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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: 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: 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: 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

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. f aa A) ( a A. case f a of Some b b B |None True)}" lemma card_of_Func_option_Func: "A BA B|" proof (unfold card_of_ordIso[symmetric], ... case f a of None True | Some b b B)}" lemma Func_Pfunc: "Func_option A BA B" unfoldingby auto lemma

23 May 2024: f x - g x) a - b) F" using tendsto_add [of f a F "λx. - ... norm (x y)xy K" using nonneg_bounded by blast obtain B where B: "B" and norm_g: "eventually (λx.

26 Jun 2024: Typically AB is not the same as BA. The commutator [A, B] captures. ... the dierence between these. Clearly [A, B] = [B, A] and the same also holds foroperators: [Ô, M̂] = [M̂, Ô].

23 May 2024: b) 'a 'b bool" for e :: 'b and r :: "'a 'b 'b" where Rec_zero: "Rec e r zero e" | Rec_succ: "Rec e r m n Rec e r (succ ... by (simp only:) qed qed qed text ‹ The recursion operator -- polymorphic!› definition rec :: "'b ('a 'b 'b) 'a 'b" where "rec e

23 May 2024: uniq_in_ya2'_parts [rule_format]: "⟦evsBB'⟧ Ciph B ⦃Agent A, Nonce NA, Nonce NB⦄evs)B' ⦃Agent A', Nonce NA', Nonce NB⦄evs) A=A' B=B' NA=NA'" apply ... simp) lemma NB_is_uniq_in_ya2': "⟦ya2' C A B NA NBevs; ya2' C' A' B' NA' NBevs; evsBB'⟧

23 May 2024: x A" | A B where "x D" "A face_of B" "B " "B 𝒢" "aff_dim AT" "x A" using x by (auto simp: _def) then have xrel: "xD" proof cases ... b" using bounded_subset_cbox_symmetric by blast define bbox where "bboxbbhave "b) b bbox" by (auto simp:intro!:

23 May 2024: a 'bwhere "Inr_Rep b x y p y = bp" definition "sum = {f. ... A <> BAB" hide_const (open) Plus ― ‹Valuable identifier› lemma InlI [intro!]: "a A Inl a A <> B" by (simp add: Plus_def) lemma InrI [intro!]: "b B Inr b A <> B"

23 May 2024: Theory Sym_Groups. ( Title: HOL/Algebra/Sym_Groups.thy Author: Paulo Emílio de Vilhena ) theory Sym_Groups imports "begin section ‹Symmetric Groups› subsection ‹Definitions› abbreviation inv' :: "('a 'b) ('b 'a)" where ... by auto obtain a b c

23 May 2024: X Y) T'" for T' proof (clarsimp; intro conjI) fix a :: "'a" and b :: "'b" assume "(a, b) T'" with that obtain U V where UV: "openin X U" "openin Y ... AX) BY)}" if "finite T" "T 𝒳 𝒴" for T using that proof induction case (insert B ) then show?

23 May 2024: 𝟬)" by (induction n) (auto) lemma scalar_coeff: "aRλb. a b) p) = (λi. ... a_p2 = "(map (λb. a b) p2)a # p1)) 𝟬)" have "a # p1))λ_.