then show "enn2real (. - f x M') = x. f x M'" by(simp add: real_lebesgue_integral_def) qed qed simp_all finally show?

23 May 2024: syntax begin lemma transfer_rule_numeral: ‹(R)› if [transfer_rule]: ‹R› ‹R› ‹(R ===> R ===> R)› for R :: ‹'a::{b::{› proof - have "(R) (λk. ... b)k) 0)› using numeral_add_unfold_funpow [where? 'a = 'b, of _ 0] by (simp add:

23 May 2024: a, 's2)where "fair_implements C ACACACA" lemma implements_trans: "A =<| B B =<| C A =<| C" by (auto simp add: ioa_implements_def) subsection ‹Modules› subsubsection ‹Execution, schedule and trace modules› ... mk_trace›› (alternative